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\begin{document}

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\title{$\mathbb{MATHEMATICS\ \ STUDY\ \ GUIDE}$}
\author{}
\date{}
\maketitle

\section{Real Field}\smallskip

\paragraph{Archimedean property}\ \\
If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$.
As a corollary, if three real numbers $a$, $x$, and $y$ satisfy the inequalities $a \leq x \leq a + \frac{y}{n}$
for every integer $n \geq 1$, then $x = a$.

\paragraph{Properties of supremum and infimum}\ \\
Let $h$ be a given positive number and let $S$ be a set of real numbers.\\
(a) If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup S - h$.\\
(b) If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf S + h$.

\paragraph{Well-ordering principle}\ \\
Every nonempty set of positive integers contains a smallest member.

\paragraph{Triangle inequality}\ \\
For arbitrary real numbers $x$ and $y$,
$|x + y| \leq |x| + |y|.$
More generally, for arbitrary real numbers $a_1$, $a_2$, \ldots, $a_n$, we have
$\left|\sum_{k=1}^n a_k\right| \leq \sum_{k=1}^n |a_k|.$

\paragraph{The Cauchy-Schwarz inequality}\ \\
If $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ are arbitrary real numbers, we have
$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$.
The equality sign holds if and only if there is a real number $x$ such that $a_k x + b_k = 0$
for each $k = 1, 2, \ldots, n$.

\bigskip\bigskip
\section{Complex Field}\smallskip

\paragraph{Field properties}\ \\
$(a,b)=(c,d)$ means $a=c$ and $b=d$\\
$(a,b)+(c,d)=(a+c,b+d)$\\
$(a,b)(c,d)=(ac-bd,ad+bc)$
$x+y=y+x$\\
$x+(y+z)=(z+y)+z$\\
$x(y+z)=xy+xz$\\
$e^{z+2n\pi i}=e^z$

\paragraph{Polar coordinates}\ \\
$x = r \cos \theta$ \hskip 10 pt $y = r \sin \theta$\\
$r$ is the modulus or absolute value of $(x,y)$, equal to $\sqrt{x^2+y^2}$.\\
$\theta$ is the angle between $(x,y)$ and the x-axis, and is called the argument of $(x,y)$,
or the principal argument if $-\pi < \theta \leq \pi$.\\
Polar form of $z$: Every complex number $z \neq 0$ can be expressed as $z=re^{i\theta}$.

\paragraph{Complex exponential}\ \\
If $z=(x,y)$, then $e^z = e^x(\cos y + i \sin y)$\\
$e^ae^b=e^{a+b}$

\paragraph{Derivatives and integrals}\ \\
If $f=u+iv$, then $f'(x)=u'(x)+iv'(x)$\\
$\int_a^bf(x)\,dx=\int_a^bu(x)\,dx+i\int_a^bv(x)\,dx$\\
$(e^{tx})'=te^{tx}$\\
$\int e^{tx}\,dx=\frac{e^{tx}}{t}$

\bigskip\bigskip
\section{Integrals}\smallskip

\paragraph{Upper and lower integrals}\ \\
Lower integral of $f$:
$\underline{I} = \sup \left\{ \int_a^b s(x)\,dx\ |\ s \leq f \right\}$ \hskip 10 pt
Upper integral of $f$:
$\bar{I} = \inf \left\{ \int_a^b t(x)\,dx\ |\ f \leq t \right\}$

\paragraph{Definition of the integral}\ \\
Every function $f$ which is boundend on $[a, b]$ has a lower integral $\underline{I}(f)$
and an upper integral $\bar{I}(f)$ satisfying the inequalities
$\int_a^b s(x)\,dx \leq \underline{I}(f) \leq \bar{I}(f) \leq \int_a^b t(x)\,dx$
for all step functions $s$ and $t$ with $s \leq f \leq t$. The function $f$ is integrable
on $[a, b]$ if and only if its upper and lower integrals are equal, in which case we have
$\int_a^b f(x)\,dx = \underline{I}(f) = \bar{I}(f).$

\paragraph{Basic properties of the integral}\ \\
Linearity: $\int_a^b [c_1 f(x) + c_2 g(x)]\,dx = c_1 \int_a^b f(x)\,dx + c_2 \int_a^b g(x)\,dx.$\\
Additivity: $\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx.$\\
Invariance under translation: $\int_a^b f(x)\,dx = \int_{a+c}^{b+c} f(x - c)\,dx.$\\
Expansion or contraction of interval of integration:
$\int_a^b f(x)\,dx = \frac{1}{k} \int_{ka}^{kb} f\left(\frac{x}{k}\right)\,dx.$

\paragraph{Definition of indefinite integral}\ \\
$A(x) = \int_a^x f(t)\,dt$, \hskip 20 pt if $a \leq x \leq b$.

\paragraph{Improper integrals}\ \\
Improper integral of first kind: $\int_a^b$ where we let $b \to +\infty$.\\
Improper integral of second kind: $\int_a^b f$ where $f$ is allowed to become unbounded at one or more points.

\paragraph{Average value of a function over an interval}\ \\
$A(f) = \frac{1}{b-a} \int_a^b f(x)\,dx.$

\paragraph{Integral of rational powers}\ \\
$\int x^n\,dx = \frac{x^{n+1}}{n+1}$

\paragraph{Integral of trigonometric functions}\ \\
$\int \sin x\,dx = -\cos x$\\
$\int \cos x\,dx = \sin x$

\paragraph{Integrals involving logarithm}\ \\
$\int \frac{du}{u} = \log |u| + C$, \hskip 10 pt where $u \neq 0$\\
$\int \frac{f'(x)}{f(x)}\,dx = \log |f(x)| + C$, \hskip 10 pt where $f(x) \neq 0$

\paragraph{Some useful integrals}\ \\
$\int \frac{dx}{x+a} = \log |x+a|$\\
$\int \frac{dx}{(x+a)^n} = \frac{(x+a)^{1-n}}{(1-n)}$, \hskip 10 pt if $n > 1$\\
For $\int \frac{x\,dx}{(x^2 + bx + c)^m}$ \hskip 3 pt and \hskip 3 pt $\int \frac{dx}{(x^2 + bx + c)^m}$,
let $u = x+ b/2$ and $\alpha = \frac{1}{2} \sqrt{4c -b^2}$ so that $x^2+bx+c = u^2+\alpha^2$.\\
Then, $\int \frac{u\,du}{(u^2+\alpha^2)} = \frac{1}{2} \log(u^2+\alpha^2)$,
and $\int \frac{u\,du}{(u^2+\alpha^2)^m} = \frac{1}{2}\frac{(u^2+\alpha^2)^{1-m}}{(1-m)}$.\\
And $\int \frac{du}{(u^2+\alpha^2)} = \frac{1}{\alpha} \arctan \frac{u}{\alpha} + C$.\\
And $\int \frac{du}{(u^2+\alpha^2)^m} =
\frac{1}{2\alpha^2(m-1)}\frac{u}{(u^2+\alpha^2)^{m-1}}+\frac{2m-3}{2\alpha^2(m-1)}\int \frac{du}{(u^2+\alpha^2)^{m-1}}$.\\
$\int e^{\alpha x}\cos \beta x\,dx = \frac{e^{\alpha x}(\alpha \cos \beta x + \beta \sin \beta x)}{\alpha^2+\beta^2}$\\
$\int e^{\alpha x}\sin \beta x\,dx = \frac{e^{\alpha x}(\alpha \sin \beta x - \beta \cos \beta x)}{\alpha^2+\beta^2}$

\bigskip\bigskip
\section{Techniques of integration}\smallskip

\paragraph{Substitution}\ \\
$\int_c^x f[g(t)]g'(t)\,dt = \int_{g(c)}^{g(x)}f(u)\,du$

\paragraph{Parts}\ \\
$\int_a^b f(x)g'(x)\,dx = f(b)g(b) - f(a)g(a) - \int_a^b f'(x)g(x)\,dx$\\
Or, $\int u\,dv = uv - \int v\,du + C$

\paragraph{Partial fractions}\ \\
The trick is to find a way to decompose a rational function into partial fractions, which are easier to integrate.\\
CASE 1: Denominator product of distinct linear factors.\\
CASE 2: Denominator product of linear factors, some of which are repeated.\\
CASE 3: Denominator contains irreducible $(b^2 - 4c < 0)$ quadratic factors, none of which are repeated.\\
CASE 4: Denominator contains irreducible quadratic factors, some of which are repeated.\\
If we have $\int R(\sin x, \cos x)\,dx$, where $R$ is a RF of two variables, use substitution $u = \tan \frac{1}{2} x$ to give an RF.\\
If we have $\int R(x, \sqrt{a^2 - x^2})\,dx$, use substitution $x = a \sin t$, \hskip 5 pt $dx = a \cos t\,dt$ to give an RF.

\bigskip\bigskip
\section{Derivatives}\smallskip

\paragraph{Definition}\ \\
$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, if this limit exists.

\paragraph{Algebra}\ \\
$(f+g)' = f' + g'$\\
$(f-g)' = f' - g'$\\
$(f \cdot g)' = f \cdot g' + g \cdot f'$\\
$(\frac{f}{g})' = \frac{g \cdot f' - f \cdot g'}{g^2}$, when $g(x) \neq 0$\\
$(c_1f + c_2g)' = c_1f' + c_2g'$\\
If $f = u \circ v$ and if $v'(x)$ and $u'(x)$ exist, then $f'(x) = u'(v(x)) \cdot v'(x)$ (Chain rule).

\paragraph{Rolle's Theorem}\ \\
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a)=f(b)$,
then there is at least one point $c$ in $(a,b)$ such that $f'(c)=0$.

\paragraph{Mean value theorem - derivatives}\ \\
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$,
then there is at least one point $c$ in $(a,b)$ such that $f(b)-f(a)=f'(c)(b-a)$.

\paragraph{Derivatives of inverse functions}\ \\
If $f$ and $g$ are inverses, and if $f$ is strictly increasing and continuous,
and if $f'(x)$ exists and is nonzero, and if $y = f(x)$, then $g'(y)$ also
exists, and the derivatives are reciprocals, i.e., $g'(y) = \frac{1}{f'(x)}$,
or, $\frac{dx}{dy} = \frac{1}{\left( \frac{dy}{dx} \right)}$.

\paragraph{Wronskian matrix}\ \\
The Wronskian matrix of $n$ functions $u_1, \dotsc, u_n$:
\begin{equation*}
W =
\begin{bmatrix}
u_1 & u_2 & \cdots & u_n \\
u_1' & u_2' & \cdots & u_n'\\
\vdots & & & \vdots\\
u_1^{(n-1)} & u_2^{(n-1)} & \cdots & u_n^{(n-1)}
\end{bmatrix}
\end{equation*}

\paragraph{Derivative of a scalar field with respect to a vector}
\begin{equation*}
f'(\bs{a}; \bs{y}) = \lim_{h\to 0} \frac{f(\bs{a}+h\bs{y}) - f(\bs{a})}{h}
\end{equation*}
If $f$ is a linear transformation, then $f'(\bs{a};\bs{y}) = f(\bs{y})$.\\
If $\lVert y \rVert = 1$, then $f'(\bs{a};\bs{y})$ is a {\it directional derivative}.\\
If $\bs{y} = \bs{e}_k$, then $f'(\bs{a};\bs{y})$ is a {\it partial derivative with respect to $\bs{e}_k$}.\\
$D_kf(\bs{a}) = f'(\bs{a};\bs{e}_k)$.

\paragraph{Mean value theorem for derivatives of scalar fields}\ \\
If $f'(\bs{a}+t\bs{y});\bs{y})$ exists for $0\leq t\leq 1$\\
...then for some real $\theta$ such that $0 < \theta < 1$,\\
...$f(\bs{a}+\bs{y})-f(\bs{a}) = f'(\bs{a}+\theta\bs{y};\bs{y})$.

\paragraph{Total derivative}\ \\
Even if directional derivatives exist for every direction at $\bs{a}$, $f$ may not be differentiable at $\bs{a}$.\\
A scalar field $f$ is ``differentiable at $\bs{a}$'' if:\\
$\phantom{x}$ (1) There is a linear transformation $T_a: \bs{R}^n \to \bs{R}$\\
$\phantom{x}$ (2) There is a scalar function $E(\bs{a},\bs{v})$\\
$\phantom{x}$ (3) $f(\bs{a}+\bs{v}) = f(\bs{a}) + T_a(\bs{v}) + \lVert\bs{v}\rVert\, E(\bs{a},\bs{v})$\\
$\phantom{x}$ (4) For all $\lVert\bs{v}\rVert < r$\\
$\phantom{x}$ (5) Where $E(\bs{a},\bs{v})\to 0$ as $\lVert\bs{v}\rVert \to 0$\\
$T_a(y) = f'(\bs{a}; \bs{y}) = \nabla f(\bs{a}) \cdot \bs{y}$.\\
$f'(\bs{a}; \bs{y}) = \sum_{k=1}^n D_k f(\bs{a}) y_k$.

\paragraph{Gradient}\ \\
$\nabla f(\bs{a}) = (D_1f(\bs{a}), \dotsc, D_nf(\bs{a}))$.

\paragraph{Continuity and differentiability of scalar fields}\ \\
If the partial derivatives exist and are continuous at $\bs{a}$, then $f$ is differentiable at $\bs{a}$.

\paragraph{Scalar field chain rule}\ \\
If $g(t) = f[\bs{r}(t)]$, then $g'(t) = \nabla f(\bs{r}(t)) \cdot \bs{r}'(t)$.

\paragraph{Level sets}\ \\
A level set is the subset of a domain of a function for which a scalar field has a contant value.\\
The gradient is perpendicular to the level set (level curves or level surfaces) / normal to the tangent plane.

\paragraph{Derivatives of vector fields}\ \\
Derivatives of vector fields are the same as for scalar fields, extended to more dimensions.\\
Jacobian matrix: $m \times n$ matrix whose $k$th row is $\nabla f_k(\bf{a})$.
Jacobian matrix: Written $Df(\bf{a})$.\\
$\bf{T}_{\bf{a}}(\bf{y}) = D\bf{f}(\bf{a})\bf{y}$.

\paragraph{Vector field chain rule}\ \\
Chain rule: $\bf{h}'(\bf{a}) = \bf{f}'(\bf{g}(\bf{a})) \circ \bf{g}'(\bf{a})$ (Composition of linear transformations).\\
Matrix form: $D\bf{h}(\bf{a}) = D\bf{f}(\bf{g}(\bf{a}))\, D\bf{g}(\bf{a})$.\\
This can also be written as a system of scalar equations:\\
$\phantom{x}$ If $f: R^n \to R^m$, $g: R^n \to R^p$, then $h = f \circ g: R^m \to R^p$.\\
$\phantom{x}$ There are then $mp$ scalar equations.\\
$\phantom{x}$ $D_jh_i(\bs{a}) = \sum_{k=1}^n D_kf_i(\bs{g}(\bs{a}))D_jg_k(\bs{a})$.\\
$\phantom{x}$ For $i = 1, 2, \dotsc, m$ and $j = 1, 2, \dotsc, p$.

\paragraph{Sufficient condition for equality of mixed partial derivatives}\ \\
If $D_1f, D_2f, and D_{2,1}f$ exist, and if $D_{2,1}$ is continuous in a neighborhood of $a$\\
Then $D_{1,2}f$ exists and the two mixed partials are equal.

\paragraph{Jacobian determinant}
\begin{equation*}
\frac{\partial(f_1,\dotsc,f_n)}{\partial(x_1,\dotsc,x_n)} = \det
\begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n}\\
\vdots & & & \vdots\\
\frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \cdots & \frac{\partial f_n}{\partial x_n}\\
\end{bmatrix}
\end{equation*}

\paragraph{Maxima, minima, and saddle points}\ \\
For a scalar field, a {\it stationary point} or {\it critical point} occurs where $\nabla f = O$.
At stationary points, the tangent plane is horizontal.\\
Three types of stationary points: maxima, minima, and saddle points.\\
Extremum: a relative (including absolute) minimum or maximum.\\
Saddle point: every $n$-ball around $\bs{a}$ contains $f(\bs{x}) > f(\bs{a})$ and $f(\bs{x}) < f(\bs{a})$.\\
If $f$ has continuous second-order partial derivatives on $B(\bs{a})$\\
$\phantom{x}$ And if $H(\bs{a})$ is Hessian matrix\\
$\phantom{x}$ And if $\bs{a}$ is a stationary point, then:\\
$\phantom{x}$ (a) If all the eigenvalues of $H(\bs{a})$ are positive, $\bs{a}$ is a relative minimum.\\
$\phantom{x}$ (b) If all the eigenvalues of $H(\bs{a})$ are negative, $\bs{a}$ is a relative maximum.\\
$\phantom{x}$ (c) If $H(\bs{a})$ has both positive and negative eigenvalues, $\bs{a}$ is a saddle point.\\
$\phantom{x}$ (d) If all eigenvalues are zero, this gives us no information.\\
There are other tests for stationary points, using the determinant of the Hessian matrix, etc.

\paragraph{Hessian matrix of a scalar field}\ \\
$H(\bs{x}) = [D_{ij}f(\bs{x})]_{i,j=1}^n$.

\paragraph{Lagrange multipliers}\ \\
Method for dealing with the following problem:\\
$\phantom{x}$ Find extreme values of a scalar field $f(\pmb{x})$ when $\pmb{x}$ is restricted to a subset of the domain.\\
Method:\\
$\phantom{x}$ (1) Scalar field $f(x_1,\dotsc,x_n)$ subject to $m < n$ constraints $g_1(x_1,\dotsc,x_n)=0$, etc.\\
$\phantom{x}$ (2) At each extremum there exist $m$ scalars $\lambda_1, \dotsc, \lambda_m$.\\
$\phantom{x}$ (3) Such that $\nabla f = \lambda_1 \nabla g_1 + \dotsb + \lambda_m \nabla g_m$.\\
$\phantom{x}$ (4) Solve the $n + m$ equations for $x_1,\dotsc,x_n$ and $\lambda_1,\dotsc,\lambda_m$.\\
$\phantom{x}$ (5) The required extrema will be among the solutions.

\bigskip\bigskip
\section{General Calculus}\smallskip

\paragraph{First fundamental theorem of calculus}\ \\
If $A(x) = \int_c^x f(t)\,dt$ for $a \leq x \leq b$, and $a \leq c \leq b$,
then $A'(x) = f(x)$.

\paragraph{Second fundamental theorem of calculus}\ \\
If $f$ is continuous on $(a,b)$, and $P$ is any primitive of $f$ on $(a,b)$,
then for every $c$, $x \in (a,b)$, $P(x) = P(c) + \int_c^x f(t)\,dt$.

\paragraph{Definition of primitive / antiderivative}\ \\
A function $P$ is called a primitive (or an antiderivative) of a function $f$ on an open interval
$I$ if the derivative of $P$ is $f$, that is, if $P'(x) = f(x)$ for all $x$ in $I$.

\bigskip\bigskip
\section{Functions, Limits, Continuity}\smallskip

\paragraph{Definition of convex and concave function}\ \\
For all $x$ and $y$ in $[a, b]$ and for $0 < \alpha < 1$:\\
Convex: $g[\alpha y + (1-\alpha)x] \leq \alpha g(y) + (1 - \alpha) g(x)$.\\
Concave: $g[\alpha y + (1-\alpha)x] \geq \alpha g(y) + (1 - \alpha) g(x)$.

\paragraph{Limit of a function}\ \\
$\lim_{x \to p} f(x) = A$ means that for every neighborhood $N_1(A)$ there is some
neighborhood $N_2(p)$ such that $f(x) \in N_1(A)$ whenever $x \in N_2(p)$ and $x \neq p$.

\paragraph{Continuity of a function at a point}\ \\
$f$ is defined at $p$, and $\lim_{x \to p} f(x) = f(p)$.

\paragraph{Limit algebra}\ \\
Let $f$ and $g$ be functions such that $\lim_{x \to p} f(x) = A$, \hskip 10 pt $\lim_{x \to p} g(x) = B$.\\
$\lim_{x \to p} [f(x) + g(x)] = A + B$\\
$\lim_{x \to p} [f(x) - g(x)] = A - B$\\
$\lim_{x \to p} f(x) \cdot g(x) = A \cdot B$\\
$\lim_{x \to p} f(x) / g(x) = A/B$, \hskip 10 pt if $B \neq 0$.

\paragraph{Continuity algebra}\ \\
If $f$ and $g$ are continuous at a point $p$, then $f + g$, $f - g$, $f \cdot g$ are also continuous at $p$.\\
If $g(p) \neq 0$, then $f/g$ is continuous at $p$.\\
If $v$ is continuous at $p$, and $u$ is continuous at $q = v(p)$,
then $f = u \circ v$ is continuous at $p$.

\paragraph{Squeezing principle}\ \\
If $f(x) \leq g(x) \leq h(x)$ for all $x \neq p$ in a neighborhood of $p$,
and $\lim_{x \to p} f(x) = \lim_{x \to p} h(x) = a$,
then $\lim_{x \to p} g(x) = a$.

\paragraph{Continuity of indefinite integrals}\ \\
If $f$ is integrable on $[a, b]$, then the indefinite integral from $a$ to $x$ is
continuous at every point of $[a, b]$.  At each endpoint, it has one-sided continuity.

\paragraph{Intermediate value theorem}\ \\
If $f$ is continuous on $[a, b]$, then for any two points $x_1, x_2 \in [a, b], f(x_1) \neq f(x_2)$,
then $f$ takes on every value between $f(x_1)$ and $f(x_2)$ in the interval $(x_1, x_2)$.

\paragraph{Inverse functions}\ \\
Let there be two functions $f$ and $g$, with $f$ having domain $A$ and range $B$,
and with $g$ having domain $B$ and range $A$.  If the value of $g$ at each point $y \in B$
is that unique $x$ in $A$ such that $f(x) = y$, then $f$ and $g$ are inverse functions.\\
Also, if $c = f(a)$ and $d = f(b)$, and $f$ is strictly increasing on $[a, b]$, then
$g$ is strictly increasing on $[c, d]$ and is continuous on $[a, b]$.\\
Left inverse: Given sets $V$ and $W$ and a function $T: V \to W$. A function $S$ is called a left
inverse of $T$ if $S: T(V) \to V$ and $S[T(x)]=x$ for all $x$ in $V$. Equivalently,
$ST = I_V$, where $I_V$ is the idenity transformation on $V$.\\
Right inverse: $TR = I_{T(V)}$.\\
Left inverses may or may not exist. If they exist, they are unique and are also right inverses.\\
Right inverses may not be unique.\\
A function has a left inverse iff it is one-to-one.\\
We call a left inverse {\it the} inverse, and denote it by $T^{-1}$.

\paragraph{Boundedness of continuous functions on any interval}\ \\
If $f$ is continuous on $[a, b]$, then $f$ is bounded on $[a,b]$.

\paragraph{Extreme value theorem}\ \\
If $f$ is continuous on $[a,b]$, then there exist points $c$, $d \in [a,b]$
such that $f(c) = \sup f$ and $f(d) = \inf f$.

\paragraph{Small span theorem}\ \\
If $f$ is continuous on $[a,b]$, then for every $\epsilon > 0$ there is a finite
partition of $[a,b]$ such that the span of $f$ in every subinterval is less than $\epsilon$.

\paragraph{Integrability of continuous functions}\ \\
If $f$ is continuous on $[a,b]$, then $f$ is integrable on $[a,b]$.

\paragraph{Mean value theorem - integrals}\ \\
If $f$ is continuous on $[a,b]$, then for some $c$ in $[a,b]$,
$\int_a^b f(x)\,dx = f(c)(b-a)$.

\paragraph{Mean value theorem - second for integrals}\ \\
If $g$ is continuous on $[a,b]$ and if $f$ has a derivative which never changes sign on $[a,b]$,
then for some $c$ in $[a,b]$, $\int_a^b f(x)g(x) = f(a)\int_a^c g(x)\,dx + f(b)\int_c^b g(x)\,dx$.

\paragraph{Mean value theorem - weighted integrals}\ \\
Let $f$, $g$ be continuous on $[a,b]$, and let $g$ never change sign in $[a,b]$.
Then for some $c$ in $[a,b]$, $\int_a^b f(x)g(x)\,dx = f(c)\int_a^b g(x)\,dx$.

\paragraph{Rational functions}\ \\
Definition: Quotient of two polynomials.\\
Proper RF: The degree of the numerator is less than that of the denominator.\\
Every RF has an integral expressible in terms of polynomials, RFs, inverse tangents, and logarithms.\\
Every proper RF can be expressed as a finite sum of fractions of the forms:\\
$\frac{A}{(x+a)^k}$ \hskip 10 pt and \hskip 10 pt $\frac{Bx + C}{(x^2 + bx + c)^m}$, \hskip 10 pt
where $A, B, C, a, b, c$ are constants, $k, m$ are positive integers, and $b^2 - 4c < 0$.\\
When an RF has been expressed in this form, it is decomposed into partial fractions.

\paragraph{Elementary functions}\ \\
Can be obtained from polynomials, exponentials, logarithms, trigonometric, or inverse trigonometric
functions in a finite number of steps by using addition, subtraction, multiplication, division,
and composition.

\paragraph{Classic non-elementary functions}\ \\
$\int_0^xe^{-t^2}\,dt$ \hskip 10 pt $\int_0^x\frac{\sin t}{t}\,dt$ \hskip 10 pt
$\int_0^x\sin(t^2)\,dt$ \hskip 10 pt $\int_0^x\sqrt{1-k^2\sin^2t}\,dt$

\paragraph{Taylor / o-notation method for determining limits}\ \\
Can replace indeterminate limit with Taylor + $o$-notation representation, which may then be easier to find.

\paragraph{Classic limits}\ \\
$\lim_{x \to 0} \frac{a^x-b^x}{x} = \log \frac{a}{b}$ \hskip 10 pt
$\lim_{x \to 0} \frac{\log(1+ax)}{x}= a$ \hskip 10 pt
$\lim_{x\to 0} (1+ax)^{1/x}=e^a$\\
$\lim_{x\to +\infty}\frac{(\log x)^b}{x^a}=0$ \hskip 10 pt for \hskip 5 pt $a>0,b>0$\\
$\lim_{x\to +\infty}\frac{x^b}{e^{ax}}=0$ \hskip 10 pt for \hskip 5 pt $a>0,b>0$.

\paragraph{L'Hopital's rule}\ \\
If $\lim_{x\to a}f(x)=0$ and $\lim_{x\to a}g(x)=0$ and $g'(x)\neq 0$, then
$lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$, if this second limit exists.

\paragraph{Limits of vector fields}\ \\
For $\bs{f}:R^n \to R^m$,\\
$\phantom{x}$ $\lim_{\bs{x}\to \bs{a}} \bs{f}(\bs{x}) = \bs{b}$\\
$\phantom{x}$ means that $\lim_{\lVert \bs{x}-\bs{a} \rVert\to 0} \lVert\bs{f}(\bs{x})-\bs{b}\rVert = 0$\\
$\phantom{x}$ or, $\lim_{\lVert \bs{h}\rVert\to 0} \lVert\bs{f}(\bs{a}+\bs{h})-b\rVert = 0$.

\paragraph{Continuity of vector fields}
\begin{equation*}
\lim_{\bs{x}\to\bs{a}}\bs{f}(\bs{x}) = \bs{f}(\bs{a})
\end{equation*}

\bigskip\bigskip
\section{o-notation}\smallskip

\paragraph{Definition}\ \\
Near $a$, $f(x)$ is small compared with $g(x)$.  More formally:\\
$f(x)=o(g(x))$ \hskip 5 pt as \hskip 5 pt $x \to a$ \hskip 5 pt means that \hskip 5 pt
$\lim_{x \to a}\frac{f(x)}{g(x)}=0$ \hskip 10 pt (assuming $g(x) \neq 0$ near $a$).

\paragraph{Algebra}\ \\
$o(g(x))\pm o(g(x))=o(g(x))$\\
$o(cg(x))=o(g(x))$ \hskip 5 pt if $c \neq 0$\\
$f(x) \cdot o(g(x))=o(f(x)g(x))$\\
$o(o(g(x)))=o(g(x))$

\bigskip\bigskip
\section{Trigonometric functions}\smallskip

\paragraph{Fundamental properties of sine and cosine}\ \\
Defined on entire real line.\\
$\cos 0 = \sin \frac{1}{2} \pi = 1, \cos \pi = -1.$\\
$\cos(y-x) = \cos y \cos x + \sin y \sin x.$\\
For $0 < x < \frac{1}{2} \pi$, $0 < \cos x < \frac{\sin x}{x} < \frac{1}{\cos x}.$

\paragraph{Additional properties of sine and cosine}\ \\
$\sin^2 x + \cos^2 x = 1.$\\
$\sin 0 = \cos \frac{1}{2} \pi = \sin \pi = 0.$\\
$\cos(-x) = \cos(x), \hskip 30 pt \sin(-x) = - \sin(x).$\\
$\sin(\frac{1}{2} \pi + x) = \cos x, \hskip 30 pt \cos(\frac{1}{2}\pi + x) = - \sin(x).$\\
$\sin(x + 2\pi) = \sin x, \hskip 30 pt \cos(x + 2\pi) = \cos x.$\\
$\cos(x+y) = \cos x \cos y - \sin x \sin y, \hskip 30 pt \sin(x+y) = \sin x \cos y + \cos x \sin y.$\\
$\sin a - \sin b = 2 \sin \frac{a - b}{2} \cos \frac{a+b}{2}, \hskip 30 pt \cos a - \cos b = -2\sin \frac{a-b}{2} \sin \frac{a+b}{2}.$\\
In the interval $[0, \frac{1}{2}\pi]$, the sine is strictly increasing, and the cosine is strictly decreasing.

\paragraph{Additional sine and cosine relationships}\ \\
If $0 < a \leq \frac{1}{2}\pi$ and $n \geq 1$,
$\frac{a}{n} \sum_{k=1}^n \cos \frac{ka}{n} < \sin a < \frac{a}{n} \sum_{k=0}^{n-1} \cos \frac{ka}{n}.$

\paragraph{Integrals}\ \\
$\int_0^a \cos x\,dx = \sin a$\\
$\int_0^a \sin x\,dx = 1 - \cos a.$

\paragraph{Other trigonometric functions}\ \\
$\tan x = \frac{\sin x}{\cos x}$ \hskip 20 pt
$\cot x = \frac{\cos x}{\sin x}$ \hskip 20 pt
$\sec x = \frac{1}{\cos x}$ \hskip 20 pt
$\csc x = \frac{1}{\sin x}$

\paragraph{Inverse trigonometric functions}\ \\
$u = \arcsin v$ \hskip 5 pt means $v = \sin u$, \hskip 10 pt
$u = \arccos v$ \hskip 5 pt means $v = \cos u$, \hskip 10 pt
$u = \arctan v$ \hskip 5 pt means $v = \tan u$\\
$\arccot x = \frac{\pi}{2} - \arctan x$ \hskip 5 pt for all real $x$, \hskip 10 pt
$\arcsec x = \arccos \frac{1}{x}$ \hskip 5 pt for $|x| \geq 1$, \hskip 10 pt
$\arccsc x = arcsin \frac{1}{x}$ \hskip 5 pt for $|x| \geq 1$

\paragraph{Inverse trigonometric derivatives and integrals}\ \\
$D \arcsin x = \frac{1}{\sqrt{1 - x^2}}$ \hskip 10 pt if $-1 < x < 1$\\
$\int_0^x \frac{1}{\sqrt{1 - t^2}}\,dt = \arcsin x$\\
$\int \arcsin x\,dx = x \arcsin x + \sqrt{1 - x^2} + C$\\
$D \arccos x = \frac{-1}{\sqrt{1-x^2}}$\\
$D \arctan x = \frac{1}{1+x^2}$\\
$\int \frac{dx}{\sqrt{1-x^2}} = -\arccos x + C$\\
$\int \frac{dx}{1+x^2} = \arctan x + C$\\
$\int \arccos x\,dx = x \arccos x - \sqrt{1-x^2} + C$\\
$\int \arctan x\,dx = x \arctan x - \frac{1}{2} \log (1+x^2) + C$

\bigskip\bigskip
\section{Exponential, logarithm, hyperbolic functions}\smallskip

\paragraph{Logarithm definition}\ \\
$L(x) = \int_1^x \frac{1}{t}\,dt$

\paragraph{Exponential definition}\ \\
$y = E(x)$ means $L(y) = x$

\paragraph{Logarithm properties}\ \\
$L(1) = 0$\\
$L'(x) = \frac{1}{x}$ for ${x > 0}$\\
$L(ab) = L(a) + L(b)$ for $a > 0$, $b > 0$

\paragraph{Exponential properties}\ \\
$E(0) = 1$\\
$E(1) = e$\\
$E'(x) = E(x)$\\
$E(a+b) = E(a)E(b)$\\
$a^x = e^{x \log a}$, for $a > 0$, $x$ real

\paragraph{Derivatives and integrals}\ \\
$(e^x)' = e^x$\\
$(e^{x \log a})' = a^x \log a$\\
$\int e^x\,dx = e^x + C$\\
$\int a^x\,dx = \frac{a^x}{\log a} + C$\\
$\int e^{f(x)}f'(x)\,dx = e^{f(x)} + C$\\
$\int a^{f(x)}f'(x)\,dx = \frac{a^{f(x)}}{\log a} + C$

\paragraph{Logarithmic differentiation}\ \\
If $g(x) = \log |f(x)|$, then $g'(x) = \frac{f'(x)}{f(x)}$.
In cases where $f'(x)$ is difficult to compute, $(\log |f(x)|)'$ may be easier to compute.
Then we know $f(x)$ and $g'(x)$, which we can multiply to give $f'(x)$.

\paragraph{Hyperbolic definitions}\ \\
$\sinh x = \frac{e^x - e^{-x}}{2}$, \hskip 10 pt
$\cosh x = \frac{e^x + e^{-x}}{2}$, \hskip 10 pt
$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$\\
$\csch x = \frac{1}{\sinh x}$, \hskip 10 pt
$\sech x = \frac{1}{\cosh x}$, \hskip 10 pt
$\coth x = \frac{1}{\tanh x}$

\paragraph{Hyperbolic properties}\ \\
$\cosh^2 x - \sinh^2 x = 1$\\
$\sinh(-x) = =\sinh x$\\
$\cosh(-x) = \cosh x$\\
$\tanh(-x) = -\tanh x$\\
$\sinh(x+y)=\sinh x \cosh y + \sinh x \sinh y$\\
$\cosh(x+y)=\cosh x \cosh y + \sinh x \sinh y$\\
$\sinh 2x = 2 \sinh x \cosh x$\\
$\cosh 2x = \cosh^2 x + \sinh^2 x$\\
$\cosh x + \sinh x=e^x$\\
$\cosh x - \sinh x = e^{-x}$\\
$(\cosh x + \sinh x)^n = \cosh nx + \sinh nx$, where $n$ is an integer\\
$2 \sinh^2 \frac{1}{2} x = \cosh x -1$\\
$2 \cosh^2 \frac{1}{2} x = \cosh x + 1$\\
$\tanh^2 x + \sech^2 x = 1$\\
$\coth^2 x - \csch^2 x = 1$

\bigskip\bigskip
\section{Taylor polynomials}\smallskip

\paragraph{Definition}\ \\
$T_n f(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k$

\paragraph{Remainder / error}\ \\
$E_n(x)=\frac{1}{n!}\int_a^x(x-t)^nf^{(n+1)}(t)\,dt$

\paragraph{Error o-notation}\ \\
$f(x)=T_nf(x) + o((x-a)^n)$ \hskip 10 pt as \hskip 5 pt $x \to a$

\paragraph{Error estimate}\ \\
If $m \leq f^{(n+1)}(t) \leq M$, then:\\
If $x>a$: $m\frac{(x-a)^{n+1}}{(n+1)!} \leq E_n(x) \leq M\frac{(x-a)^{n+1}}{(n+1)!}$\\
If $x<a$: $m\frac{(a-x)^{n+1}}{(n+1)!} \leq (-a)^{n+1}E_n(x) \leq M\frac{(a-x)^{n+1}}{(n+1)!}$

\paragraph{Algebra}\ \\
Linearity: $T_n(c_1f+c_2g) = c_1T_n(f)+c_2T_n(g)$\\
Differentiation: $(T_nf)' = T_{n-1}(f')$\\
Substitution: If $g(x)=f(cx)$, then $T_ng(x;a)=T_nf(cx;ca)$\\
Integration: An indefinite integral of a Taylor polynomial of $f$ is a Taylor
polynomial of an indefinite integral of $f$.\\
Integration, precisely: If $g(x)=\int_a^xf(t)\,dt$,
then $T_{n+1}g(x)=\int_a^xT_nf(t)\,dt$.

\bigskip\bigskip
\section{Linear Differential Equations}\smallskip

\paragraph{First order}\ \\
$\left\{f'(x)=f(x),\ f(0)=C \right\}$ \yields $f(x)=Ce^x$\\
$\left\{y'+P(x)y=0,\ f(a)=b \right\}$ \yields $f(x)=be^{-A(x)},\ A(x)=\int_a^xP(t)\,dt$\\
$\left\{y'+P(x)y=Q(x),\ f(a)=b \right\}$ \yields $f(x)=be^{-A(x)}+e^{-A(x)}\int_a^xQ(t)e^{A(t)}\,dt$

\paragraph{Second order - homogeneous, constant coefficients}\ \\
Form: $y'' + ay' + by = 0$\\
Let $d = a^2-4b$. Every solution has the form:\\
$y=e^{-ax/2}[c_1u_1(x)+c_2u_2(x)]$ \hskip 10 pt where $c_1$ and $c_2$ are constants, and $u_1$ and $u_2$:\\
If $d=0$, then $u_1(x)=1$ and $u_2(x)=x$\\
If $d>0$, then $u_1(x)=e^{kx}$ and $u_2(x)=e^{-kx}$ \hskip 10 pt where $k=\frac{1}{2}\sqrt{d}$\\
If $d<0$, then $u_1(x)=\cos kx$ and $u_2(x)=\sin kx$ \hskip 10 pt where $k=\frac{1}{2}\sqrt{-d}$

\paragraph{Second order - nonhomogeneous, constant coeffcients}\ \\
Form: $y'' + ay' + by = R$\\
Let $v_1(x)$ and $v_2(x)$ be a basis of solutions to $L(y)=0$ given by
$v_1(x)=e^{-ax/2}u_1(x)$ \hskip 5 pt and \hskip 5 pt $v_2(x)=e^{-ax/2}u_2(x)$.\\
Let $W(x)=v_1(x)v_2'(x)-v_2(x)v_1'(x)$. \hskip 5 pt (Wronskian of $v_1$ and $v_2$)\\
Then $L(y)=R$ has a particular solution given by $y_1(x)=t_1(x)v_1(x)+t_2(x)v_2(x)$\\
$t_1(x)=-\int v_2(x)\frac{R(x)}{W(x)}\,dx$, \hskip 10 pt $t_2(x)=\int v_1(x)\frac{R(x)}{W(x)}\,dx$\\
The general solution is given by $y=c_1v_1+c_2v_2+y_1$.

\paragraph{Second order - nonhomogeneous, constant coefficients - special methods}\ \\
$R$ is a polynomial (of degree $n$) and $b \neq 0$:\\
$\phantom{x}$ (1) Take $Ax^n+Bx^{n-1}+\cdots+D$ and differentiate twice\\
$\phantom{x}$ (2) Plug into the differential equation, and equate coeffs of like powers to find $y_1$.\\
$R(x)=p(x)e^{mx}$, $p(x)$ is polynomial of degree $n$, and $m$ is constant.\\
$\phantom{x}$ (1) Use the substitution $y=u(x)e^{mx}$\\
$\phantom{x}$ (2) This transforms the diff eq into $u''+(2m+a)u'+(m^2+am+b)u=p$, where $p$ is a polynomial\\
$\phantom{x}$ (3) This can be solved as in the previous case

\paragraph{Order $\boldsymbol{n}$}\ \\
Form: $P_0(x)y^{(n)} + P_1(x)y^{(n-1)} + \dotsb + P_n(x)y = R(x)$\\
Singular point: where $P_0(x) = 0$ (often introduce complications)\\
Form ($P_0(x) \neq 0$): $y^{(n)} + P_1(x)y^{(n-1)} + \dotsb + P_n(x)y = R(x)$\\
Operator notation: $L(f) = f^{(n)} + P_1f^{(n-1)} + \dotsb + P_nf$\\
Operator notation (alternate): $L = D^n + P_1 D^{n-1} + \dotsb + P_n$\\
$L$ is a linear differential operator of order $n$\\
Homogeneous: $L(y) = 0$\\
Nonhomogeneous: $L(y) = R$\\
Solution of the homogeneous equation is the null space of $L$, and $\dim N(L) = n$\\
Homogeneous existence-uniqueness: there is one and only one solution satisfying an order $n$ eqn and initial conditions\\
For $n$ independent solutions, all linear combinations of these solutions is the general solution\\
General solution of nonhomogeneous is one particular solution of nonhomogeneous, plus linear combinations of homogeneous

\paragraph{Order $\boldsymbol{n}$ - constant coefficients}\ \\
Constant-coefficient operator: $A = a_0D^n + a_1D^{n-1} + \dotsb + a_{n-1}D + a_n$\\
The set of all constant-coefficient operators is a linear space\\
Constant-coefficient operators commute, since $D^rD^s = D^sD^r$\\
Characteristic polynomial of $A$: $p_A(r) = a_0r^n + a_1r^{n-1} + \dotsb + a_n$\\
Let $A$ and $B$ be constant-coefficient operators and $\lambda$ real:\\
$\phantom{x}$ (a) $A = B$ iff $p_A = p_B$\\
$\phantom{x}$ (b) $p_{A+B} = p_A + p_B$\\
$\phantom{x}$ (c) $p_{AB} = p_A \cdot p_B$\\
$\phantom{x}$ (d) $p_{\lambda A} = \lambda \cdot p_A$\\
Factorizations of a characteristic polynomial are also factorizations of the operators\\
If you factor an operator, the solution space of the operator contains the solution space of each factor\\
Annihilation: If an operator maps an element to $O$, it is said to {\it annihilate} that element\\
If a factor annihilates $u$, then $L$ annihilates $u$ as well\\
Solutions:\\
$\phantom{x}$ (a) $p_L(r) = 0$ has real distinct roots: $y = \sum_{k=1}^n c_k e^{r_k x}$\\
$\phantom{x}$ (b) Real roots, some repeated: $u_1(x) = e^{rx}, u_2(x) = xe^{rx}, u_3 = x^2e^{rx}$ etc.\\
$\phantom{x}$ (c) Complex roots $\alpha + i \beta$: $u_1(x) = e^{\alpha x}\cos \beta x$, $u_2(x) = e^{\alpha x}\sin \beta x$\\
$\phantom{x}$ (d) Repeated complex roots: Add $x^{m-1}$ to each solution

\paragraph{Determining a particular solution of a nonhomogeneous equation - Variation of parameters}\ \\
Want to determine one solution of $L(y) = R$\\
Let $u_1, \dotsc, u_n$ be $n$ independent solutions of the homogeneous equation $L(y) = 0$\\
Define an $n \times 1$ column matrix $v$ as:
\begin{equation*}
v(x) = \int_c^x R(t) W(t)^{-1}
\begin{bmatrix}
0 \\ \vdots \\ 0 \\ 1
\end{bmatrix}
\, dt
\end{equation*}
where $W$ is the Wronskian matrix of $u_1, \dotsc, u_n$ and $c$ is any point in $J$.\\
Then $y_1(x) = \sum_{k=1}^n u_k(x) v_k(x)$\\
The Wronskian matrix of $n$ independent solutions is always nonsingular.

\paragraph{Determining a particular solution of a nonhomogeneous equation - Reduction to first-order system}\ \\
Works if: constant coefficients\\
Make a substitution, say $u = (D - 2)y$\\
This reduces the order of the equation by one.\\
When you find the solution to one equation, substitute it back into other equations.

\paragraph{Determining a particular solution of a nonhomogeneous equation - Annihilator method}\ \\
Works if:\\
$\phantom{x}$ Constant coefficients and $R$ is annihilated by a constant-coefficient differential operator\\
Steps:\\
$\phantom{x}$ (1) Apply the annihilator operator to both sides, producing a homogeneous constant-coefficient equation.\\
$\phantom{x}$ (2) Find the general solution to this equation.  The particular solution is one of these.\\
$\phantom{x}$ (3) Plug the general solution with undetermined coefficients back into $L(y) = R$.\\
$\phantom{x}$ (4) Equate coefficients of like terms to determine the coefficients of this particular solution.\\
Functions and corresponding annihilators:\\
\begin{tabular}{ll}
$y = x^{m-1}$ & $D^m$\\
$y = e^{\alpha x}$ & $D - \alpha$\\
$y = x^{m-1}e^{\alpha x}$ & $(D - \alpha)^m$\\
$y = \cos \beta x$ or $y = \sin \beta x$ & $D^2 + \beta^2$\\
$y = x^{m-1} \cos \beta x$ or $y = x^{m-1} \sin \beta x$ & $(D^2 + \beta^2)^m$\\
$y = e^{\alpha x} \cos \beta x$ or $y = e^{\alpha x} \sin \beta x$ & $D^2 - 2 \alpha D + (\alpha^2 + \beta^2)$\\
$y = x^{m-1} e^{\alpha x} \cos \beta x$ or $y = x^{m-1} e^{\alpha x} \sin \beta x$
& $[D^2 - 2 \alpha D + (\alpha^2 + \beta^2)]^m$
\end{tabular}

\paragraph{Second order - analytic coefficients}\ \\
A function is {\it analytic} if it has a power series expansion in an interval.\\
For $y'' + P_1(x)y' + P_2(x)y$, if $P_1$ and $P_2$ are analytic, then the equation has two independent analytic solutions.\\
More generally, a homogeneous linear equation of order $n$ with analytic coefficients has $n$ independent analytic solutions.\\
Finding power series solutions:\\
$\phantom{x}$ (1) Write a general possible solution, $y = \sum_{n=0}^\infty a_n (x-x_0)^n$\\
$\phantom{x}$ (2) Differentiate this possible solution twice to find $y'$ and $y''$\\
$\phantom{x}$ (3) Plug these into the differential equation\\
$\phantom{x}$ (4) Determine what the coefficients $a_n$ would have to look like (recursion formula) to satisfy the equation\\
$\phantom{x}$ (5) Verify the determined power series converges in the given interval\\
$\phantom{x}$ (6) Determine specific coefficients to satisfy initial conditions or to find independent solutions

\paragraph{Legendre equation and polynomials}\ \\
Legendre equation: $(1 - x^2)y'' - 2xy' + \alpha (\alpha + 1) y = 0$, where $\alpha$ is any real constant.\\
Legendre equation (Sturm-Liouville operator): $[(x^2 - 1)y']' = \alpha(\alpha + 1)y$.\\
$\to$ So $T(y) = \lambda y$, where $T(f) = (pf')'$ and $p(x) = x^2 - 1$ and $\lambda = \alpha(\alpha + 1)$.\\
$\to$ So the nonzero solutions are eigenfunctions belonging to $\alpha(\alpha + 1)$.\\
General solution on $(-1, 1)$:
\begin{equation*}
y = a_0u_1(x) + a_1u_2(x)
\end{equation*}
\begin{equation*}
u_1(x) = 1 + \sum_{n=1}^\infty (-1)^n \frac{\alpha (\alpha - 2) \dotsb (\alpha - 2n + 2) \cdot
(\alpha + 1) (\alpha + 3) \cdot (\alpha + 2n - 1)}{(2n)!} x^{2n}\\
\end{equation*}
\begin{equation*}
u_2(x) = x + \sum_{n=1}^\infty (-1)^n \frac{(\alpha - 1)(\alpha - 3) \dotsb (\alpha - 2n + 1) \cdot
(\alpha + 2)(\alpha + 4) \dotsb (\alpha + 2n)}{(2n + 1)!} x^{2n+1}
\end{equation*}
When $\alpha$ is zero or a positive integer, the equation has polynomial solutions called Legendre polynomials:
\begin{equation*}
P_n(x) = \frac{1}{2^n} \sum_{r=0}^{[n/2]} \frac{(-1)^r(2n-2r)!}{r!(n-r)!(n-2r)!} x^{n-2r}
\end{equation*}
where $[n/2]$ is the greatest integer $\leq n/2$.\\
These are the same (except for scalar factors) as the result of the Gram-Schmidt process on $\{1, x, x^2, x^3, \dotsc\}$.\\
$\to$ So, they are special in that they are orthogonal in Euclidean space with inner product $\int_{-1}^1 fg\,dx$.\\
Rodrigues' formula for the Legendre polynomials: $P_n(x) = \frac{1}{2^nn!} \frac{d^n}{dx^n} (x^2 - 1)^n$.\\
$P_n(-x) = (-1)^nP_n(x)$, so $P_n$ is even when $n$ is even and odd when $n$ is odd.

\paragraph{Method of Frobenius}\ \\
Works if:\\
$\phantom{x}$ (1) Differential equation has the form: $(x-x_0)^2y''+(x-x_0)P(x)y'+Q(x)y=0$.\\
$\phantom{x}$ (2) $P$ and $Q$ are analytic in some open interval about $x_0$.\\
$\phantom{x}$ (Note) We say $x_0$ is a {\it regular} singular point of the equation.\\
Preliminaries:\\
$\phantom{x}$ Indicial equation of the differential equation: $t(t-1)+P(x_0)t+Q(x_0)=0$.\\
$\phantom{x}$ Denote the two roots of the indicial equation by $\alpha_1$ and $\alpha_2$.\\
First case: $\alpha_1 - \alpha_2$ is not an integer\\
$\phantom{x}$ There are two independent solutions, $u_1$ and $u_2$.\\
$\phantom{x}$ $u_1(x) = |x - x_0|^{\alpha_1} \sum_{n=0}^\infty a_n(x-x_0)^n$, with $a_0 = 1$.\\
$\phantom{x}$ $u_2(x) = |x - x_0|^{\alpha_2} \sum_{n=0}^\infty b_n(x-x_0)^n$, with $b_0 = 1$.\\
Second case: $\alpha_1 - \alpha_2 = N$, a nonnegative integer\\
$\phantom{x}$ $u_1$ is the same as in the first case.\\
$\phantom{x}$ $u_2 = |x-x_0|^{\alpha_2}\sum_{n=0}^\infty b_n(x-x_0)^n + C u_1(x)\log|x-x_0|$, with $b_0 = 1$.\\
$\phantom{x}$ $\to C$ is nonzero if $N = 0$; otherwise it may or may not be zero.

\paragraph{Gamma function}\ \\
For complex $z$ with real part $> 0$:\\
$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\, dt$\\
It can also be extended to be defined for the rest of the complex plane except for negative integers.

\paragraph{Bessel equation and functions}\ \\
Bessel equation: $x^2y'' + xy' + (x^2 - \alpha^2)y = 0$, where $\alpha \geq 0$\\
Bessel function of the first kind of order $\alpha$:\\
$\phantom{x}$ $J_a(x) =
\bigl(\frac{x}{2}\bigr)^\alpha \sum_{n=0}^\infty \frac{(-1)^n}{n!\Gamma(n+1+\alpha)}\bigl(\frac{x}{2}\bigr)^{2n}$\\
$\phantom{x}$ Valid for $x>0$\\
Bessel function of the first kind when $\alpha = p$ is a nonnegative integer:\\
$\phantom{x}$ $J_p(x) = \sum_{n=0}^\infty \frac{(-1)^n}{n!(n+p)!} \bigl(\frac{x}{2}\bigr)^{2n+p}$\\
$\phantom{x}$ Valid for $x \neq 0$\\
Bessel function valid if $\alpha$ is not zero or a positive integer:\\
$\phantom{x}$ $J_{-\alpha}(x) =
\bigl(\frac{x}{2}\bigr)^{-\alpha} \sum_{n=0}^\infty \frac{(-1)^n}{n!\Gamma(n+1-\alpha)}\bigl(\frac{x}{2}\bigr)^{2n}$\\
$\phantom{x}$ Valid for $x>0$\\
If $\alpha$ is not an integer, $J_\alpha(x)$ and $J_{-\alpha}(x)$ are linearly independent and general solution is:\\
$\phantom{x}$ $y = c_1 J_\alpha(x) + c_2 J_{-\alpha}(x)$\\
$\phantom{x}$ Valid for $x>0$\\
Bessel function of the second kind of order $p$:\\
$\phantom{x}$ $Y_p(x) = J_p(x) \log x + x^{-p} \sum_{n=0}^\infty C_n x^n$\\
$\phantom{x}$ With last term determined:
$Y_p(x) = J_p(x) \log x - \frac{1}{2}\bigl(\frac{x}{2}\bigr)^{-p} \sum_{n=0}^{p-1} \frac{(p-n-1)!}{n!}
\bigl(\frac{x}{2}\bigr)^{2n} - \frac{1}{2}\bigl(\frac{x}{2}\bigr)^p \sum_{n=0}^\infty
(-1)^n \frac{h_n + h_{n+p}}{n!(n+p)!}\bigl(\frac{x}{2}\bigr)^{2n}$\\
$\phantom{x}$ Where $h_0 = 0$ and $h_n = 1 + \frac{1}{2} + \dotsb + 1/n$ for $n \geq 1$\\
$\phantom{x}$ Valid when $\alpha = p$ is a nonnegative integer and $x > 0$\\
If $\alpha$ is a nonnegative integer, general solution is:\\
$\phantom{x}$ $y = c_1 J_p(x) + c_2 Y_p(x)$\\
$\phantom{x}$ Valid for $x>0$

\bigskip\bigskip
\section{Systems of differential equations}\smallskip

\paragraph{Concepts}\ \\
Any order differential equation can be rewritten as a system of first-order equations.\\
A general linear system has the form $Y' = P(t)Y + Q(t)$\\
...where $Y'$ is a column matrix, $P(t)$ is a coefficient matrix, and $Q(t)$ is a column matrix.

\paragraph{Basic exponential differential equation}\ \\
If $F'(t) = F(t)A$ or $F'(t) = FE(t)$, then $F(t) = e^{tA}$.\\
$F'(t) = AF(t)$ and $F(0) = B$ \yields $F(t) = e^{tA}B$.\\
$F'(t) = F(t)A$ and $F(0) = B$ \yields $F(t) = Be^{tA}$.

\paragraph{Exponential matrices}\ \\
$e^O = I$
\begin{equation*}e^A = \sum_{k=0}^\infty \frac{A^k}{k!}\end{equation*}
A series of matrices is convergent if the series for each element is convergent.
\begin{equation*}\text{Matrix norm: } \lVert A \rVert = \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|\end{equation*}\\
If $\sum_{k=1}^\infty \lVert C_k \rVert$ converges, then $\sum_{k=1}^\infty C_k$ also converges.\\
In general two matrices commute under multiplication if they are both diagonalizable\\
...but there are other circumstances where they commute as well, for example $\lambda I$ commutes with any matrix.\\
$e^{tA}$ is always nonsingular, and its inverse is $e^{-tA}$.\\
If $A$ and $B$ commute, then $e^{A+B} = e^A + e^B$.

\paragraph{Homogeneous linear systems with constant coefficients}\ \\
For $Y'(t) = A Y(t)$, $Y(a) = B$ \yields $Y(t) = e^{(t-a)A}B$.

\paragraph{Calculating $e^{tA}$}\ \\
Diagonal matrix: If $A=\diag(\lambda_1,\dotsc,\lambda_n)$, then $e^{tA}=\diag(e^{t\lambda_1},\dotsc,e^{t\lambda_n}).$\\
If $A$ can be diagonalized ($A = CDC^{-1}$): $e^{tA} = Ce^{tD}C^{-1}$.\\
Putzer's method: can be used for any $n \times n$ matrix.\\
There are other special methods for matrices where eigenvalues are identical, distinct, or
of multiplicity $1$ and $n-1$.

\paragraph{Nonhomogeneous systems with constant coefficients}\ \\
Form: $Y'(t) = AY(t) + Q(t)$, $Y(a) = B$.\\
Solution: $Y(x) = e^{(x-a)A}B + e^{xa} \int_a^x e^{-tA} Q(t)\, dt$.

\paragraph{Cayley-Hamilton theorem}\ \\
An $n \times n$ matrix satisfies its own characteristic equation.\\
For an $n \times n$ matrix $A$,\\
...characteristic polynomial is $f(\lambda)=\det(\lambda I-A)=\lambda^n+c_{n-1}\lambda^{n-1}+\dotsb+c_1\lambda + c_0$.\\
...Then $f(A) = O$\\
...Or, $A^n + c_{n-1}A^{n-1} + \dotsb + c_1A + c_0I = O$.

\paragraph{Homogeneous linear systems}\ \\
If $A(t)$ is analytic, you can attempt to find a power series solution.

\paragraph{General linear system}\ \\
Form:\\
$\phantom{x}$ $Y'(x) = P(x)Y(x) + Q(x)$, $Y(a) = B$\\
There is one and only one solution to this initial value problem.\\
Solution:\\
$\phantom{x}$ $Y(x) = F(x)^{-1}Y(a) + F(x)^{-1} \int_a^x F(t)Q(t)\, dt$\\
$F(x)$ is the transpose of the matrix whose $k$th column is the solution of the initial-value problem:\\
$\phantom{x}$ $Y'(x) = -P(x)^t Y(x)$, $Y(a) = I_k$, where $I_k$ is the $k$th column of $I$.\\
Note: this is not always a useful formula since determining $F(x)$ can be difficult.

\paragraph{Method of successive approximations}\ \\
Basically, start with something, apply an operation, plug that back into the original, apply the operation again, etc.\\
Can be used in an existence theorem for homogeneous linear systems $Y'(t) = A(t) Y(t)$, $Y(a) = B$.\\
Process:\\
$\phantom{x}$ (1) Start with $Y(x) = B$.\\
$\phantom{x}$ (2) Plug into equation, $Y'(t) = A(t)B$.\\
$\phantom{x}$ (3) Solve to get $Y_1(x) = B + \int_a^x A(t)B\, dt$.\\
$\phantom{x}$ (4) Plug $Y_1(x)$ back into original equation, repeat.\\
$\phantom{x}$ (5) Come up with a recursive formula describing all terms, determining a series.\\
$\phantom{x}$ (6) Prove that the series converges on the required interval.\\
Note: In practice, it is often very difficult to compute the successive approximations,\\
...so this method is often more useful for existence proofs than solving problems.

\bigskip\bigskip
\section{Nonlinear Differential Equations}\smallskip

\paragraph{First order - separable}\ \\
If $y'=Q(x)R(y)$, this can be written as $A(y)y'=Q(x)$.
Then, if $y=Y(x)$ is a solution, and if $G$ is any primitive of $A$, then we have
an implicit formula for y: $G(y)=\int Q(x)\,dx+C$, \hskip 5 pt for some $C$.

\paragraph{First order - homogeneous}\ \\
For $y'=f(x,y)$, {\it homogeneous} means that $f(tx,ty)=f(x,y)$ for all $x,y,t \neq 0$.
In this case, we can use the substitution $v=y/x$, which gives us
$x\frac{dv}{dx}=f(1,v)-v$, which is a first-order separable equation for $v$.

\paragraph{Using power series to solve}\ \\
Technique:  Assume there is a power series expansion of the solution about a point (say $0$).
Then, differentiate the power series sum function as needed, and plug into the original equation,
equate coefficients of like terms, etc.  Can also make use of the uniqueness theorem for power
series which says that two power series with the same sum function about the same point must
be equal, term by term.  Then, I think you verify convergence of the resulting series.  Normally,
will have undetermined coefficients, which gives a general class of solutions to the diff eq.
Can also use the fact the $a_n=\frac{f^{(n)}(0)}{n!}$, if the expansion is about $0$.

\paragraph{Nonlinear systems}\ \\
Form:\\
$\phantom{x}$ $Y'(x) = F(x, Y(x))$, $Y(a) = B$\\
Lipschitz condition:\\
$\phantom{x}$ For a vector mapping $F(x, Y)$,\\
$\phantom{x}$ $\lVert F(x, Y) - F(x, Z) \rVert \leq A \lVert Y - Z \rVert$\\
$\phantom{x}$ where $A$ is a positive constant.\\
If $\lVert F(x, Y) \rVert \leq M$ where $M$ is a positive constant,\\
...and if $G(x) = F(x, Y(x))$ is continuous on $(a-h, a+h)$,\\
...and if $F$ satisfies the Lipschitz condition\\
...and let $S$ be a k-cell $\{(x, Y) | \quad |x-a| \leq h, \lVert Y - B \rVert \leq k\}$\\
...where $h$ and $k$ are positive constants\\
...then there is one and only one function defined on $(a-c,a+c)$ with $c = \min\{h,k/M\}$\\
...which satisfies the nonlinear system given above.

\bigskip\bigskip
\section{Partial Differential Equations}\smallskip

\paragraph{General concepts}\ \\
It often happens that in solving a PDE, an equation of one variable is introduced\\
that can be anything - and so the solution space is often infinite-dimensional.

\bigskip\bigskip
\section{Fixed points and contraction operators}\smallskip

\paragraph{Fixed point of an operator}\ \\
If $T(Y) = Y$, then $Y$ is a {\it fixed point} of $T$.

\paragraph{Norms}\ \\
A function in a linear space such that:\\
$\phantom{x}$ (1) $\lVert x \rVert \geq 0$\\
$\phantom{x}$ (2) $\lVert cx \rVert = |c| \lVert x \rVert$\\
$\phantom{x}$ (3) $\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert$\\
$\phantom{x}$ (4) $\lVert x \rVert = 0$ implies $x = O$
A linear space with a norm defined is called a {\it normed linear space}.\\
The max norm:\\
$\phantom{x}$ For a function $\varphi \in C(J)$, the max norm is $\lVert \varphi \rVert = \max_{x\in J} |\varphi(x)|$.

\paragraph{Contraction operators}\ \\
An operator is a {\it contraction operator} if:\\
$\phantom{x}$ $T: C(J) \to C(J)$\\
$\phantom{x}$ $\lVert T(\varphi) - T(\psi) \rVert \leq \alpha \lVert \varphi - \psi \rVert$\\
$\phantom{x}$ with contraction constant $\alpha$ such that $0 \leq \alpha < 1$.\\
Every contraction operator has one and only one fixed point such that $T(\varphi) = \varphi$.

\bigskip\bigskip
\section{Sequences and series}\smallskip

\paragraph{Definitions}\ \\
Infinite sequence: a function whose domain is the set of all positive integers.\\
Sequence convergence: can be for real or complex sequences. In the complex case, each component must converge separately.\\
Series or infinite series: a sequence $\{s_n\}$ of partial sums of an infinite sequence such that $s_n = \sum_{k=1}^n a_k$.\\
If $\lim_{n \to \infty} s_n = S$, for some finite $S$, then the series converges and $S$ is its sum.

\paragraph{Properties}\ \\
A monotonic sequence converges iff it is bounded.\\
Linearity: $\sum_{n=1}^\infty(\alpha a_n + \beta b_n) = \alpha \sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n$.\\
If $\sum a_n$ conveges and $\sum b_n$ diverges, then $\sum(a_n+b_n)$ diverges.\\
If $\{a_n\}$ and $\{b_n\}$ are two sequences of complex numbers st $a_n = b_n - b_{n+1}$, then
$\sum a_n$ converges iff $\{b_n\}$ converges, in which case $\sum_{n=1}^\infty a_n=b_1-\lim_{n \to \infty} b_n$.\\
For complex $x$, if $|x|<1$, $\sum_{n=0}^\infty x^n$ converges and equals $\frac{1}{1-x}$. Otherwise, it diverges.\\
Finite sums can be rearranged without affecting the sum. This is not necessarily true for series.\\
Absolutely convergent series can be rearranged without affecting the sum.\\
If $\sum a_n$ is a real conditionally convergent series, then for any real number $S$, there is a
rearrangement $\sum b_n$ of $\sum a_n$ which converges to $S$.

\paragraph{Convergence and divergence}\ \\
If $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 0$.\\
If all $a_n$ are nonnegative, then $\sum a_n$ converges iff $\{a_n\}$ is bounded above.\\
If all $a_n$ and $b_n$ are nonnegative, and if there is some $c$ st $a_n\leq cb_n$ for all $n$,
then convergence of $\sum b_n$ implies convergence of $\sum a_n$.\\
If $\{a_n\}$ is a monotonic decreasing sequence with limit $0$, then $\sum_{n=1}^\infty(-1)^{n-1}a_n$ converges.
Also, $0 < (-1)^n(S-s_n)<a_{n+1}$, for $n\geq 1$.\\
If $\sum |a_n|$ converges, then $\sum a_n$ also converges, and $\left|\sum_{n=1}^\infty a_n\right|\leq\sum_{n=1}^\infty |a_n|$.

\paragraph{Series Tests}\ \\
Comparison test:\\
$\phantom{x}$ If $a_n$ and $b_n$ nonnegative, and $a_n \leq cb_n$ for some positive constant $c$\\
$\phantom{x}$ Then convergence of $\sum b_n$ implies convergence of $\sum a_n$.\\
$\phantom{x}$ When the inequality is satisfied, $\sum b_n$ {\it dominates} $\sum a_n$.\\
Limit comparison test:\\
$\phantom{x}$ If all $a_n$ and $b_n$ are positive\\
$\phantom{x}$ And if $\lim_{n\to\infty}\frac{a_n}{b_n}=1$ (asymptotically equal)\\
$\phantom{x}$ Then $\sum a_n$ and $\sum b_n$ converge or diverge together.\\
Integral test:\\
$\phantom{x}$ If $f(x)$ is a positive decreasing fn defined on $x\geq1$\\
$\phantom{x}$ And if $s_n=\sum_{k=1}^nf(k)$ and $t_n=\int_1^nf(x)\,dx$\\
$\phantom{x}$ Then $\{s_n\}$ and $\{t_n\}$ converge or diverge together.\\
Root test:\\
$\phantom{x}$ $a_n$ nonnegative and $\lim_{n\to\infty}\sqrt[n]{a_n}=R$\\
$\phantom{x}$ $R<1$: Converges\\
$\phantom{x}$ $R>1$: Diverges\\
$\phantom{x}$ $R=1 or limit doesn't exist$: Inconclusive\\
Ratio test:\\
$\phantom{x}$ $a_n$ positive and $\frac{a_{n+1}}{a_n}\to L$ as $n \to \infty$\\
$\phantom{x}$ $L < 1$: Converges\\
$\phantom{x}$ $L > 1$: Diverges\\
$\phantom{x}$ $L = 1 or limit doesn't exist$: Inconclusive\\
Dirichlet's test:\\
$\phantom{x}$ If the partial sums of $\sum a_n$ form a bounded sequence\\
$\phantom{x}$ And if $\{b_n\}$ is a decreasing sequence which converges to $0$\\
$\phantom{x}$ Then $\sum a_nb_n$ converges.\\
Abel's test:\\
$\phantom{x}$ If $\sum a_n$ is a convergent complex series\\
$\phantom{x}$ And if $\{b_n\}$ is a monotonic convergent real sequence\\
$\phantom{x}$ Then $\sum a_nb_n$ converges.

\paragraph{Converging series}\ \\
$\sum\frac{1}{n^2+n)}$\\
$\sum\frac{1}{n^2}$\\
$\sum\frac{1}{n^s}$, for $s>1$

\paragraph{Diverging series}\ \\
$\sum\frac{1}{n}$

\paragraph{Miscellaneous}\ \\
Harmonic series: $\sum_{k=1}^n \frac{1}{k}$\\
Geometric series: $\sum_{k=0}^n x^k$\\
Power series: $\sum_{n=0}^\infty a_nx^n$ \hskip 5 pt or \hskip 5 pt $\sum_{n=0}^\infty a_n(z-a)^n$\\
Riemann zeta-function: $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$, if $s>1$

\paragraph{Abel's partial summation formula}\ \\
Let $\{a_n\}$ and $\{b_n\}$ be sequences of complex numbers, and let $A_n=\sum_{k=1}^na_k$.
Then, $\sum_{k=1}^na_kb_k=A_nb_{n+1}+\sum_{k=1}^nA_k(b_k-b_{k+1})$.

\bigskip\bigskip
\section{Sequences and series of functions}\smallskip

\paragraph{General}\ \\
In general, the limit of integrals is not equal to the integral of the limit.\\
For a {\it uniformly convergent} sequence, limit and integration can be interchanged.

\paragraph{Pointwise vs. Uniform convergence}\ \\
Pointwise vs. uniform convergence\\
If each function $f_n$ is continuous at a point $p$, and if $f_n \to f$ uniformly, then $f$ is also continuous at $p$.\\
If a series of functions converges uniformly, and if each term is continuous at a point, then the sum is also continuous at that point.\\
If $f \to f_n$ uniformly on an interval, then $\lim_{n\to\infty}\int_a^xf_n(t)\,dt=\int_a^x\lim_{n\to\infty}f_n(t)\,dt$.\\
If $\sum u_k$ converges uniformly, then $\sum_{k=1}^\infty\int_a^xu_k(t)\,dt=\int_a^x\sum_{k=1}^\infty u_k(t)\,dt$.

\paragraph{Tests for uniform convergence}\ \\
Weierstrass M-Test: If $\sum u_n$ converges pointwise to $f$ on $S$, and if there is a convergent series of
positive constants $\sum M_n$ such that $0 \leq |u_n(x)| \leq M_n$ for every $n \geq 1$ and $x \in S$,
then $\sum u_k$ converges uniformly on S.

\paragraph{Circle of convergence for complex power series}\ \\
For every complex power series $\sum_{n=0}^\infty a_n(z-a)^n$, there is a circle of convergence.  Inside the circle
of convergence, the series converges absolutely for any fixed $z$, and converges uniformly for variable
$z$ with domain inside the circle.  The series diverges outside the circle.  On the boundary, the series
may or may not converge.

\paragraph{Power series expansion / representation}\ \\
Power series expansion of $f$ about $a$: $f(x)=\sum_{n=0}^\infty a_n(x-a)^n$

\paragraph{Properties of power-series represented functions}\ \\
$\bullet$ Assume a function $f$ is represented by the power series $f(x)=\sum_{n=0}^\infty a_n(x-a)^n$ in
an open interval $(a-r, a+r)$. Then $f$ is continuous on this interval, and its integral over any
closed subinterval may be computed by integrating the series term by term. In particular, for
every $x$ in $(a - r, a + r)$, we have
$\int_a^xf(t)\,dt = \sum_{n=0}^\infty a_n\int_a^x (t-a)^n\,dt = \sum_{n=0}^\infty \frac{a_n}{n+1}(x-a)^{n+1}$\\
$\bullet$ If $f$ is represented by a power series as above in the interval of convergence $(a-r,a+r)$, then
the derivative $f'(x)$ exists for each $x$ in the interval of convergence, and is given by the
differentiated series $f'(x)=\sum_{n=1}^\infty na_n(x-a)^{n-1}$, which also has radius of convergence $r$.\\
$\bullet$ The sum function of a power series has derivatives of {\it every} order, which may be obtained by
repeated term-by-term differentiation of the power series.\\
Uniqueness theorem for power series expansions: If a function has a power series expansion, it has the form
$f(x)=\sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k$.  In other words, $a_n = \frac{f^{(n)}(a)}{n!}$.\\
$\bullet$ A function has a power series expansion about $a$ if and only if it is infinitely differentiable in
the interval, and the error term of its Taylor polynomial about $a$ tends to $0$ as $n\to\infty$.\\
$\bullet$ Sufficient condition for Taylor's series converging to $f(x)$ in an interval:  If $f$ is infinitely
differentiable in an open interval, and if $|f^{(n)}(x)| \leq A^n$ for $n\geq 1$ and $x\in I$, for
some positive constant $A$.

\paragraph{Classic power series expansions}\ \\
$\sin x = x - \frac{x^3}{3!}+\frac{x^t}{5!}-\frac{x^7}{7!}+\cdots+(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}+\cdots$\\
$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots+(-1)^n\frac{x^{2n}}{(2n)!}+\cdots$\\
$e^x=1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}+\cdots$

\bigskip\bigskip
\section{General mathematics}\smallskip

\paragraph{Binomial theorem}\ \\
$(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k$\\
Also, for $|x|<1$, $(1+x)^\alpha=\sum_{n=0}^\infty\binom{\alpha}{n}x^n$, with $\alpha$ an arbitrary real number.

\paragraph{Euler's constant}\ \\
$\gamma = \lim_{n\to\infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n} - \log n \right)$\\
$\gamma$ is approximately equal to $0.5772156649$

\bigskip\bigskip
\section{Vector algebra}\smallskip

\paragraph{Basic Properties}\ \\
Cauchy-Schwarz Inequality:  For $A$ and $B$ (vectors in $V_n$), we have $(A \cdot B)^2 \leq (A\cdot A)(B \cdot B)$.  The equality sign
holds iff one of the vectors is a scalar multiple of the other.\\
Length or norm:  $\lVert A \rVert = (A \cdot A)^{1/2}$.\\
Triangle inequality:  $\lVert A+B \rVert^2 \leq \lVert A \rVert + \lVert B \rVert$.  The equality sign holds
iff $A = O$, $B = O$, or if $B = cA$ for some $c > 0$.\\
Projection of $A$ along $B$:  $tB$, where $t=\frac{A \cdot B}{B \cdot B}$.\\
Angle between two vectors:  $\theta = \arccos \frac{A \cdot B}{\lVert A\rVert \lVert B\rVert}$, $0 \leq\theta\leq\pi$.\\
Dot product for $V_n(C)$:  $A \cdot B = \sum_{k=1}^n a_k \bar{b}_k$.\\
Two vectors $A$ and $B$ in $V_n$ are linearly dependent iff they lie on the same line through the origin.\\
Eqn for a line: $L(P; A) = \{P + tA\ | t {\text real}\}$.\\
Cartesian eqn for line: $y-q=\frac{b}{a}(x-p)$, or $(X-P)\cdot N = 0$, where $N = (b, -a)$, and is normal to the line.
Or, $X \cdot N = P\cdot N$.\\
Eqn of a plane in $V_n$, with $A$ and $B$ independent: $M=\{P+sA+tB\}$.\\
Three vectors $A, B, C$ are linearly dependent iff they lie on the same plane through the origin.\\
Alternate description of plane in $V_3$: $(X-P)\cdot N = 0$, where $N=A\times B$.

\paragraph{Concepts}\ \\
Linear span:  All vectors which can be represented by a linear combination of the set of vectors.\\
Linear independence:  Spans the $O$ vector in one way only (the trivial representation).\\
Basis:  A set $S$ of vectors is called a {\it basis} of $V_n$ if $S$ spans every vector in $V_n$ uniquely.

\paragraph{Cross product}\ \\
$A \times B = (a_2b_3=a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1)$\\
$A\times B = -(B\times A)$\\
$A\times (B+C) = (A\times B)+(A\times C)$\\
$c(A\times B) = (CA) \times B$\\
$a\cdot(A\times B)= 0$\\
$B\cdot(A\times B)=0$\\
$\lVert A\times B\rVert^2=\lVert A\rVert^2\lVert B\rVert^2 - (A\cdot B)^2$\\
$A\times B = O$ iff $A$ and $B$ are linearly dependent\\
Cross product is {\it not} associative\\
$\lVert A\times B\rVert = \lVert A\rVert \lVert B\rVert \sin \theta$\\
The length of $A \times B$ is equal to the area of the parallelogram determined by $A$ and $B$.\\
Alternate defintion of cross product:
\begin{equation*}
A\times B = \begin{vmatrix}\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}
\end{equation*}

\paragraph{Scalar triple product}\ \\
$A\cdot B\times C = A\cdot (B\times C)$.\\
$A, B, C$ are linearly dependent iff $A\cdot B\times C = 0$.\\
Scalar triple product is equal to the volume of the parallellipiped defined by $A, B, C$.\\
Alternate notation for scalar triple product: $A\cdot B\times C = [ABC]$.

\paragraph{Conic sections}\ \\
Definition 1: Slicing right circular cones with a plane.\\
Definition 2: Ellipse: $d_1+d_2$ constant, Hyperbola: $\lvert d_1-d_2\rvert$ constant,
Parabola $d_1 = d_2$ (focus, directrix).\\
Definition 3: Eccentricity ($e$): constant ratio of distance from a fixed point and fixed line.
Ellipse ($0<e<1$), Parabola ($e=1$), Hyperbola ($e>1$)\\
Vector eqn for conic section: $\lVert X-F\rVert = e\, d(X,L)$, where $N$ is a normal to $L$ and
$d(X,L)=\frac{\lvert(X-P)\cdot N\rvert}{\lVert N\rVert}$.\\
If $N$ is a unit normal, eqn for conic section becomes $\lVert X-F\rVert = e\lvert (X-F)\cdot N - d\rvert$.\\
Cartesian eqn for ellipse and hyperbola: $\frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)}=1$, where
$a=\frac{ed}{(1-e^2)}$, where $d$ is distance between $F$ and $L$.

\bigskip\bigskip
\section {Vector calculus}\smallskip

\paragraph{Algebra + differentiation}\ \\
$(F+G)' = F' + G'$\\
$(uF)' = u'F = uF'$\\
$(F\cdot G)' = F'\cdot G + F\cdot G'$\\
$(F\times G)' = F'\times G + F\times G'$\\
$(F \circ u)' = F'[u(t)]u'(t)$

\paragraph{Properties}\ \\
If $F$ has constant length on $I$, then $F\cdot F' = 0$ on $I$.\\
$C\cdot \int_a^bF(t)\,dt = \int_a^b C\cdot F(t)\,dt$\\
If $F$ and $\lVert F\rVert$ are integrable, we have $\lVert\int_a^bF(t)\,dt\rVert\leq\int_a^b\lVert F(t)\rVert\,dt$.\\
Unit tangent: $T(t)=\frac{X'(t)}{\lVert X'(t)\rVert}$\\
Principal normal: $N(t) = \frac{T'(t)}{\lVert T'(t)\rVert}$\\
Relationship b/w acceleration, $T(t)$, $N(t)$: $\pmb{a}(t)=v'(t)T(t)+v(t)T'(t)
= v'(t)T(t)+v(t)\lVert T'(t)\rVert N(t)$\\
Acceleration vector is linear combination of $T(t)$ and $N(t)$\\
Osculating plane: plane determined by $T(t)$ and $N(t)$\\
Rectifiable curve: $|\pi(P)|\leq M$ for a positive $M$ and for all partitions $P$.\\
Arc length: $\Lambda(a,b) = \sup |\pi(P)| = \int_a^bv(t)\,dt = \int_a^b\lVert X'(t)\rVert\,dt$.\\
Arc length function: $s(t)=\Lambda(a,t)$, for $t>a, s(a) = 0$.\\
Curvature vector: rate of change of the unit tangent with respect to arc length.\\
$\frac{dT}{ds}=\frac{dt}{ds}\frac{dT}{dt}=\frac{1}{s'(t)}T'(t)=\frac{1}{v(t)}T'(t)
=\frac{\lVert T'(t)\rVert}{v(t)}N(t)$.\\
Curvature of a curve at $t$: $\kappa(t)=$ length of curvature vector $= \frac{\lVert T'(t)\rVert}{v(t)}$.\\
Acceleration in terms of radial and transverse components:
$\pmb{a}=\left(\frac{d^2r}{dt^2}-r\left(\frac{d\theta}{dt}\right)^2\right)\pmb{u_r}
+ \left(r\frac{d^2\theta}{dt^2}+2\frac{dr}{dt}\frac{d\theta}{dt}\right)\pmb{u_\theta}$

\bigskip\bigskip
\section{Linear spaces}\smallskip

\paragraph{Concepts}\ \\
Linear space: a set which has operations ``multiplication'' and ``addition'' defined, which obey certain
axioms, such as closure under addtion and closure under multiplication by scalars.\\
Subspace: subset of linear space which is itself a linear space, i.e. follows closure axioms.\\
Dependence: A set $S$ is dependent if it spans the zero element in more than one way.\\
Finite basis for $V$: independent and spans $V$.\\
Dimension: the number of elements in a basis.\\
Euclidean space: linear space with an inner product defined.\\
Orthogonal set: every pair of distinct elements has $\langle x,y \rangle = 0$.\\
Orthonormal set: orthogonal set with each element having norm $1$.\\
Every finite dimensional Euclidean space has an orthogonal / orthonormal basis.\\
Projection of $x$ along $y$: $\frac{\langle x, y \rangle}{\langle y, y \rangle} y$, if $y \neq O$.\\
$S^\bot$: the set of all elements orthogonal to $S$.\\
$S^\bot$ is always a subspace. If $S$ is also a subspace, then $S^\bot$ is called the orthogonal complement of $S$.\\
Projection of $x$ on the subspace $S$: $s = \sum_{i=1}^n \langle x, e_i \rangle e_i$, where $\{e_1, \dotsc, e_n\}$
is an orthonormal basis for $S$.\\
The projection of $x$ on $S$ is nearer to $X$ than any other element of $S$.

\paragraph{Important applications}\ \\
Trigonometric polynomials:\\
$\phantom{x}$ Elements of a subspace $S$ of a linear space $V = C(0, 2\pi)$,\\
$\phantom{x}$ with inner product $\int_0^{2\pi} f(x)\, g(x)\, dx$,\\
$\phantom{x}$ spanned by the orthonormal basis $\varphi_0(x) = \frac{1}{\sqrt{2\pi}}$,
$\varphi_{2k-1}(x) = \frac{\cos kx}{\sqrt{\pi}}$, $\varphi_{2k}(x) = \frac{\sin kx}{\sqrt{\pi}}$.\\
Projection of $f \in C(0, 2\pi)$ on $S$ (trig polys):\\
$\phantom{x}$ $f_n = \sum_{k=0}^{2n} \langle f, \varphi_k \rangle \varphi_k$,\\
$\phantom{x}$ where $\langle f, \varphi_k \rangle = \int_0^{2\pi} f(x)\, \varphi_k(x)\, dx$.\\
Fourier coefficients of $f$:\\
$\phantom{x}$ The number $\langle f, \varphi_k \rangle$ so defined.\\
Alternate notation for projection of $f$:\\
$\phantom{x}$ $f_n(x) = \frac{1}{2} a_0 + \sum_{k=1}^n (a_k \cos kx + b_k \sin kx)$,\\
$\phantom{x}$ where $a_k = \frac{1}{\pi} \int_0^{2\pi} f(x)\, \cos kx\, dx$\\
$\phantom{x}$ and $b_k = \frac{1}{\pi} \int_0^{2\pi} f(x)\, \sin kx\, dx$.\\
Legendre polynomials for approximating functions:\\
$\phantom{x}$ If $f \in C(-1, 1)$, with inner product $\int_{-1}^1 f(x)\, g(x)\, dx$,\\
$\phantom{x}$ then the polynomial of degree $n$ which ``best approximates'' $f$\\
$\phantom{x}$ is the projection of $f$ on the subspace of polynomials of degree $n$,\\
$\phantom{x}$ which can be obtained by the projection formula using the normalized Legendre polynomials.

\paragraph{Inner product}\ \\
$(1)\ \langle x,y\rangle =\overline{\langle y,x\rangle }$\\
$(2)\ \langle x, y+z\rangle  = \langle x,y\rangle  + \langle x, z\rangle $\\
$(3)\ c\langle x,y\rangle =\langle cx,y\rangle $\\
$(4)\ \langle x,x\rangle  > 0$ if $x \neq O$

\paragraph{Norms}\ \\
$\lVert x \rVert = \langle x,x\rangle^{1/2}$\\
$\lVert cx \rVert = |c|\; \lVert x \rVert$\\
Triangle inequality: $\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert$

\paragraph{Cauchy-Schwarz Inequality}\ \\
In a Euclidean space: $|\langle x,y\rangle|^2 \leq \langle x,x\rangle \langle y,y \rangle$.\\
Or, equivalently: $|\langle x, y\rangle| \leq \lVert x\rVert\;\lVert y\rVert$.\\
The equality holds iff $x$ and $y$ are dependent.

\paragraph{Examples of linear spaces}\ \\
Real numbers, complex numbers, $V_n$, $V_n(C)$, all functions defined on an interval, all polynomials,
all functions continuous or differentiable or integrable on an interval.

\paragraph{General properties}\ \\
If an element $x$ is described as a linear combination of elements of some orthogonal basis $x=\sum_{i=1}^nc_ie_i$,
then its components relative to that basis are $c_j=\frac{\langle x, e_j\rangle}{\langle e_j, e_j \rangle}$.\\
If the basis is an orthonormal basis, then $c_j = \langle x, e_j \rangle$.\\
Parseval's formula: If $V$ has a finite orthonormal basis, then
$\langle x, y \rangle = \sum_{i=1}^n \langle x, e_i \rangle \overline{\langle y, e_i \rangle}$.\\
Parseval's for $y = x$: $\lVert x \rVert^2 = \sum_{i=1}^n |\langle x, e_i \rangle|^2$.\\
Gram-Schmidt process: A process for taking a finite or infinite sequence of elements, and
constructing a sequence of elements which is orthogonal and which has the same span as the sequence.

\paragraph{Legendre polynomials}\ \\
Take the linear space of all polynomials, with inner product $\langle x, y \rangle = \int_{-1}^1 x(t)\, y(t)\, dt$,
and take the infinite sequence $x_0, x_1, x_2, \dotsc$, where $x_n(t) = t^n$.  If the orthogonalization theorem /
Gram-Schmidt process is applied to this sequence, a new sequence of orthogonal polynomials is obtained. These
are the Legendre polynomials.  If each polynomial is divided by its norm, we have the normalized Legendre polynomials.

\bigskip\bigskip
\section{Linear transformations and matrices}\smallskip

\paragraph{Concepts}\ \\
Linear transformation: A function from one linear space to another, with the properties of additivity and homogeneity.\\
Basically, a linear transformation is equivalent to a series of linear equations with no constant terms.\\
If there are constant terms, you have instead an {\it affine} transformation.\\
Range of a linear transformation is always a subspace of the target space.\\
Linear transforms always map zero element to zero element.\\
Null space of $T$: All elements of source space that map to zero element in target space (also called kernel of T).\\
The null space of $T$ is denoted by $N(T)$.\\
Null space is always a subspace of source space.\\
Nullity: the dimension of the null space.\\
Rank: the dimension of the range.\\
Nullity $+$ Rank $=$ dimension of the source space.\\
${\mathscr L}(V,W)$: set of all linear transformations from $V$ to $W$, where $V$ and $W$ have the same scalars.
${\mathscr L}(V, W)$ is itself a linear space, with addition and multiplication by scalars as defined below.\\
The composition of two linear transformations is also linear (for same scalar spaces).\\
There is one and only one linear transformation that maps each of the $n$ basis elements of $V$ onto any $n$ arbitrary
elements of $W$.\\
If $T(e_k) = u(k)$, and $x = \sum_{k=1}^n x_k e_k$, then $T(x) = \sum_{k=1}^n x_k u_k$.\\
If $T(e_k) = \sum_{i=1}^m t_{ik} w_i$, then the matrix representation of $T$ is $m \times n$ and each
column represents the components of the target space basis elements $T$ maps a given source space basis
element to.\\
$T(x) = \sum_{i=1}^m y_i w_i$, where $y_i = \sum_{k=1}^n t_{ik} x_k$, for $i = 1, 2, \dotsc, m$.\\
Diagonal matrix: A matrix with all non-diagonal elements $= 0$.\\
For any finite linear transformation, you can create a diagonal matrix representation with $1$'s along the diagonal
equal to the rank of the transformation, and $0$'s on the diagonal after that.  This is done by choosing the
bases appropriately.\\
Linear space of matrices: $M_{m,n}$, with appropriately defined addtion and scalar multiplication.\\
Linear transformations and matrices are isomorphic.\\
Matrix multiplication: If $A$ is $m \times p$, and $B$ is $p \times n$, then $AB$ is such that $c_{ij} = \sum_{k=1}^p a_{ik} b_{kj}$.\\
Matrix multiplication corresponds to linear transformation composition.\\
Computing a transformation corresponds to matrix multiplication between the transformation matrix and a column matrix of $x$.\\
If we have a system of linear equations such that $T(x) = c$, then if we can find $k = \nullity(T)$ independent solutions of the
homogeneous system $T(x) = O$, which we denote $v_1, \dotsc, v_k$, and if we can find one particular solution of the
nonhomogeneous system, which we denote by $b$, then the general solution of the system is
$x = b + t_1 v_1 + \dotsb + t_k v_k$, where $t_1, \dotsc, t_n$ are arbitrary scalars. In other words, the solution is any
particular solution, plus any element of the image of the null space of $T$.\\
A matrix is nonsingular iff $T$ is invertible.\\
You can obtain a matrix inverse by applying the Gauss-Jordan method to a matrix augmented with an identity matrix.\\
For the linear system $AX = C$ (square), if $A$ is nonsingular there is a unique solution of the system: $X = A^{-1}C$.\\
Transpose of a matrix $A^t$ has its $i, j$ entry as $a_{ji}$.\\
Invariance: a subspace $U$ of $S$ is called invariant under $T$ if $T$ maps every element of $U$ onto an element of $U$.

\paragraph{Gauss-Jordan elimination method}\ \\
You can perform the following operations and have an equivalent system: 1) Interchanging two rows, 2) multiplying all
terms of a row by a nonzero scalar, and 3) adding a multiple of one row to another row.\\
Process: 1) obtain a $1$ in the top left position, 2) make all other elements of column $1$ to be $0$,
3) Continue to get an upper triangular matrix, 4) Continue to make a diagonal matrix.

\paragraph{One-to-one linear transformations}\ \\
$T$ is one-to-one $\Longleftrightarrow$ $T$ is invertible and $T^{-1}$ is linear $\Longleftrightarrow$ $N(T)$ contains only $O$.\\
If $V$ is finite-dimensional, then: $T$ is one-to-one $\Longleftrightarrow$ Independent elements in $V$ map to
independent elements in $T(V)$ $\Longleftrightarrow$ $\dim T(V) = \dim V$

\paragraph{Algebra of linear transformations}\ \\
$(S+T)(x) = S(x) + T(x)$\\
$(cT)(x) = cT(x)$\\
Composition: $(ST)(x) = S[T(x)]$, with domains and ranges appropriately defined.\\
$R(ST) = (RS)T$\\
$T^0 = I$, and $T^n = TT^{n-1}$ for $n \geq 1$.\\
$R(S + T) = RS + RT$ \hskip 5 pt and \hskip 5 pt $R(cS) = c(RS)$

\paragraph{Algebra of matrices}\ \\
$A(BC) = (AB)C$\\
$(A+B)C = AC + BC$\\
$C(A+B) = CA + CB$\\
For a square matrix, $A^0 = I$, and $A^n = AA^{n-1}$, for $n \geq 1$.\\
$(AB)_i = A_i B$, where $A_i$ denotes the $i$th row of $A$.

\paragraph{Examples of linear transformations}\ \\
Identity transformation, zero transformation, multiplication by a scalar, linear eqns with no constant terms,
inner product with a fixed element, projection on a subspace, differentiation, integration.

\bigskip\bigskip
\section{Determinants}\smallskip

\paragraph{Concepts}\ \\
In 3-space, the scalar triple product is the determinant of the matrix whose rows are the vectors:
$A_1 \times A_2 \cdot A_3$.\\
In 3-space, this is the area of the parallelipiped determined by the three vectors.\\
Only defined for square matrices.\\
The determinant is unique (in satisfying the axioms).\\
$A_{kj}'$ denotes the matrix obtained from $A$ by replacing row $A_k$ with unit coordinate vector $I_j$.\\
Cofactor of $a_{kj} = \det A_{kj}'$.\\
Minor: The $kj$ minor of $A$ is the matrix obtained from $A$ by deleting the $k$th row and the $j$th column from $A$.\\
The minor is denoted by $A_{kj}$.\\
Cofactor matrix: $\cof A = (\cof a_{ij})$

\paragraph{Axioms}\ \\
(1) Homogeneity in each row: $d(\dotsc, tA_k, \dotsc) = t\, d(\dotsc, A_k, \dotsc)$.\\
(2) Additivity in each row: $d(\dotsc, A_k + C, \dotsc) = d(\dotsc, A_k, \dotsc) + d(\dotsc, C, \dotsc)$.\\
(3) Determinant is zero if any two rows are equal.\\
(4) Determinant of identity matrix is $1$.

\paragraph{Properties}\ \\
(1) If any row is $O$, the determinant is $0$.\\
(2) If any two rows are interchanged, the determinant changes sign.\\
(3) If the rows are dependent, the determinant is $0$.\\
(4) A set of $n$ vectors in $n$-space is independent iff their determinant is $0$.

\paragraph{Calculating determinants}\ \\
Determinant of $1 \times 1$ matrix is just the element.\\
Determinant of $2 \times 2$ matrix is $\bigl[ \begin{smallmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{smallmatrix} \bigr]$
$ = a_{11}a{22} - a_{12}a_{21}$.\\
Determinant of a diagonal matrix is the product of its diagonal elements.\\
Determinant of an upper triangular matrix is the product of its diagonal elements.

\paragraph{Gauss-Jordan method for finding determinant}\ \\
Use Gauss-Jordan to obtain an upper triangular matrix, then take the product of the diagonal elements.\\
(1) Interchange two rows (changes sign).\\
(2) Multiply a row by a nonzero scalar (is multiplied by the scalar).\\
(3) Add a scalar multiple of one row to another row (no change).

\paragraph{Algebra}\ \\
$\det (AB) = (\det A)(\det B)$\\
For $A$ nonsingular, $\det A^{-1} = \frac{1}{\det A}$\\
$\det \bigl[ \begin{smallmatrix} A & O \\ O & B \end{smallmatrix} \bigr] = (\det A)(\det B)$, where $A$ and $B$
form part of a block diagonal matrix.\\
Expansion by cofactors: $\det A = \sum_{j=1}^n a_{kj} \cof a_{kj}$ for any row $k$.\\
Relationship between cofactor and determinant of the minor: $\cof a_{kj} = (-1)^{k+j} \det A_{kj}$.\\
$\det A = \det A^t$\\
$A(\cof A)^t = (\det A) I$, for any square matrix with $n \geq 2$.\\
$A^{-1} = \frac{1}{\det A} (cof A)^t$.  This is a good way to compute the inverse of A.

\paragraph{Cramer's rule}\ \\
For $n$ linear equations in $n$ unknowns, we have $AX = B$.\\
Solution: $x_j = \frac{1}{\det A} \sum_{k=1}^n b_k \cof a_{kj}$.

\bigskip\bigskip
\section{Eigenvalues and eigenvectors}\smallskip

\paragraph{Concepts}\ \\
Eigenvalue / eigenvector: When a linear tranformation has $T(x) = \lambda x$ for some scalar $\lambda$,
$\lambda$ is an eigenvalue and $x$ is an eigenvector, as long as $x \neq 0$.\\
For $T: V \to V$, $T \in {\mathscr L}(V, V)$, $\dim V = n$, if there are $n$ independent elements $u_k$
such that $T(u_k) = \lambda_k u_k$, then $A = \diag(\lambda_1, \dotsc, \lambda_n)$ is a respresentation
of $T$ relative to the $u_k$. The converse is also true.\\
Eigenspace: the linear subspace consisting of all elements such that $T(x) = \lambda x$ for a given $\lambda$
(including zero).  This is a linear subspace.\\
Zero {\it can} be an eigenvalue.\\
In a real linear space, $T$ preserves the ``direction'' of an eigenvector.  This is not generally true for
complex linear spaces.\\
Eigenvectors corresponding to distinct eigenvalues are independent.\\
$\lambda_1 \dotsm \lambda_n = \det A$.
Trace of $A$: Sum of the roots of $f(\lambda)$, which is equal to the sum of the diagonal elements of $A$.

\paragraph{Characteristic polynomials}\ \\
Eigenvalues must satisfy $T(x) = \lambda x \quad \Longrightarrow \quad (\lambda I - T)(x) = O$ for $x \neq 0$.\\
If this is satisfied, it means that some $x$ other than $0$ is mapped to $0$, which means that $(\lambda I - T)$ is not one-to-one.\\
This means that $(\lambda I - T)$ has no inverse, and thus that the determinant of its matrix $A$ is $0$.\\
So $\det(\lambda I - A) = 0$.\\
We thus need to find the roots of the polynomial $f(\lambda) = \det(\lambda I - A)$.\\
This is a polynomial in $\lambda$ of degree $n$, the term or highest degree is $\lambda^n$, and the constant term is $(-1)^n \det A$.\\
The eigenvalues are the roots of the characteristic polynomial which lie in $F$ (real or complex scalars).\\
Once eigenvalues are found, you can find corresponding eigenvectors by solving $AX = \lambda X$.

\paragraph{Similar matrices}\ \\
If $A$ and $B$ are $n \times n$ matrices representing the same linear transformation, there is some matrix $C$
such that $B = C^{-1}AC$.\\
If $A$ is relative to basis $E$, and $B$ is relative to basis $U$, the bases are related by $U = EC$.\\
Similar matrices: square matrices related by $B = C^{-1}AC$.\\
Similar matrices have the same determinant, same characteristic polynomial, and the same eigenvalues.

\bigskip\bigskip
\section{Eigenvalues and eigenvectors in Euclidean spaces}\smallskip

\paragraph{General concepts}\ \\
$\lambda = \frac{\langle T(x), x \rangle}{\langle x, x \rangle}$ \hskip 5pt and \hskip 5pt
$\bar{\lambda} = \frac{\langle x, T(x) \rangle}{\langle x, x \rangle}$\\
An eigenvalue is real or 0 iff $\langle T(x), x \rangle = \langle x, T(x) \rangle$.\\
An eigenvalue is pure imaginary or 0 iff $\langle T(x), x \rangle = - \langle x, T(x) \rangle$.\\
Hermitian/symmetric transformation: $\langle T(x), y \rangle = \langle y, T(x) \rangle$\\
skew-Hermitian/skew-symmetric transformation: $\langle T(x), y \rangle = - \langle y, T(x) \rangle$\\
We use ``Hermitian'' terminology if the Euclidean space is complex, and ``symmetric'' if real.\\
Hermitian transformations have real eigenvalues.\\
Skew-Hermitian transformations have pure imaginary eigenvalues.\\
Diagonal elements for: Hermitian(real), skew-Hermitian(pure imaginary), skew-symmetric(zero).\\
Distinct eigenvalues of Hermitian or skew-Hermitian transformations have orthogonal eigenvectors.\\
If $T$ operates on a finite-dimensional complex space and is Hermitian or skew-Hermitian, then
there exist $n$ orthonormal eigenvectors (which form a basis for $V$).\\
$T$ can be determined to be (skew-)Hermitian based only on its action on any basis.\\
Adjoint of a matrix: The transpose of the conjugate, $\bar{A}^t$, denoted by $A^*$.
Hermitian matrix: $a_{ij} = \bar{a}_{ji}$, or $A = A^*$.\\
Skew-Hermitian matrix: $a_{ij} = - \bar{a}_{ji}$, or $A = - A^*$.\\
For (skew-)Hermitian matrices, the diagonalizing matrix $C$ (where $\Lambda = C^{-1}AC$) has $C^{-1} = C*$, or $C\, C^* = I$.\\
For real matrices, you always have $A^t = A^*$.\\
Unitary matrix: $A\, A^* = I$.\\
Orthogonal matrix: $A\, A^t = I$.\\
Every real unitary matrix is also orthogonal, since $A^* = A^t$.\\
Symmetric matrix: $A = A^t$.\\
Skew-symmetric matrix: $A = - A^t$.\\
Every square matrix $A$ can be expressed as a sum of a Hermitian matrix ($\frac{1}{2}(A + A^*)$)
and a skew-Hermitian matrix ($\frac{1}{2}(A - A^*)$).\\
Every square matrix $A$ can be expressed as a sum of a symmetric matrix ($\frac{1}{2}(A + A^t)$)
and a skew-symmetric matrix ($\frac{1}{2}(A - A^t)$).\\
Orthogonal matrices have determinant $= \pm 1$.

\paragraph{How to find a diagonalizing matrix for (skew-)Hermitian matrices}\ \\
(1) Find an orthonormal set of eigenvectors $u_1, \dotsc, u_n$\\
(2) Use the components of $u_j$ (relative to the basis of unit coordinate vectors) as entries of the $j$th column.

\paragraph{Sturm-Liouville operators}\ \\
Let $V$ be subspace of $C(a, b)$ with continuous second derivative on $[a, b]$.\\
Let $p$ and $q$ be fixed functions in $C(a, b)$, with $p$ having a continuous derivative on $[a, b]$.\\
Let $f \in V$.\\
Let $p$ and $q$ satisfy the boundary conditions $p(a)f(a) = 0$ and $p(b)f(b) = 0$.\\
Let $T: V \to C(a, b)$ be the following operator: $T(f) = (pf')' + qf$.\\
$T$ is then called a {\it Sturm-Liouville} operator.\\
Sturm-Liouville operators are symmetric.

\paragraph{Quadratic forms}\ \\
For a symmetric operator (on a real space), quadratic form associated with $T$ is: $Q(x) = \langle T(x), x \rangle$.\\
If $A$ is a representation of $T$ relative to an orthonomal basis, then $Q(x) = \sum_{i=1}^n \sum_{j=1}^n a_{ij}x_ix_j$.\\
Where $x_k$ is the component of $x$ relative to $e_k$.\\
You can actually associate a quadratic form with {\it any} square matrix (not just symmetric).\\
Diagonal form: If $A$ is a diagonal matrix, $Q(x) = \sum_{i=1}^n a_{ii} x_i^2$.\\
The quadratic form can also be written as $Q(x) = X\, A\, X^t$, where $X$ is a row matrix.\\
$X\, A\, X^t = X\, [\frac{1}{2} (A + A^t)]\, X^t$.

\paragraph{Converting a real quadratic form to diagonal form}\ \\
If $A$ is a real symmetric matrix, and $C$ is a diagonalizing matrix (so $\Lambda = C^t A C$),
then $X A X^t = \sum_{i=1}^n \lambda_i y_i^2$, where $Y = XC$, and $\lambda_1, \dotsc, \lambda_n$
are the eigenvalues of $A$.  This is because $X A X^t = Y \Lambda Y^t$.

\paragraph{Relation between eigenvalues of a symmetric transformation and its quadratic form}\ \\
For eigenvectors $x$ with norm $1$, $Q(x) = \lambda$.\\
So, any eigenvalues of $T$ are found on the values $Q$ takes on the unit sphere.\\
If $Q$ has a maximum or minumum on the unit sphere at $x$, then $x$ is an eigenvector, and $Q(x)$ is its eigenvalue.\\
If you find the minimum value of $Q$ on the unit sphere, this is the smallest eigenvalue.\\
You can then look at the subset of the unit sphere in $S^\bot$ - all elements not spanned by the found eigenvector.\\
The minimum of $Q$ on this unit sphere subset will be the next smallest eigenvalue, etc.

\paragraph{Unitary transformations}\ \\
Unitary transformation: $\langle T(x), T(y) \rangle = \langle x, y \rangle$.\\
When $E$ is a real space, a unitary transformation is also called an {\it orthogonal} transformation.\\
Unitary transformations preserve inner products, orthogonality, norms, and distances.\\
Unitary transformations are invertible, and their inverses are unitary on $T(V)$.

\paragraph{Unitary transformations and eigenvectors}\ \\
Any eigenvalues of $T$ have $| \lambda | = 1$.\\
Distinct eigenvalues have corresponding eigenvectors which are orthogonal.\\
For $T: V \to V$ and finite dimensional $V$, an orthonormal $T$-eigenvector basis of $V$ exists,\\
and the matrix of $T$ relative to this basis is the diagonal matrix $\Lambda$.

\bigskip\bigskip
\section{Probability}\smallskip

\paragraph{Concepts}\ \\
Finitely additive set function: $f(A \cup B) = f(A) + f(B)$, when $A$ and $B$ are disjoint.\\
Countably additive set function: Also holds for unions of countably infinite subsets.\\
Boolean algebra of sets: closed under union and complement (also closed under intersection and difference).\\
Boolean $\sigma$-algebra: also closed under countably infinite union.\\
Finitely additive measure: a finitely additive set function that is nonnegative.\\
Probability measure: set function which is finitely additive, nonnegative, and $P(S) = 1$.\\
Probability space: $S$ (sample space), ${\mathscr B}$ (boolean algebra of subsets of $S$), $P$ (probability measure).\\
Experiment: Various definitions.\\
Descriptive vs. Inferential statistics.\\
Number of distinct subsets of size $k$ from a set of size $n$: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.\\
Sampling with replacement: $n^k$.\\
Sampling without replacement: $n(n-1)(n-2) \dotsb (n-k+1) = \frac{n!}{(n-k)!}$.\\
Total number of subsets: $\sum_{k=0}^n \binom{n}{k} = 2^n$.\\
Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.\\
Independence: $P(A \cap B) = P(A) P(B)$.\\
Random variable: a real vector-valued function on $S$.\\
Distribution function: $F(t) = P(X \leq t)$.\\
Probability mass function: $p(t) = P(X = t)$.\\
If $S$ is uncountable, there can be at most countably many points at which $P > 0$.\\
Probability density function: $f(t)$, where $P(X \leq t) = \int_{-\infty}^t f(u)\, du$.\\
Mean squared error: the mean squared difference between an estimator and the true value.\\
Standard error: $\sqrt{\text{MSE}}$.\\
Sampling distribution: distribution of an estimator of a parameter.

\paragraph{Estimation}\ \\
Maximum likelihood fn: A joint probability distribution, viewed as fixed observed values and variable parameter(s).\\
Maximum likelihood estimator: A parameter estimator which maximizes the maximum likelihood function.\\
Invariance: $g(\hat{\theta})$ is the MLE of $g(\theta)$.\\
Normal distribution: $\hat{\mu} = \bar{X}_n$, $\hat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n(X_i-\bar{X}_n)^2$.

\paragraph{Sampling distributions}\ \\
Normal distribution, $\bar{X}_n$: mean $= \mu$, var $= \sigma^2/n$.\\
Normal distribution, $(1/n)\sum_{i=1}^n(X_i-\bar{X}_n)^2$: $\chi^2$ distribution with $n-1$ df.\\
For normal distributions, sample mean and sample variance are independent RV's.

\paragraph{Chi-Square distribution}\ \\
Gamma distribution with $\alpha = n/2$ and $\beta = 1/2$.
\begin{equation*}f(x) = \frac{1}{2^{n/2}\Gamma(n/2)}x^{(n/2)-1}e^{-x/2}\end{equation*}
$n$ is number of ``degrees of freedom''.\\
$E(X) = n$, $\text{Var}(X) = 2n$.\\
The sum of $k$ independent $\chi^2$ distributed RV's has a $\chi^2$ distribution with $df = n_1 + \dotsb + n_k$.\\
If $X$ has a normal distribution, then $Y = X^2$ has a $\chi^2$ distribution with one degree of freedom.\\
If $k$ RV's are independent and normally distributed, then $X_1^2 + \dotsb + X_k^2$ are $\chi^2$ with $k$ df.

\paragraph{t distribution}\ \\
If $Y$ is standard normal and $Z$ is $\chi^2$ with $n$ df, then
\begin{equation*}X=\frac{Y}{\left(\frac{Z}{n}\right)^{1/2}}\end{equation*}
has a $t$ distribution with $n$ df.
\begin{equation*}
f(x)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{(n\pi)^{1/2}\Gamma\left(\frac{n}{2}\right)}
% \left(1+\frac{x^2}{n}\right)^{-(n+1)/2}
\end{equation*}
When $n = 1$, the $t$ distribution is a Cauchy distribution.\\
When $n \to \infty$, the $t$ distribution approaches the normal distribution.

\paragraph{Bernoulli trials}\ \\
Definition: Sequence of repeated, independent trials, $p$ = probability of success, $q$ = failure.\\
$P$($k$ successes in $n$ trials): $\binom{n}{k}p^kq^{n-k}$.\\
$P$(at least $r$ successes in $n$ trials): $\sum_{k=r}^n \binom{n}{k} p^k q^{n-k}$.

\paragraph{Cauchy distribution}\ \\
$F(t) = \frac{1}{2} + \frac{1}{\pi} \arctan t$.\\
$f(t) = \frac{1}{\pi (1 + t^2)}$.

\paragraph{Exponential distributions}
\begin{equation*}
F(t) =
\begin{cases}
1 - e^{-\lambda t} & \text{for } t \geq 0 \\
0 & \text{for } t < 0
\end{cases}
\end{equation*}
\begin{equation*}f(x) = \begin{cases}
\lambda e^{-\lambda t} & \text{for } t \geq 0\\
0 & \text{for } t < 0
\end{cases}\end{equation*}

\paragraph{Normal distributions}
\begin{equation*}F(t) = \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^t e^{-[(u-m)/\sigma]^2/2}\, du\end{equation*}
\begin{equation*}f(t) = \frac{1}{\sigma \sqrt{2\pi}} e^{-[(t-m)/\sigma]^2/2}\end{equation*}

\paragraph{Approximating binomial distribution with normal}
\begin{equation*}
\sum_{k=a}^b \binom{n}{k} p^k q^{n-k} \thicksim \Phi \left(\frac{b-np+\frac{1}{2}}{\sqrt{npq}}\right)
- \Phi \left( \frac{a-np-\frac{1}{2}}{\sqrt{npq}} \right)
\end{equation*}

\paragraph{Expectation}
\begin{equation*}
E(X) = \int_{-\infty}^{+\infty} t f_X(t)\, dt
\end{equation*}
\begin{equation*}
E(X) = \sum_{k=1}^\infty x_k p_k
\end{equation*}

\paragraph{Variance}
\begin{equation*}
\text{Var}(X) = \int_{-\infty}^{+\infty} [t = E(X)]^2 f_X(t)\, dt
\end{equation*}
\begin{equation*}
\text{Var}(X) = \sum_{k=1}^\infty [x_k - E(X)]^2 p_k
\end{equation*}

\paragraph{Expectation of $Y = \varphi(X)$}
\begin{equation*}
E(Y) = \int_{-\infty}^{+\infty} \varphi(t) f_X(t)\, dt
\end{equation*}

\paragraph{Confidence intervals}\ \\
To determine confidence interval:\\
$\phantom{x}$ (1) Find a way that say $0.95$ of the area of a relevant distribution falls within boundaries.\\
$\phantom{x}$ (2) Make an inequality.\\
$\phantom{x}$ (3) Manipulate the inequality to isolate the parameter of interest.\\
For $X_1, \dotsc, X_n$ from a normal distribution,
\begin{equation*}
\text{Pr}(\bar{X}_n - \frac{c\sigma'}{\sqrt{n}} < \mu < \bar{X}_n + \frac{c\sigma'}{\sqrt{n}}) = 0.95
\end{equation*}
where $c$ is a constant such that
\begin{equation*}\int_{-c}^c g_{n-1}(x)\, dx = 0.95\end{equation*}
and $g_{n-1}(x)$ is a $t$ distribution with $n-1$ degrees of freedom.

\paragraph{Unbiased estimators}\ \\
An estimator is {\it unbiased} if its expected value equals the parameter for all parameter values.\\
Unbiased estimator of $\sigma^2$: $\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X}_n)^2$.

\paragraph{Hypothesis testing}\ \\
Type I Error: Rejecting $H_0$ when it is true ($\alpha$).\\
Type II Error: Accepting $H_0$ when it is false ($\beta$).\\
Can test two alternate values of a parameter, or one-sided alternatives.\\
Unbiased vs. biased tests (they are almost always biased).

\paragraph{Neyman-Pearson Lemma}\ \\
A test procedure of the form: ``Accept $H_0$ if $f_a(x)>kf_1(x)$''\\
...where $H_0$ and $H_a$ are two point hypotheses\\
...and where $f_0(x)$ is the joint pdf of the sample observations when $H_0$ is true\\
...and with specified $\alpha$\\
...will minimize $\beta$.\\
Likelihood ratio: $f_a(x)/f_0(x)$.

\paragraph{Uniformly Most Powerful tests}\ \\
For a test procedure $\delta *$\\
...and a fixed level of significance $\alpha_0$\\
...and for any other procedure $\delta$\\
...$\delta$ is UMP if $\pi(\theta | \delta) \leq \pi(\theta | \delta *)$\\
...for every value of $\theta \in \Omega_a$.

\paragraph{Monotone likelihood ratio}\ \\
$f_n(\boldsymbol{x}\,|\,\theta)$ has a MLR in the statistic $T = r(\boldsymbol{x})$\\
...if for any two values $\theta_1 \in \Omega$ and $\theta_2 \in \Omega$\\
...with $\theta_1 < \theta_2$\\
...the ratio $f_n(\boldsymbol{x}\,|\,\theta_2)/f_n(\boldsymbol{x}\,|\,\theta_1)$\\
...depends on $\boldsymbol{x}$ only through $r(\boldsymbol{x})$\\
...and this ratio is an increasing function of $r(\boldsymbol{x})$.

\paragraph{t test, one-sided}\ \\
$H_0$: $\mu \leq \mu_0$\\
$H_a$: $\mu > \mu_0$\\
Normal distribution, $\mu$ and $\sigma^2$ unknown.
\begin{equation*}
\text{Likelihood ratio:} \qquad \frac{\sup_{(\mu,\sigma^2)\in\Omega_a}{f_n(\boldsymbol{x}\,|\,\mu, \sigma^2)}}
{\sup_{(\mu,\sigma^2)\in\Omega_0}f_n(\boldsymbol{x}\,|\,\mu, \sigma^2)}
\end{equation*}
$H_0$ should be rejected if $r(\boldsymbol{x}) \geq k$.\\
There is a version of this ratio that differs by a constant factor (and translated) and which has a $t$ distribution:
\begin{equation*}
U = \frac{n^{1/2}(\overline{X}_n - \mu_0)}{\left[\frac{S_n^2}{n-1}\right]^{1/2}}
\end{equation*}
This has $n-1$ degrees of freedom.\\
Thus, $H_0$ should be rejected if $U \geq c$, where $c$ is chosen to have desired $\alpha$ value.\\
Comparing two means, equal sample size, equal variance:
\begin{equation*}
U = \frac{\overline{X}_1 - \overline{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{2}{n}}}
\text{ where } S_{X_1X_2} = \sqrt{\frac{S_{X1}^2 + S_{X2}^2}{2}}
\text{ and with } 2n - 2 \text{ degrees of freedom}
\end{equation*}
Comparing two means, unequal sample size, equal variance:
\begin{equation*}
U = \frac{\overline{X}_1 - \overline{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}
\text{ where } S_{X_1X_2} = \sqrt{\frac{(n_1-1)S_{X_1}^2 + (n_2-1)S_{X_2}^2}{n_1+n_2 - 2}}
\text{ and with } n_1 + n_2 - 2 \text{ degrees of freedom}
\end{equation*}
Comparing two means, dependent:
\begin{equation*}
U = \frac{\overline{X}_D - \mu_0}{s_D / \sqrt{N}}
\text{ with } N - 1 \text{ degrees of freedom}
\end{equation*}

\paragraph{Distributions of functions of random variables}\ \\
If you know the distribution of a random variable (or the joint distribution of multiple random variables),
then you can determine the distribution of functions of those random variables [described in Apostol].

\paragraph{Chebyshev's inequality}\ \\
For every positive number $c$:
\begin{equation*}P[|X-E(X)| > c] \leq \frac{\text{Var}(X)}{c^2}\end{equation*}

\paragraph{Weak law of large numbers}\ \\
For $n$ random variables with the same expectation ($m$) and variance, $\lim_{n\to\infty} P(|\overline{X}-m| > \epsilon) = 0$.

\paragraph{Central limit theorem}\ \\
Central limit property:\\
Take a sequence of random variables $X_1, X_2, \dotsc, X_n$\\
where each $X_k$ has a finite expectation $m_k$ and a finite variance $\sigma_k^2$.\\
Take $S_n = \sum_{k=1}^n (X_k - m_k)$ and $T_n = \frac{S_n}{\sqrt{\text{Var}(S_n)}}$.\\
This sequence is said to satisfy the central limit property if, for all $a \leq b$,
\begin{equation*}\lim_{n\to\infty}P(a\leq T_n\leq b)=\frac{1}{\sqrt{2\pi}}\int_a^b e^{-u^2/2}du\end{equation*}
If the variables are independent with a common distribution, this condition is satisfied.\\
More generally, another condition, the Lindeberg condition, is necessary and sufficient for independent variables to satisfy the CLP.\\
The CLP can also be satisfied for dependent variables, but the theory on this is still incomplete.

\bigskip\bigskip
\section{Line Integrals}\smallskip

\paragraph{General concepts}\ \\
Continuous path in $n$-space: $\bs{\alpha}: [a,b] \in R^1 \to R^n$, continuous on $[a,b]$.\\
Smooth path: $\bs{\alpha}'$ exists and is continuous on $(a,b)$.\\
If $\bs{\alpha}$ and $\bs{\beta}$ trace out the same path in the same direction, the line integrals are equal.\\
If $\bs{\alpha}$ and $\bs{\beta}$ trace out same path in opposite directions, the line integrals have opposite sign.\\
Independence of path: If a line integral depends only on the endpoints.\\
Potential function: If a scalar field $\varphi$ has a gradient $\bs{f}$, $\varphi$ is a potential function for $\bs{f}$.

\paragraph{Definitions}\ \\
Line integral of a scalar field with respect to arc length:\\
$\phantom{x}$ (a) $\varphi$: $R^n \to R^1$.\\
$\phantom{x}$ (b) $\bs{\alpha}: piecewise smooth path in $n$-space defined on $[a,b]$.\\
$\phantom{x}$ (c) Defined as: \int_C \varphi\, ds = \int_a^b \varphi[\bs{\alpha}(t)]s'(t)\, dt$.\\
$\phantom{x}$ (d) Where $s(t) = \int_a^t \lVert\bs{\alpha}'(u)\rVert\, du$.\\
$\phantom{x}$ (e) Where $s'(t) = \lVert\bs{\alpha}'(t)\rVert$.
Line integral of a scalar field with respect to a domain variable:\\
$\phantom{x}$ (a) $\int_C f(x, y)\, dx = \int_a^b f(x(t), y(t)) x'(t)\, dt$.\\
Line integral of a vector field:\\
$\phantom{x}$ (a) $\bs{f}: R^n \to R^n$.\\
$\phantom{x}$ (b) $\bs{\alpha}$: piecewise smooth path in $n$-space defined on $[a,b]$.\\
$\phantom{x}$ (c) Defined as: $\int f_1d\alpha_1 + \dotsb + f_nd\alpha_n$.\\
$\phantom{x}$ (d) Written: $\int \bs{f} \cdot d\bs{\alpha}$.\\
$\phantom{x}$ (e) Equivalent to: $\int_a^b \bs{f}[\bs{\alpha}(t)] \cdot \bs{\alpha}'(t)\, dt$.\\
$\phantom{x}$ (f) $\int_C \bs{f}\cdot d\bs{\alpha}=\int_C\bs{f}\cdot \bs{T}\,ds$,
$\bs{T}(t)=\frac{\bs{\alpha}'(t)}{\lVert\bs{\alpha}'(t)\rVert}$.

\paragraph{Properties}\ \\
(a) $\int (a\bs{f} + b\bs{g}) \cdot d\bs{\alpha} = a \int \bs{f} \cdot d\bs{\alpha}+b\int \bs{g}\cdot d\bs{\alpha}$.\\
(b) $\int_C \bs{f} \cdot d\bs{\alpha} = \int_{C_1}\bs{f}\cdot d\bs{alpha}+\int_{C_2}\bs{f}\cdot d\bs{\alpha}$.

\paragraph{First fundamental theorem for line integrals}\ \\
If the line integral of $\bs{f}$ is path independent,\\
And if $\varphi(\bs{x}) = \int_{\bs{a}}^{\bs{x}} \bs{f}\cdot d\bs{\alpha}$,\\
Then $\nabla\varphi(\bs{x}) = \bs{f}(\bs{x})$.

\paragraph{Second fundamental theorem for line integrals}\ \\
For a scalar field $\varphi$ with a continuous gradient $\nabla \varphi$ on an open connected set in $R^n$:\\
$\int_{\bs{a}}^{\bs{b}} \nabla\varphi\cdot d\bs{\alpha} = \varphi(\bs{a}) - \varphi(\bs{b})$.\\
This means the line integral of a gradient is path independent.

\paragraph{Conditions for a vector field to be a gradient (conservative)}\ \\
A vector field is {\it conservative} if it is a gradient of some scalar field.\\
The following are equivalent:\\
$\phantom{x}$ (a) $\bs{f}$ is the gradient of some potential function.\\
$\phantom{x}$ (b) The line integral of $\bs{f}$ is independent of the path.\\
$\phantom{x}$ (c) The line integral of $\bs{f}$ is zero around every piecewise smooth closed path.\\
Necessary for a vector field to be a gradient:\\
$\phantom{x}$ (a) $D_if_j(\bs{x}) = D_jf_i(\bs{x})$.\\
$\phantom{x}$ (b) If the domain of $\bs{f}$ is convex, then this is also sufficient.\\
$\phantom{x}$ (c) If the domain of $\bs{f}$ is open and simply connected, then this is sufficient.

\paragraph{Constructing potential functions}\ \\
If $\bs{f}$ is a gradient, a potential function can be constructed.\\
Take the line integral of $\bs{f}$ from any point $\bs{a}$ to an arbitrary point $\bs{x}$, using any path.

\bigskip\bigskip\
\section{Multiple Integrals}\smallskip

\paragraph{Repeated or iterated integration}\ \\
If the given integrals exist and $Q = [a,b] \times [c,d]$,
\begin{equation*}
\iint\limits_Q f(x,y)\,dx\,dy = \int_c^d\left[\int_a^b f(x,y)\,dx\right]\,dy
\end{equation*}

\paragraph{Integrability of continuous functions}\ \\
If $f$ is continuous on $Q$, it is integrable, and the integral can be obtained by iterated integration.\\
If $f$ is defined and bounded on $Q$, and if its set of discontinuities is of content zero, then it is integrable.

\paragraph{More general regions}\ \\
Type I:\\
$\phantom{x}$ $\{(x,y)\quad | \quad a \leq x \leq b \qquad \text{and} \qquad \varphi_1(x) \leq y \leq \varphi_2(x)\}$.\\
$\phantom{x}$ Can be evaluated by iterated integration, first on $y$, then on $x$.\\
Type II:\\
$\phantom{x}$ $\{(x,y)\quad | \quad c \leq y \leq d \qquad \text{and} \qquad \psi_1(y) \leq x \leq \psi_2(x)\}$.\\
$\phantom{x}$ Can be evaluated by iterated integraion, first on $x$, then on $y$.

\paragraph{Green's Theorem}\ \\
If $P$ and $Q$ are scalar fields that are continuously differentiable\\
...and if $C$ is a piecewise smooth Jordan curve [but more generally rectifiable Jordan I think],\\
...and if the below line integrals are traversed in a counterclockwise direction, then:
\begin{equation*}
\oint_C P\,dx+Q\,dy
= \iint\limits_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy
\end{equation*}
\begin{equation*}
\oint_C Q\,dy = \iint\limits_R \frac{\partial Q}{\partial x}\,dx\,dy
\end{equation*}
\begin{equation*}
\oint_C P\,dx = - \iint\limits_R \frac{\partial P}{\partial y}\,dx\,dy
\end{equation*}
[These last two don't seem necessarily to follow from Green's Theorem.]

\paragraph{Green's Theorem for multiply connected regions}\ \\
If appropriate conditions are met:
\begin{equation*}
\oint_{C_1}(P\,dx+Q\,dy)-\sum_{k=2}^n\oint_{C_k}(P\,dx+Q\,dy)
= \iint\limits_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dx\,dy
\end{equation*}

\paragraph{Winding number}\ \\
If $\bs{\alpha}(t) = X(t)\bs{i} + Y(t)\bs{j}$ for $a \leq t \leq b$, and for a point $P_0$,
And if $C$ is a piecewise smooth closed curve in the plane, then
\begin{equation*}
W(\bs{\alpha};P_0)=\frac{1}{2\pi}\int_a^b\frac{[X(t)-x_0]Y'(t)-[Y(t)-y_0]X'(t)}{[X(t)-x_0]^2-[Y(t)-y_0]^2}\,dt
\end{equation*}
If $C$ is a Jordan curve (simple), then $W$ is:\\
$\phantom{x}$ (a) Zero for every $P_0$ outside of $C$\\
$\phantom{x}$ (b) +1 for every $P_0$ inside $C$ if $\bs{\alpha}$ traces $C$ in a positive / counterclockwise diretion\\
$\phantom{x}$ (c) -1 for every $P_0$ inside $C$ if $\bs{\alpha}$ traces $C$ in a negative / clockwise diretion

\paragraph{Change of variables}\ \\
For single integrals:
\begin{equation*}
\int_{g(c)}^{g(d)} f(x)\,dx = \int_c^d f[g(t)]g'(t)\,dt
\end{equation*}
For double integrals:
\begin{equation*}
\iint\limits_S f(x,y)dx\,dy = \iint\limits_T f[X(u,v),Y(u,v)] \left| \frac{\partial(X,Y)}{\partial(u,v)} \right| du\,dv
\end{equation*}
Where $x = X(u,v)$ and $y = Y(u,v)$ and $\bs{r}(u,v) = X(u,v)\bs{i} + Y(u,v)\bs{j}$ and $\bs{r}(T) = S$.

\paragraph{Higher dimension integrals}\ \\
Pretty much everything follows analagously from two dimensions.

\bigskip\bigskip\
\section{Surface Integrals}\smallskip

\paragraph{Concepts}\ \\
Simple parametric surface: If $\bs{r}$ is one-to-one on $T$.\\
Regular point of $\bs{r}$: $\partial\bs{r}/\partial u$ and $\partial\bs{r}/\partial v$ are continuous, and fundamental vector product is nonzero.\\
Smooth surface: A surface $\bs{r}(T)$ is smooth if all of its points are regular points.

\paragraph{Parametric representations of surfaces}\ \\
Sphere with radius $a$: $x=a\cos u\cos v \qquad y = a\sin u\cos v \qquad z = a \sin v$

\paragraph{Fundamental vector product}\ \\
Take $\bs{r}(u,v) = X(u,v)\bs{i} + Y(u,v)\bs{j} + Z(u,v)\bs{k}$, where $(u,v) \in T$.
\begin{equation*}
\frac{\partial \bs{r}}{\partial u} = \frac{\partial X}{\partial u}\bs{i} + \frac{\partial Y}{\partial u}\bs{j} + \frac{\partial Z}{\partial u}\bs{k}
\end{equation*}
\begin{equation*}
\frac{\partial \bs{r}}{\partial v} = \frac{\partial X}{\partial v}\bs{i} + \frac{\partial Y}{\partial v}\bs{j} + \frac{\partial Z}{\partial v}\bs{k}
\end{equation*}
\begin{equation*}
\text{Fundamental vector product} = \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v}
= \begin{pmatrix}\bs{i} & \bs{j} & \bs{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & \frac{\partial Z}{\partial u} \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & \frac{\partial Z}{\partial v} \end{pmatrix}
= \frac{\partial(Y,Z)}{\partial(u,v)}\bs{i} + \frac{\partial(Z,X)}{\partial(u,v)}\bs{j} + \frac{\partial(X,Y)}{\partial(u,v)}\bs{k}
\end{equation*}
The length of the fundamental vector product can be thought of as a sort of local magnification factor for areas.\\
The fundamental vector product is normal to the surface of $\bs{r}(T)$.

\paragraph{Surface area}
\begin{equation*}
a(S) = a(\bs{r}(T)) = \iint\limits_T \lVert \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v} \rVert du\, dv
\end{equation*}

\paragraph{Definition of surface integral}\ \\
For a scalar field $f$:
\begin{equation*}
\iint\limits_{\bs{r}(T)} f\, dS = \iint\limits_T f[\bs{r}(u, v)] \lVert \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v} \rVert du\, dv
\end{equation*}

\paragraph{Change of parametric representation}\ \\
If $\bs{r}(u, v) = A$ and $\bs{R}(s, t) = B$, and if $G(s, t) = U(s, t,)\bs{i} + V(s, t)\bs{j}$ so $\bs{r}$ and $\bs{R}$ are smoothly equivalent, then
\begin{equation*}
\frac{\partial \bs{R}}{\partial s} \times \frac{\partial \bs{R}}{\partial t}
= \left( \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v} \right)
\frac{\partial(U, V)}{\partial(s, t)}
\end{equation*}
In other words, fundamental vector products of smoothly equialent functions have the same direction and only differ in magnitude.\\
Which means that surface integrals are invariant under change of smoothly equivalent parametric representations.

\paragraph{Alternate notation}\ \\
Sometimes we want to take a surface integral of a vector field, in the direction normal to the f.v.p.:
\begin{equation*}
\iint\limits_S \bs{F} \cdot \bs{n}\, dS
= \iint\limits_T \bs{F}[\bs{r}(u, v)] \cdot \bs{n}(u, v) \lVert \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v} \rVert du\, dv
= \iint\limits_T \bs{F}[\bs{r}(u, v)] \cdot \frac{\partial \bs{r}}{\partial u} \times \frac{\partial \bs{r}}{\partial v} du\, dv
\end{equation*}
where $\bs{n} = \frac{\bs{N}}{\lVert \bs{N} \rVert}$.\\
This effectively turns this into a scalar field, where the value is the magnitude of the vector in the direction normal to the surface.\\
For $\bs{f}(x, y, z) = P(x, y, z)\bs{i} + Q(x, y, z)\bs{j} + R(x, y, z)\bs{k}$ and $\bs{r}(u, v) = X(u, v)\bs{i} + Y(u, v)\bs{j} + Z(u, v)\bs{k}$:
\begin{multline*}
\iint\limits_S \bs{F} \cdot \bs{n}\, dS
= \iint\limits_T P[\bs{r}(u, v)] \frac{\partial(Y, Z)}{\partial(u, v)}
+ \iint\limits_T Q[\bs{r}(u, v)] \frac{\partial(Z, X)}{\partial(u, v)}
+ \iint\limits_T R[\bs{r}(u, v)] \frac{\partial(X, Y)}{\partial(u, v)}\\
= \iint\limits_S P(x, y, z) dy \wedge dz + \iint\limits_S Q(x, y, z) dz \wedge dx + \iint\limits_S = R(x, y, z) dx \wedge dy
\end{multline*}
So basically,
\begin{equation*}
\iint\limits_S P\, dy \wedge dz = \iint\limits_T P[\bs{r}(u, v)] \frac{\partial(Y, Z)}{\partial(u, v)} du\, dv
\end{equation*}

\paragraph{Stokes' theorem}
\begin{equation*}
\oint_C P\, dx + Q\, dy + R\, dz
= \iint\limits_S \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) dy \wedge dz
+ \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) dz \wedge dx
+ \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx \wedge dy
\end{equation*}
Or,
\begin{equation*}
\oint_C \bs{F} \cdot d\bs{\alpha} = \iint\limits_S (\curl \bs{F}) \cdot \bs{n}\, dS
\end{equation*}
where $\bs{\alpha}(t) = \bs{r}(\bs{\gamma}(t)]$, where $\bs{\gamma}(t)$ parameterizes $\Gamma$.

\paragraph{Stokes' theorem extended}\ \\
For non-simply connected regions:
\begin{equation*}
\oint_C\bs{F}\cdot d\bs{\rho}-\sum_{k=1}^n\oint_{C_k}\bs{F}\cdot d\bs{\rho_k}
= \iint\limits_S (\curl\bs{F})\cdot\bs{n}\, dS
\end{equation*}
where $\bs{\rho_k}(t) = \bs{r}[\bs{\gamma_k}(t)]$ and where $\bs{\gamma_k}(t)$ is the parameterization of $\Gamma_k$.

\paragraph{Curl}\ \\
Take $\bs{F}(x, y, z) = P(x, y, z)\bs{i} + Q(x, y, z)\bs{j} + R(x, y, z)\bs{k}$.
\begin{equation*}
\curl \bs{F} = \left(\parfrac{R}{y} - \parfrac{Q}{z} \right)\bs{i} + \left(\parfrac{P}{z}-\parfrac{R}{x}\right)\bs{j}+\left(\parfrac{Q}{x}-\parfrac{P}{y}\right)\bs{k}
\end{equation*}
\begin{equation*}
\curl \bs{F} = \begin{pmatrix}\bs{i}&\bs{j}&\bs{k}\\ \parfrac{}{x}&\parfrac{}{y}&\parfrac{}{z}\\ P&Q&R\end{pmatrix}
\end{equation*}

\paragraph{Divergence}\ \\
$\text{div} \bs{F} = \tr D\bs{F}(x, y, z) \qquad$ (The trace of the Jacobian matrix.)

\paragraph{Laplacian}\ \\
$\nabla^2 \varphi = \text{div}(\grad \varphi) = \frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2}+\frac{\partial^2\varphi}{\partial z^2}$.\\
Harmonic function: $\nabla^2\varphi = 0$.

\paragraph{Symbolic formulas}\ \\
$\nabla = \parfrac{}{x}\bs{i} + \parfrac{}{y}\bs{j} + \parfrac{}{z}\bs{k}$\\
$\curl \bs{F} = \nabla \times \bs{F}$\\
$\text{div} \bs{F} = \nabla \cdot \bs{F}$\\
$\grad \varphi = \nabla \varphi$\\
$\bs{F}$ is a gradient on an open convex set iff $\curl \bs{F} = O$.

\paragraph{Other relationships}\ \\
If $\varphi$ is a scalar field with continuous second-order mixed partial derivatives, $\curl(\grad \varphi) = O$.\\
Irrotational: A vector field with $\curl \bs{F} = 0$.\\
If $\bs{F}$ is a vector field with continous mixed partial dervatives, $\text{div}(\curl \bs{F}) = 0$ and $\curl(\curl \bs{F}) = \grad(\text{div} \bs{F}) - \nabla^2 \bs{F}$.\\
Curl and divergence are linear operators.\\
Let $\bs{F}$ be continously differentiable on an open interval $S$ in 3-space.\\
$\phantom{x}$ Then there exists a vector field $\bs{G}$ such that $\curl \bs{G} = \bs{F}$ iff $\text{div} \bs{F} = 0$ everywhere in $S$.\\
Solenoidal: A vector field with $\text{div} \bs{F} = 0$.

\paragraph{Divergence theorem}\ \\
Let $V$ be a solid in 3-space bounded by an orientable closed surface $S$.\\
Let $\bs{n}$ be the unit outer normal to $S$.\\
Let $\bs{F}$ be a continously differentiable vector field defined on $V$.
\begin{equation*}
\iint\limits_S \bs{F} \cdot \bs{n}\, dS
= \iiint\limits_V (\text{div} \bs{F}) dx\, dy\, dz
\end{equation*}

\end{document}