Example. Compute the surface integral

$$
\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma
$$

where $\Sigma-$ is the upper hemisphere

$$
x^{2}+y^{2}+z^{2}=9, \quad z \geq 0
$$
SOLUTION.

1. We introduce curvilinear coordinates on the given surface (sphere)

$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \sin \theta \\
y=\varrho \sin \varphi \sin \theta \\
z=\varrho \cos \theta
\end{array}\right.
$$

In these coordinates, the surface is defined by the conditions

$$
\Sigma=\left\{\begin{array}{ll} 
& \varrho=3 \\
(\varrho, \theta, \varphi): & 0 \leq \theta \leq \pi / 2 \\
& 0 \leq \varphi \leq 2 \pi
\end{array}\right\}
$$

2. Since $d \sigma=9 \sin \theta d \theta d \varphi$ and $f(x, y)=x^{2}+y^{2}=9 \sin ^{2} \theta$, we have

$$
\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma=\int_{0}^{2 \pi} d \varphi \int_{0}^{\pi / 2} 9 \sin ^{2} \theta \cdot 9 \sin \theta d \theta
$$

3. We compute the double integral:

$$
\int_{0}^{2 \pi} d \varphi \int_{0}^{\pi / 2} 9 \sin ^{2} \theta \cdot 9 \sin \theta d \theta=-162 \pi \int_{0}^{\pi / 2}\left(1-\cos ^{2} \theta\right) d \cos \theta=108 \pi
$$

Answer. $\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma=108 \pi$.

## PROBLEM CONDITIONS. Compute the surface integrals.

1. $\quad \int_{\Sigma} x d \sigma$

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=7, \quad z \geq 0\right\}
$$

2. $\quad \int_{\Sigma} z d \sigma$

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=4, \quad z \geq 0\right\}
$$

3. $\quad \int_{\Sigma}(2 x-5 y) d \sigma$

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=5, \quad z \geq 0\right\}
$$

4. $\quad \iint_{\Sigma}\left(z^{2}+1\right) d \sigma$

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=2, \quad z \geq 0\right\}
$$

5. $\quad \iint_{\Sigma} z^{4} d \sigma$

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=1, \quad z \geq 0\right\}
$$

6. $\quad \iint_{\Sigma} \sqrt{x^{2}+y^{2}} d \sigma$,

$$
\Sigma=\left\{x^{2}+y^{2}+z^{2}=1, \quad z \geq 0\right\}
$$

7. $\quad \iint_{\Sigma}\left(x^{2}+y^{2}-2 z\right) d \sigma, \quad \Sigma=\left\{x^{2}+y^{2}+z^{2}=9, \quad z \geq 0\right\}$.
8. $\quad \int_{\Sigma}\left(x^{2}+y^{2}-2\right) d \sigma, \quad \Sigma=\left\{x^{2}+y^{2}+z^{2}=3, \quad z \geq 0\right\}$.
9. $\quad \iint_{\Sigma}(21 x-21 y+z) d \sigma, \quad \Sigma=\left\{x^{2}+y^{2}+z^{2}=10, \quad z \geq 0\right\}$. 10. $\iint_{\Sigma}\left(x y+z^{2}\right) d \sigma, \quad \Sigma=\left\{x^{2}+y^{2}+z^{2}=1, \quad z \geq 0\right\}$.

Answers. 1. $0.2 .2 \pi$ 3. $8 \pi$ 4. $20 \pi / 3$. 5. $2 \pi / 5.6 . \quad \pi^{2} / 2$. 7. $54 \pi$. 8. 0. $9.10 \sqrt{10} \pi$. 10. $2 \pi / 3$.

## Chapter 14

## FIELD THEORY

In studying the topic FIELD THEORY (VECTOR ANALYSIS), you will become familiar with the concepts of vector fields, vector lines, flux of a vector field through a surface, circulation of a vector field, divergence, and curl of a vector field. You will learn how to compute the flux of a vector field as a surface integral, as well as use the Ostrogradsky formula to compute the flux through a closed surface. You will learn how to compute the work of a vector field as a line integral of the second kind, and apply the Stokes' theorem to compute circulation.

Using the package SOLVER.VM, you can compute derivatives, the unit normal vector to a surface, the divergence and curl of a vector field, definite and repeated integrals, perform all numerical calculations, and verify the correctness of your results.

## 14.1. Vector Lines

PROBLEM STATEMENT. Find the vector lines of vector fields

$$
\vec{a}=P(x, y) \vec{i}+Q(x, y) \vec{j}
$$

or

$$
\vec{a}=P(x, z) \vec{i}+R(x, z) \vec{k}
$$

or

$$
\vec{a}=Q(y, z) \vec{j}+R(y, z) \vec{k}
$$

PLAN OF SOLUTION.

1. Write the differential equations of the vector lines

$$
\frac{d x}{P(x, y)}=\frac{d y}{Q(x, y)} \quad \text { when } \quad z=C
$$

or

$$
\frac{d x}{P(x, y)}=\frac{d z}{R(x, y, z)} \quad \text { when } \quad y=C
$$

or

$$
\frac{d y}{Q(x, y)}=\frac{d z}{R(x, y, z)} \quad \text { when } \quad x=C
$$

We have taken into account that in the first case $d z=0$, in the second case $d y=0$, and in the third case $d x=0$, since the corresponding coordinate of the vector field is zero.

2. Solving the corresponding differential equation, we obtain that the vector lines (in space) are defined by the systems of equations

$$
\left\{\begin{array}{l}
F(x, y)=C_{1} \\
z=C_{2}
\end{array}\right.
$$

or

$$
\left\{\begin{array}{l}
F(x, z)=C_{1} \\
y=C_{2}
\end{array}\right.
$$

or

$$
\left\{\begin{array}{l}
F(y, z)=C_{1} \\
x=C_{2}
\end{array}\right.
$$