Ask HN: How do I learn math/physics in my thirties?
Music:<p>Get a piano, look up the basics of how to read music, find the keys on the piano, see my post on music theory and the Bach cello piece, get a recording of some relatively simple music you do like, get the sheet music, and note by note learn to play it.  After 3-4 such pieces, get an hour of piano instruction and continue on.<p>Violin:  Much the same except need more help at the start.  From my music theory post, learn how to tune a violin.  Get a good shoulder rest -- the most popular is, IIRC, from Sweden and is excellent.  Look at images of violinists and see what rests they are using.  Get Ivan Galamian&#x27;s book on violin.  Start in the key of A major and then branch out to E major and D major.  Get some good advice on how to hold the violin and the bow; look at pictures of Heifetz, etc.  Learn some scales and some simple pieces, get some lessons, and continue.<p>Math:  High school 1st and 2nd year algebra, plane geometry (with proofs), trigonometry, and hopefully also solid geometry.  Standard analytic geometry and calculus of one variable.<p>For calculus of several variables and vector analysis, I strongly recommend<p>Tom M.\ Apostol,
{\it Mathematical Analysis:
A Modern Approach to Advanced Calculus,\&#x2F;}
Addison-Wesley,
Reading, Massachusetts,
1957.\ \<p>Get a used copy -- I did.  Actually, it&#x27;s not &quot;modern&quot; and instead is close to what you will see and need in applications in physics and engineering.  There, relax any desire for really careful proofs; really careful proofs with high generality are too hard, and the generality is nearly never even relevant in applications so far.  Maybe do the material again if want to do quantum gravity at the center of black holes or some such; otherwise, just stay with what Apostol has.  For <i>exterior algebra of differential forms</i>, try hard enough to be successful ignoring that stuff unless you later insist on high end approaches to differential geometry and relativity theory.<p>Linear algebra, done at least twice and more likely several times.  Start with a really easy book that starts with just Gauss elimination for systems of linear equations -- actually a huge fraction of the whole subject builds on just that, and that is close to dirt simple once you <i>see it</i>.<p>Continue with an intermediate text.  I used E. Nearing, student of Artin at Princeton.  Nearing was good but had a bit too much, and his appendix on linear programming was curious but otherwise awful -- linear programming can be made dirt simple, mostly just Gauss elimination tweaked a little.<p>Mostly you want linear algebra over just the real or complex numbers, but nearly all the subject can also be done over any <i>algebraic field</i> -- Nearing does this.  Actually, might laugh at linear algebra done over finite fields, but the laughter is not really justified:  E.g., algebraic coding theory, e.g., R. Hamming, used finite fields.  But if you just stay with the real and complex numbers, likely you will be fine and can go back to Nearing or some such later if wish.<p>So, concentrate on eigen values and eigen vectors, the standard inner product, orthogonality, the Gram-Schmidt process, orthogonal, unitary, symmetric, and Hermitian matrices.  The mountain peak is the polar decomposition and then singular value decomposition, etc.  Start to make the connections with convexity and the normal equations in multi-variate statistics, principle components, factor analysis, data compression, etc.<p>Then, of course, go for P. Halmos, <i>Finite Dimensional Vector Spaces</i>, grand stuff, written as an introduction to Hilbert space theory at the knee of von Neumann.  Used in Harvard&#x27;s Math 55.  Commonly given to physics students as their source on Hilbert space for quantum mechanics.  Likely save the chapter on multi-linear algebra for later!<p>For more, get into numerical methods and applications.  You can do linear programming, non-linear programming, group representation theory, multi-variate Newton iteration, differential geometry.  Do look at W. Fleming, <i>Functions of Several Variables</i> and there the inverse and implicit function theorems and their applications to Lagrange multipliers and the eigenvalues of symmetric or Hermitian matrices.  The inverse and implicit function theorems are just local, non-linear versions of what you will see with total clarity at the end of applying Gauss elimination in the linear case.<p>Physics<p>Work through a famous text of freshman physics and then one or more of the relatively elementary books on E&amp;M and Maxwell&#x27;s equations.  Don&#x27;t get stuck:  Physics people commonly do math in really obscure ways; mostly they are thinking intuitively; generally you can just set aside after a first reading what they write, lean back, think a little about what they likely really do mean, derive a little, and THEN actually understand.  E.g., in changing the coordinates of the gradient of a function, that&#x27;s not what they are doing!  Instead they are getting the gradient of a surface, NOT the function, as the change the coordinates of the surface.  They are thinking about the surface, not the function of the surface in rectangular coordinates.<p>For more than that, you will have to start to specialize.  Currently a biggie is a lot in probability theory.  There the crown jewels are the classic limit theorems, that is, when faced with a lot of randomness, can make the randomness go away and also say a lot about it.<p>For modern probability, that is based on the 1900 or so approach to the integral of calculus, the approach due to H. Lebesgue and called <i>measure theory</i>.  In the simple cases, it&#x27;s just the same, gives the same numerical values for, the integral of freshman calculus but otherwise is much more powerful and general.  One result of the generality is that it gives, via A. Kolomogorov in 1933, the currently accepted approach to advanced probability, stochastic processes, and statistics.<p>That&#x27;s a start.