Math 32A/H Final Review Materials

I have started compiling some review materials for math 32A and 32AH for this quarter. They consist of

I will post solutions to the exercises before the review session on Thursday. As always, let me know in the comments below if anything is unclear or incorrect.

Update I have posted the solutions to the practice problems. Also, a student pointed out a typo in statement of the final problem in the review. Spherical coordinates should be given by
x = \rho \cos \theta \sin \varphi, \quad y = \rho \sin \theta \sin \varphi, \quad z = \rho \cos \varphi.
These changes have been made in the version of the problems and solutions online.

Update 2 There was a mistake in the solution of problem 10 involving curvature. This has (hopefully) been fixed.

  • Anonymous

    I think that equation (24) should have one g(x, y) and the other h(x, y) when its referring to the squeeze theorem. In the document both are g(x, y)

    • will

      You are absolutely right. Thanks for spotting that typo — it has been fixed.

  • Akash

    umm can we solve the 14th by assuming y,z to be the functions of x ( let g(x) = ( y, z)) and then the given equations to be level set at 0 of a function f(x,g(x))–> (equation 1, equation 2) which is also the graph of g(x) and then using the theorem to find g'(x) which give us the tangent vector ?

    • will

      In principle, that technique will definitely work as well, as long as the hypotheses of the implicit function theorem hold (so that you can guarantee that you can find such a function g(x)). The tangent vector you find will be
      (1, g'(1)).