# 22.1.1: i. Review Exercises

## Q1

Transform (using the coordinate system provided below) the following functions accordingly:

a. from Cartesian to spherical polar coordinates

$3x + y -4z =12$

b. from Cartesian to cylindrical coordinates

$y^2 + z^2 = 9$

c. from spherical polar to Cartesian coordinates

$r = 2\sin \theta \cos \phi$

## Q2

Perform a separation of variables and indicate the general solution for the following expressions:

a. $$9x + 16y \dfrac{\partial y}{\partial x} = 0$$

b. $$2y + \dfrac{\partial y}{\partial x} + 6 = 0$$

## Q3

Find the eigenvalues and corresponding eigenvectors of the following matrices:

1. $$\begin{bmatrix} -1 & 2 \\ 2 & 2 \end{bmatrix}$$
2. $$\begin{bmatrix} -2 & 0 & 0 \\ 0 & -1 & 2 \\ 0 & 2 & 2 \end{bmatrix}$$

## Q4

For the hermitian matrix in review exercise 3a show that the eigenfunctions can be normalized and that they are orthogonal.

## Q5

For the hermitian matrix in review exercise 3b show that the pair of degenerate eigenvalues can be made to have orthonormal eigenfunctions.

## Q6

Solve the following second order linear differential equation subject to the specified "boundary conditions":

$\dfrac{d^2 x}{dt^2} + k^2x(t) = 0$

where $$x(t=0) = L$$ and $$\dfrac{dx}{dt} \bigg|_{t=0} = 0.$$