Review Sheet 2 - Due Thursday April 21st

Warning - One hour lab. exam next week (April 28)!!

Use Mathematica to solve all parts of the problems

Vectors

*Problem 1.*
Show, for any three-dimensional vectors **, ****, **, that

(i)

(ii)
**
**

Lists and Matrices

*Problem 2.* Display the following matrix in matrixform.

Find its eigenvalues, eigenvectors and determinant. Find the sum of the eigenvalues (= the trace of the matrix) and the product of the eigenvalues (= the determinant of the matrix). Notice that both the trace and the determinant are real even though the eigenvalues themselves are complex.

Function definitions **
**

*Problem 3.* Define the function
*f*(*x*) = *x* - 2 *tanh*(*x*). Find
the three real roots(use FindRoot), *x*_{r}**, of ***f*(*x*) and check that one of the
non-zero roots satisfies *f*(*x*_{r})=0**.
**

Plotting

*Problem 4.* Define
*x*(*t*) = *Cos*[ *t*] , *y*(*t*) = *Sin*[*t*], *z*(*t*) = *t*.
Do a parametric plot of
(*x*(*t*),*y*(*t*),*z*(*t*)).
What sort of motion is this ?

Ordinary differential equations

*Problem 5.* An ideal pendulum (
*m* = 1 *kg*, *l*=2*m*)
oscillates in air. The drag coefficient of
air is *b*=0.001. Consider a small amplitude
initial displacement of 1 *cm* and initial velocity of zero.
Solve the linear differential equation for this problem.
(
). Plot
.

Partial differential equations

*Problem 6.* Show that
*y* (*x*,*t*) = *A Cos*(*x* - *v t*) solves the
wave equation

Plot