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 $\vec{a}$, $\vec{b}$, $\vec{c}$, that
(i) $\vec{a}.(\vec{a} \wedge \vec{b}) = 0$
(ii) $\vec{a}.(\vec{b} \wedge \vec{c}) = (\vec{a} \wedge \vec{b}).\vec{c}$

Lists and Matrices

Problem 2. Display the following matrix in matrixform.

$$matrix = \{\{0.2,-0.4,0.1\},\{0.4,0.2,-0.3\},\{0.6,-0.3,-0.1\}\}$$

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 $f[x\_]$

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=2m) 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. ( $\theta''(t) + 0.001 \theta'(t) + 5 \theta (t) = 0$). Plot $\theta(t)$.
Partial differential equations
Problem 6. Show that y (x,t) = A Cos(x - v t) solves the wave equation

$${\partial y^2 (x,t) \over \partial t^2 } = v^2 {\partial^2 y (x,t) \over \partial x^2},$$

Plot y[0,t] for the velocity v=1.