Review sheet 1 (Due April 14th)

Vectors

Problem 1. Verify that for vectors in three dimensions,

$(\vec{A} \wedge \vec{B}). (\vec{C} \wedge \vec{D}) = (\vec{A} \cdot \vec{C}) (\vec{B} \cdot \vec{D}) - (\vec{A}\cdot \vec{D}) (\vec{B}\cdot \vec{C})$.

Lists and Matrices

Problem 2. Using matrix operations, find the values of x,y,z which solve the following set of equations.

1.2 x + 4.5 y + 2.3 z = -1.3

-0.6 x + 3.1 y - 0.3 z = 0.6

1.9 x + 0.5 y + 1.3 z = 1.7

Function definitions $f[x\_]$

Problem 3. Define the function f(x) = 0.8 x + x² - 3.2 x⁴. Find the four roots, 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) = 10 t , y(t) = 10 t - 5 t². Plot x(t), y(t) and also do a parametric plot of (x(t),y(t)). What sort of motion is this ?

Ordinary differential equations

Problem 5. A mass(m= 1.1 kg)/spring(k = 2.2 N/m) system hangs vertically at equilibrium in Earth's gravity. It is displaced from equilirium by a small amount and then oscillates. It experiences a damping of $b \vec{v}$, where b=0.1. Consider a small amplitude initial displacement of 1 cm and initial velocity of zero. Solve the linear differential equation for this problem. ( x''(t) + 0.1 x'(t) + 2 x (t) = 0). Plot x(t).

Partial differential equations
Problem 6. Show that $\rho (x,t) = {A \over t^{1/2}} Exp(-a x^2/t)$ solves the diffusion equation

$${\partial \rho (x,t) \over \partial t } = D {\partial^2 \rho (x,t) \over \partial x^2},$$

provided that D is related to a. What is that relationship?