1. Prove that div(~

u × ~

v) = rot ~

u ◦ ~

v − ~

u ◦ rot ~

v.

2. Let ˆ be the Fourier’s transform and g

a

(x) = f (x − a). Prove that ˆ

g

a

(ξ) = e

−iaξ

ˆ

f (ξ).

3. Let u be a solution to the heat equation

u

t

(t, x) =u

xx

(t, x),

t > 0, x ∈ (0, 1)

u(t, 0) =u(t, 1) = 0,

t > 0

u(0, x) =u

0

(x).

Prove that F (t) =

R

1

0

[u

x

(t, x)]

2

dx is monotonically decreasing.

4. Solve the following problem by using the separation of variables method:

u

tt

(t, x) =u

xx

(t, x),

t > 0, x ∈ (0, 2)

u(t, 0) =u(t, 1) = 0,

t > 0

u(0, x) =2 sin

 1

2

πx



+ 3 sin(3πx)

u

t

(0, x) =5 sin

 3

2

πx



.

5. Suppose that u solves the wave equation u

tt

= u

xx

. Let a, b, c, d be the consecutive vertices of a

rectangle in the (x, t)-space, with edges parallel to the lines x = t and x = −t. Prove that
u(a) + u(c) = u(b) + u(d).