1. Prove that div(u × v) = rot u ć% v - u ć% rot v.
Ć
2. Let Ć be the Fourier s transform and ga(x) = f(x - a). Prove that %1Å„a(¾) = e-ia¾f(¾).
3. Let u be a solution to the heat equation
ut(t, x) =uxx(t, x), t > 0, x " (0, 1)
u(t, 0) =u(t, 1) = 0, t > 0
u(0, x) =u0(x).
1
Prove that F (t) = [ux(t, x)]2 dx is monotonically decreasing.
0
4. Solve the following problem by using the separation of variables method:
utt(t, x) =uxx(t, x), t > 0, x " (0, 2)
u(t, 0) =u(t, 1) = 0, t > 0
1
u(0, x) =2 sin Ä„x + 3 sin(3Ä„x)
2
3
ut(0, x) =5 sin Ä„x .
2
5. Suppose that u solves the wave equation utt = uxx. 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).