NAME:_________________________________

Instructions: Write the answers where indicated (or points will be taken off). If you do not understand what is being asked for, ask Prof. Harrell for a clarification.

1. Find the solution of Laplace's equation

grad²u = 0

on the rectangle 0 < x < 1, 0 < y < 3, with the boundary conditions::

u(0,y) = 0

u(1,y) = 1

uy(x,0) = 0

uy(x,3) = x

ANSWER: __________________________________________________________

2. A torus is represented parametrically as

x = (2 + cos( \theta)) cos(\phi)

y = (2 + cos( \theta)) sin(\phi)

z = sin( \theta),

where \theta and \phi are both angular variables. This means that if f is any function of \theta or \phi , then f(\theta+2 \pi,\phi ) = f( \theta, \phi +2 \pi) = f( \theta+2 \pi, \phi +2 \pi) = f( \theta, \phi ). Suppose that the diffusion equation in terms of the coordinates \theta and \phi is:

u_t = u_{\phi \phi} + 4 u_{\theta \theta}.

a) Find (explicitly) all solutions of the form T(t) \Theta (\theta) \Phi(\phi ) to this equation:

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b) Write down the general solution:

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c) If at time 0, u(0, \theta, \phi ) = 20 cos²(\phi ), find

3. a) Construct an orthogonal set of three functions on the interval -1 < x < 1, the span of which is the same as the span of {f₁(x) := x², f₂(x) := x³, f₃(x) := 1+|x|}. Call the resulting set {g₁(x), g₂(x), g₃(x)}.

g₁(x) = ______________________________________

g₂(x) = ______________________________________

g₃(x) = ______________________________________

b) Find the function f(x) in the span of these three functions, which is closest in the r.m.s. sense to the function |x|.

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

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