6.1
LAPLACES EQUATION
MATH 124B Solution Key HW 03
6.1
LAPLACES EQUATION
4. Solve u x x + u y y + uzz = 0 in the spherical shell 0 < a < r < b with the boundary
conditions u = A on r = a and u = B on r = b, where A and B are constants. Hint: Look
for a so
5.3
ORTHOGONALITY AND GENERAL FOURIER SERIES
MATH 124B Solution Key HW 02
5.3
ORTHOGONALITY AND GENERAL FOURIER SERIES
1. (a) Find the real vectors that are orthogonal to the given vectors (1, 1, 1) and (1, 1, 0).
(b) Choosing an answer to (a), expand the
7.1
GREENS FIRST IDENTITY
MATH 124B Solution Key HW 05
7.1
GREENS FIRST IDENTITY
1. Derive the 3-dimensional maximum principle from the mean value property.
SOLUTION. We aim to prove that if u is harmonic in the bounded set D 3 and u is
continuous on D =
6.3
POISSONS FORMULA
MATH 124B Solution Key HW 04
6.3
POISSONS FORMULA
1. Suppose that u is a harmonic function in the disk D = cfw_r < 2 and that u = 3 sin 2 + 1
for r = 2. Without nding the solution, answer the following questions.
(a) Find the maximum
5.1
THE COEFFICIENTS
MATH 124B Solution Key HW 01
5.1
THE COEFFICIENTS
1. In the expansion 1 =
the sum
n odd (4/n) sin nx,
1
1 1 + 1 13 + +
5
9
1
3
valid for 0 < x < , put x = /4 to calculate
1
1 + 11 +
7
1
= 1 + 3 1 1 + 1 +
5
7
9
Hint: Since each of t
5.3
ORTHOGONALITY AND GENERAL FOURIER SERIES
MATH 124B Solution Key HW 02
5.3
ORTHOGONALITY AND GENERAL FOURIER SERIES
1. (a) Find the real vectors that are orthogonal to the given vectors (1, 1, 1) and (1, 1, 0).
(b) Choosing an answer to (a), expand the
7.2
GREENS SECOND IDENTITY
MATH 124B Solution Key HW 06
7.2
GREENS SECOND IDENTITY
1. Derive the representation formula for harmonic functions (7.2.5) in two dimensions.
u(x0 ) =
1
u(x)
2
bdy D
n
(log | x x0 |)
u
n
log | x x0 |
ds.
SOLUTION. Let x0 be an
9.2
THE WAVE EQUATION IN SPACE-TIME
MATH 124B Solution Key HW 08
9.2
THE WAVE EQUATION IN SPACE-TIME
1. Prove that (u) = (u) for any function; that is, the laplacian of the average is the
average of the laplacian. Hint: Write u in spherical coordinates an
7.3
GREENS FUNCTIONS
MATH 124B Solution Key HW 07
PRELIMINARIES: All of our arguments rely on the fact that the function
1
for n = 1,
2 |x x 0 |
1
log | x x0 |
for n = 2,
v(x) = v(x; x0 ) =
2
1
| x x0 |2n for n 3
(2 n) An
where x and x0 represent disti