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
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
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) a
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 harm
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
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 + 1
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) a
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
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 t