Math 603-601 - Fall 2002
Homework
Assignment 9
- Problem 8, pg. 101, in Borisenko and Tarapov.
- Problem 9, pg. 101, in Borisenko and Tarapov.
- Let q1=r, q2 =θ, and
q3=φ be the radius, co-latitude, and azimuthal angle in
spherical coordinates. Find the following.
- The basis vectors: e1, e2, and
e3, in terms of {i,j,k}.
- The reciprocal basis vectors: e1,
e2, e3, in terms of
{i,j,k}.
- The metric tensor g and the square of the arc length,
ds2
- Repeat the previous exercise for parabolic cylinder coordinates
q1, q2, and q3, which are defined by the
equations below.
x = ½((q1)2 -
(q2)2)
y = q1q2
z = q3
- Let q1, q2, and q3 be a set of
generalized coordinates and let x1, x2, and
x3 be rectangular coordinates.
- Show that [dx1 dx2
dx3]T = K[dq1 dq2
dq3]T, where
K = [
[e1]B
[e2]B
[e3]B], B = {i,j,k}
- Show that the metric tensor g = KTK, and use the properties of
determinants to show that
G := det(g) = (det(K))2
- Use part b above to show that the volume dV of the parallelepiped
with edges e1dq1,
e2dq2, e3dq3
is given by
dV = G½dq1dq2dq3
Due Thursday, 21 November.