Homework 2 Vector
Bundles
(Note that I have been inventive about some of the symbols)
11. Show that for a k-form, **A = +/- A.
Compute
the sign for a two form in R4, a one-form in R2,
and a one form in R1,1.
12. Show that for A , B k-forms,
A^*B = +/- B^ *A = +/- (A,B)G .
13. Compute the eigenvalues and eigen(co)vectors for
* mapping E*--> E* where E = R1,1. Find a set
of coordinates whose differentials are these eigen (co)vectors.
14. Do the same problem for E* = R2. In this
case, you will have to formally complexify and work with z = x1+ix2,
z(bar) = x1 - ix2.
15. Find the eigenvalues and eigen(co)vectors for * mapping
^2E*-->^2E* when E = R4.
16. If A is a one-form in E=R3, show that *dA =
curl A. How should we define curl in
R1,2? Compute explicity in terms of
coordinates a0dx0 + a1dx1
+ a2dx2. How does the signature affect the
computation?
17. Write the Maxwell's equations discussed in the lecture in Rx Rn
in terms of div and
curl when n = 3. Recall that the four equations were:
(i) d*E =
4 j ; - d/dt E + d* B = 4
J.
(ii) dB =
0 ; d/dt B + dE = 0.
18. Show that if j = J = 0, then wave E
= wave B = 0 where wave =-
(d/dt)2 + laplace.
19. Suppose E = -d/dt U, and B = dU, for U a one form on R3.
Show that two of the four Maxwell equations (in space) are
satisfied, and write the other two equations in terms of the one-form U.
20. Suppose V = v1e1 = v2e2
and W = w1e1 + w2e2 are
two complex vectors written in terms of basis vectors.
Define the inner product (V,W) = Re (v1(w1(bar))+
v2(w2(bar)). Compute explicitly in terms
of complex numbers all the 2x2 matrices of complex numbers A which
satisfy (AV,AW) = (V,W) for all complex vectors V,W. What
extra restriction on the numbers do you get if you require det(A) =
1? What is the dimension of the space of these matrices?
What is the topology? Show the space of all such A is a group.