Physics 263: Problem Set #15

1. (BTM 8.1.2.) Once you've written the transformation in matrix form, which will be analogous to Eq. (8.1.25) except for a sign difference, the demonstration here is a simple matrix multiplication followed by an application of hyperbolic trig identities. It's easy to verify the latter using the exponential form of sinh and cosh.
2. (BTM 8.2.2.) The rotation matrix R_theta is given in Eq. (8.1.25) and the formula for the inverse of a 2x2 matrix (which you should learn if you don't know it) is in (8.2.11). This problem is just an exercise in carrying it out and interpreting the result. If a determinant vanishes, then there is no inverse according to (8.2.11). How do you "undo" a rotation?
3. (BTM 8.2.3 and 8.2.5.) The first is a bread-and-butter application of (8.2.11), following the example in eqs. (8.2.2) through (8.2.14). If this is unfamiliar, follow through that example first. The second problem works the same way, except you don't get any further than calculating the determinant.
4. (BTM 8.2.7.) Just imitate the proof for Theorem 8.2, but now with three matrices. This is not a difficult proof, but this is a result to engrave on your brain.
5. (BTM 8.3.9.) This problem recalls problem 7.1.3. The "box product" is the same as what we call the "triple product" or "scalar triple product". You'll want to think of x, y, and z each as functions of u,v,w; that is x = x(u,v,w). Then if we change u from u to u+du, then x, y, and z each change according to the partial derivatives with respect to u (since v and w are held fixed). That tells us how the vector rvec = x ihat + y jhat + z khat changes, which in turn defines dr_u. And similarly with the others.