homework1.nb

Probelm 3:

Definition of Rk(θ)

$RK[θ_] := {<br /> {kx^2 (1 - Cos[θ]) ... bsp; kz^2 (1 - Cos[θ]) + Cos[θ]}} ;$

Solution

To prove eq. (2) of the homework, we can differentiate Rk(θ) with respect to θ two different ways: (1) element by element, and (2) by applying eq. (5) of the homework. If the two results are equivalent, than eq. (2) must be true.

( 2 ... kx kz sin(θ) - ky cos(θ) kx cos(θ) + ky kz sin(θ) (kz - 1) sin(θ)

{{ky (kx kz (1 - cos(θ)) - ky sin(θ)) - kz (kx ky (1 - cos(θ)) + kz sin(θ) ... ), kx (ky kz (1 - cos(θ)) - kx sin(θ)) - ky (kx kz (1 - cos(θ)) + ky sin(θ))}}

The two results certainly don't look the same. Let's try to simplify the above result...

( 2 2 ... kx kz sin(θ) - ky cos(θ) kx cos(θ) + ky kz sin(θ) -(kx + ky ) sin(θ)

Now, all but the diagonal terms of the two equations are identical. However, since k is a unit vector, kx^2 + ky^2 + kz^2 = 1. Let's make some substitutions:

$result2 = result2 /. {(ky^2 + kz^2) -> (1 - kx^2), (kx^2 + kz^2) -> (1 - ky^2), (kx^2 + ky^2) -> (1 - kz^2)} ; result2 = result2 // FullSimplify ; result2 // TraditionalForm$

( 2 ... kx kz sin(θ) - ky cos(θ) kx cos(θ) + ky kz sin(θ) (kz - 1) sin(θ)

Finally, we can test if the two results are the same:

Created by Mathematica (September 29, 2003)