Homework Problems

1. For the two vectors $\vec{A}=\hat{x}+3\hat{y}-2\hat{z}$ and $\vec{B}=-2\hat{x}+\hat{y}+\hat{z}$, find $\vert\vec{A}-\vec{B}\vert$, the component of $\vec{B}$ along $\vec{A}$, the angle between $\vec{A}$ and $\vec{B}$, $\vec{A}\times\vec{B}$, and $(\vec{A}-\vec{B})\times(\vec{A}+\vec{B})$.

2. The gradient operator $\vec{\nabla}={\partial\over\partial x_i}\hat{e}_i$ transforms like a vector. Use this to write the equations of electromagnetism with the Einstein summation convention, that is, with no vector symbols. To make it interesting, here are the EM equations in Rationalized Heaviside-Lorentz units.

   $$\vec{\nabla}\cdot\vec{B} = 0$$

   $$\vec{\nabla}\times\vec{E}+{1\over c}{\partial \vec{B}\over\partial t}=0$$

   $$\vec{\nabla}\cdot\vec{E}=\rho$$

   $$\vec{\nabla}\times\vec{B}-{1\over c}{\partial \vec{E}\over\partial t}={1\over c}\vec{j}$$

   $$\vec{F}=-e\left(\vec{E}+{1\over c}\vec{v}\times\vec{B}\right)$$

   As an example, we can write the first equation.
   $${\partial\over\partial x_i}B_i=0$$

3. Use the operator $\vec{\nabla}={\partial\over\partial x_i}\hat{e}_i$ to show that the divergence of a curl is zero.
   $$\vec{\nabla}\cdot\left(\vec{\nabla}\times\vec{A}\right)=0$$
   Also calculate the curl of the curl.
   $$\vec{\nabla}\times\left(\vec{\nabla}\times\vec{A}\right)=?$$

4. Use the totally antisummetric tensor to derive the identity $(\vec{A}\times\vec{B})\times (\vec{C}\times\vec{D})=((\vec{A}\times\vec{B})\cdot\vec{D})\vec{C}-((\vec{A}\times\vec{B})\cdot\vec{C})\vec{D}$.

5. Calculate the rotation matrix for a passive rotation about the $\hat{z}$ axis through an angle $\phi$ followed by a rotation about the new $\hat{x}$ axis by an angle $\theta$.

6. Calculate the rotation matrix for a passive rotation through an angle of 120 degrees about an axis making equal angles with the original three coordinte axes.

7. Calculate the time derivative of a rotation matrix through an angle $\omega t$ about the $z$ axis. Remembering that the axial vector $\vec{\omega}$ is related to an antisymmetric tensor, relate the time derivative of the rotation matrix (evaluated at $t=0$) to $\vec{\omega}$.

8. Show that for matrices $\mathbb{A}$ and $\mathbb{B}$, $(\mathbb{AB})^T=\mathbb{B}^T\mathbb{A}^T$, and that $(\mathbb{AB})^{-1}=\mathbb{B}^{-1}\mathbb{A}^{-1}$.