%begin preamble
\documentclass[11pt,titlepage]{article}
\usepackage[margin=1in,noheadfoot]{geometry}
\usepackage[parfill]{parskip}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{wrapfig}
\usepackage{calligra}
\DeclareMathAlphabet{\mathcalligra}{T1}{calligra}{m}{n}
\DeclareFontShape{T1}{calligra}{m}{n}{<->s*[2.2]callig15}{}
\newcommand{\scriptr}{\hspace{-.1ex}\ensuremath{\mathcalligra{r}}\,}
\newcommand{\dslash}{d \hspace{-0.5ex}\rule[1.2ex]{0.6ex}{.1ex}}

\newcommand{\e}[1]{\times 10^{#1}}
\newcommand{\q}{\mathrm}
\newcommand{\where}[1][where]{\qquad\mbox{#1}\qquad}
\newcommand{\bs}[1]{\boldsymbol{#1}}
\newcommand{\bh}[1]{\hat{\boldsymbol{#1}}}
\newcommand{\bv}[1]{\vec{\boldsymbol{#1}}}
\newcommand{\bt}[1]{\tilde{\boldsymbol{#1}}}
\newcommand{\flow}[1]{\mathcal{E}_{#1}}
\newcommand{\flux}[1]{\Phi_{#1}}
%end preamble

\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#1, Rev. A, due Friday, 2014-09-05 }
\end{center}

\vspace{.2 in}

{\bf 1. a)} Show graphically that the following equations
define a set of points $\{\bs x\}$ \mbox{on a {\bf line} or {\bf plane},}
\begin{center}
  \begin{tabular}{r|@{\qquad}l@{\qquad}l}
    & relational & parametric \\ \hline
    line & $\bs a \times \bs x = \bs d$  &  $\bs x = \bs x_1 + \bs a \alpha$ \\
    plane& $\bs A \cdot \bs x = D$       &  $\bs x = \bs x_2 + \bs b \beta + \bs c \gamma$ \\
  \end{tabular}
\end{center}
where $\bs a, \bs d, \bs A, D$ are constants, $\bs x_1$ is a fixed point on
the line, $\bs x_2$ is a fixed point on the plane, $\bs x$ is an arbitrary
point on the line or plane, and $\alpha, \beta, \gamma$ are parameters that
vary along the line/plane (they uniquely parametrize points in the
line/plane).  

Technically, the set of points $\{\bs x ~|~ \bs a \times \bs x = \bs d\}$ or
$\{\bs x = \bs x_1 + \bs a \alpha ~|~ \alpha \in \mathbb{R}\}$ each form a
line, while the sets $\{\bs x ~|~ \bs A \cdot \bs x = D\}$ or $\{\bs x = \bs
x_2 + \bs b \beta + \bs c \gamma ~|~ \beta,\gamma \in \mathbb{R}\}$ each form
a plane.

{\bf \quad b)} What constraint between $\bs a$ and $\bs d$ is implicit in the
formula $\bs a \times \bs x = \bs d$?  Likewise, what relations between $\bs
b$, $\bs c$, and $\bs A$ must be satisfied if both equations describe the same
plane?

{\bf \quad c)} For the line and the plane, substitute $\bs x$ from the
parametric form into the relational form to show that they are consistent.
What are $\bs d$ and $D$ in terms of $\bs a, \bs x_1$ and $\bs A, \bs x_2$,
respectively?

{\bf \quad d)} Let us define $\bt a \equiv \bs A /( \bs a{\cdot} \bs A)$.  It
is parallel to $\bs A$ but {\it normalized} in the sense that $\bs a \cdot \bt
a = 1$.  Using the BAC-CAB rule, show that $\bs x = \bs a (\bt a\cdot \bs x) -
\bt a \times (\bs a \times \bs x)$ for any $\bs x$.  This is a non-orthogonal
projection of $\bs x$ into a vector parallel to the line and a vector parallel
to the plane.  Illustrate this projection and show which term corresponds to
each.

{\bf \quad e)} Using d), calculate the point $\bs x_0$ at the intersection of the
line and plane in terms of $\bs a, \bs d, \bs A, D$.

{\bf \quad f)} Verify e) by showing $\bs x_0$ satisfies the relational equation for
both the line and plane.

{\bf \quad g)} Let ~$\bt a = \frac{\bs b\times \bs c}{\bs a \cdot \bs b \times \bs
  c}$, ~$\bt b = \frac{\bs c\times \bs a}{\bs a \cdot \bs b \times \bs
  c}$,~and ~$\bt c = \frac{\bs a\times \bs b}{\bs a \cdot \bs b \times \bs
  c}$.  This definition of $\bt a$ is consistent with above if we require
$\bs A = \bs b \times \bs c$.  Calculate the nine combinations of $(\bs a, \bs
b, \bs c)^T \cdot (\bt a, \bt b, \bt
c)$.  %, i.e. $\bs a \cdot\bt a$, $\bs a \cdot\bt b$, \ldots.
Note that $(\bt a, \bt b, \bt c)$ is called the {\it dual} or {\it reciprocal}
basis of $(\bs a, \bs b, \bs c)$ because it is orthonormal to it \mbox{in the above
sense.}

{\bf \quad h)} The {\it contravariant components} of $\bs x$ are defined as the
components $(\alpha,\beta,\gamma)$ that satisfy the equation $\bs x = \bs a
\alpha + \bs b \beta + \bs c \gamma$.  In other words, $(\bs a, \bs b, \bs
c)$ is the contravariant basis.  Using $\bs x \cdot \bt a$, etc., calculate
the three contravariant components of $\bs x$ in terms of dot products.

\quad Bonus: How is this equivalent to the method of finding components by
matrix inversion, as in the
\href{http://www.pa.uky.edu/~crawford/phy416_fa14/notes_01_1a.pdf}{class
  notes}, first paragraph?

{\bf \quad i)} Find the {\it covariant components}
$(\tilde\alpha,\tilde\beta,\tilde\gamma)$ of $\bs x$.  As above, they are
defined as the components $(\tilde\alpha,\tilde\beta,\tilde\gamma)$ which
satisfy the equation $\bs x = \bt a \tilde\alpha + \bt b \tilde\beta + \bt c
\tilde\gamma$; i.e. $(\bt a, \bt b, \bt c)$ is the covariant basis.

\quad Bonus: How does everything simplify if $(\bs a,\bs b,\bs c)$ are an
orthonormal basis?

\newpage

{\bf 2. Clifford algebra}.  The complete 3-vector algebra including dot and
cross products can be implemented using the identity ($I$) as the unit scalar
and Pauli matrices ($\bs\sigma$) as unit vectors:
\begin{equation}
  1           = I           \equiv \pmatrix{1 & 0 \cr 0 & 1 } \quad
  \hat{\bs x} = \bs\sigma_x \equiv \pmatrix{0 & 1 \cr 1 & 0 } \quad
  \hat{\bs y} = \bs\sigma_y \equiv \pmatrix{0 & -i \cr i & 0 } \quad
  \hat{\bs z} = \bs\sigma_z \equiv \pmatrix{1 & 0 \cr 0 & -1 }.
  \label{eq:sigma}
\end{equation}
The dot and cross products can be represented by matrix multiplication, as in
the formula
\begin{equation}
  \bs\sigma_i\bs\sigma_j = I
  (\bs\sigma_i\cdot\bs\sigma_j) + i (\bs\sigma_i\times\bs\sigma_j) = I \delta_{ij} +
  i\epsilon_{ijk}\bs\sigma_k,
  \label{eq:dotcross}
\end{equation}
where the dot and cross products are interpreted in the usual sense of unit
vectors.  Note the difference between the imaginary $i$ and the index $_i$.
Often $I$ is implicit, so that scalars are actually multiples of the identity
matrix.  This algebra generalizes to space-time, where the Dirac matrices
$\gamma_\mu$ are used instead of the Pauli matrices.

{\bf \quad a)} Verify this formula for all nine products
$\bs\sigma_i\bs\sigma_j$. Which products are symmetric and which are
antisymmetric?  While in general, a product can have both symmetric and
antisymmetric parts, this partition of symmetry into $i=j$ and $i\neq j$ is
the defining feature of a Clifford algebra.

{\bf \quad b)} Show that any linear product $\bs a\circ \bs b$, can be
decomposed into the sum $\bs a\circ \bs b = \{\bs a\circ \bs b\} + \langle \bs
a\circ \bs b\rangle$ of symmetric $\{\bs a\circ \bs b\} \equiv \frac12(\bs
a\circ \bs b + \bs b\circ \bs a)$ and antisymmetric $\langle \bs a\circ \bs
b\rangle \equiv \frac12(\bs a\circ \bs b - \bs b\circ \bs a)$ parts, with
respect to exchange of $\bs a$ and $\bs b$.  Show that $\langle \bs a\circ \bs
a \rangle = 0$ always.  Why are the diagonal elements of an antisymmetric
matrix zero?  Apply this decomposition to the product
$\bs\sigma_i\bs\sigma_j$.

{\bf \quad c)} The imaginary $i$ in the above formula is not present in the
ordinary cross product.  It distinguishes [axial] pseudovectors from [polar]
vectors.  Using part a), calculate the value of the pseudoscalar
$\bs\sigma_i\bs\sigma_j\bs\sigma_k$ for all values of $i \neq j \neq k$.  What
is the dimension of this space?  Use this result to distinguish between
pseudoscalars and scalars.  What is the analog of this triple product in terms
of vector products?

Bonus: These matrices have the additional capability of adding scalars and
vectors, which is not meaningful in basic vector spaces.  Use this property to
simultaneously solve the two equations $\bs a \cdot \bs x = D$ and $\bs a
\times \bs x = \bs d$ from Problem \#1 (with $\bs A=\bs a$) as a single matrix
equation.

\vspace{.166 in}

Also, Griffiths 3ed or 4ed chapter 1, problems \#%
1,  %p4,  distributive \qquad
%2. associative p4\\
4,  %p7,  normal to the plane \qquad
5,  %p8,  BAC-CAB \qquad
%6,  % Jacobi identity
8,  %p12, orthogonal \qquad
10, %p12, transformations pseudo \qquad
12, %p15, hill gradient \qquad
15, %p18, calc divergence \qquad
16, %p18, rhat/$r^2$\qquad
18. %p20, calc curl\qquad

\end{document}
