\documentclass[11pt,titlepage]{article}
\usepackage[margin=1in,noheadfoot]{geometry} % See geometry.pdf to learn the layout options. There are lots.
%\geometry{letterpaper}               % ... or a4paper or a5paper or ... 
\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{hyperref}

\newcommand{\e}[1]{\times 10^{#1}}
\newcommand{\q}{\mathrm}
\newcommand{\where}[1][where]{\qquad\mbox{#1}}
\newcommand{\bs}[1]{\boldsymbol{#1}}            % italics, works well
\newcommand{\bt}[1]{\tilde{\boldsymbol{#1}}}

\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#1, due Friday, 2011-09-02 }
\end{center}

\vspace{.3 in}

1. a) Show graphically that the following equations
define a set of points $\{\bs x\}$ on a line or plane,
\begin{center}
  \begin{tabular}{r@{\qquad}l@{\qquad}l}
    & relational & parametric \\
    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.

\quad b) What constraints between $\bs a$ and $\bs d$ are implicit in the
above formulas, assuming the relational and parametric equations both describe
the same line?  Likewise, what are the constraints between $\bs b$, $\bs
c$, and $\bs A$ if both equations describe the same plane?

\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 $\bs x_1$ and $\bs x_2$?

\quad d) Let us define $\bt a = \bs A /( \bs a{\cdot} \bs A)$.  It is parallel
to $\bs A$ but ``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 projection of $\bs
x$ into a vector parallel to the line plus a vector parallel to the plane.
Show which term corresponds to each.

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

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

\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 that
$\bs A = \bs b \times \bs c$.  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 in the above
sense.

\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. \\
Bonus: How is this equivalent to the method of finding components by matrix
inversion found in the
\href{http://www.pa.uky.edu/~crawford/phy416_fa11/notes_01_1.pdf}{class
  notes}, first paragraph?

\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 j) How does everything simplify if $(\bs a,\bs b,\bs c)$ are an
orthonormal basis?  \vspace{.166 in}

\newpage

2. The complete vector algebra including dot and cross products can be
impleted using the Identity and Pauli matrices for the unit scalar and unit
vectors respectively,
\begin{equation}
  1           = I        = \pmatrix{1 & 0 \cr 0 & 1 } \quad
  \hat{\bs x} = \sigma_x = \pmatrix{0 & 1 \cr 1 & 0 } \quad
  \hat{\bs y} = \sigma_y = \pmatrix{0 & -i \cr i & 0 } \quad
  \hat{\bs z} = \sigma_z = \pmatrix{1 & 0 \cr 0 & -1 }.
  \label{eq:sigma}
\end{equation}
Both both the dot and cross product are implemented by matrix multiplication.
We will prove and work with the formula
\begin{equation}
  \sigma_i\sigma_j = I
  (\sigma_i\cdot\sigma_j) + i (\sigma_i\times\sigma_j) = I \delta_{ij} +
  i\epsilon_{ijk}\sigma_k.
  \label{eq:dotcross}
\end{equation}
Note the difference between the imaginary $i$ and
the index $_i$.  Often the $I$ is implied and omitted from formulas.

\quad a) Show that for any product $a\circ b$, the combination $\{a\circ b\}
\equiv \frac12(a\circ b + b\circ a)$ is symmetric and the combination $\langle
a\circ b\rangle \equiv \frac12(a\circ b - b\circ a)$ is antisymmetric which
respect to exchange of $a$ and $b$.

\quad b) Show that $\langle a\circ a \rangle = 0$ always.  Equivalently, the
diagonal elements of an antisymmetric matrix are zero.

\quad c) Show that $a\circ b = \{a\circ b\} + \langle a\circ b\rangle$.

\quad d) Thus we can isolate the dot and cross products as symmetric and
antisymmetric portions of the single matrix product.  Identify terms of
$\sigma_i\sigma_j = \{\sigma_i\sigma_j\} + \langle \sigma_i\sigma_j \rangle$
with Eq.~\ref{eq:dotcross}.

\quad e) Verify that $\{ \hat{\bs x} ,\hat{\bs y} , \hat{\bs z} \}$ form an
orthonormal basis using the dot product: $\{\sigma_i\sigma_j\}=I
\sigma_i\cdot\sigma_j$.

\quad f) Show that $\langle \sigma_i\sigma_j \rangle = i \hat{\bs e}_i \times
\hat{\bs e}_j$ for all six combinations of $i,j$.  Thus $\sigma_i\sigma_j$ is
antisymmetric for $i\neq j$.

\quad g) The imaginary $i$ in the above formula is not present in the ordinary
cross product.  This distinguishes pseudovectors from vectors.  Calculate the
value of the pseudoscalar $\sigma_i\sigma_j\sigma_k$ for all values of $i,j,k$
using the results above to find the distinction between pseudoscalars and
scalars.

\quad h) 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 as a single matrix equation, for
the case that $\bs A=\bs a$).

3. a) Show that the equation $(\bs r - \bs a)\cdot\bs r = \bs 0$ defines a
sphere.  What is the interpretation of $\bs a$?

\quad b) Show that the 2-d matrix equation $(\bs r-\bs{r_0})^T A (\bs r- \bs
r_0)=d$ defines a conic section (an elipse, circle, parabola, or hyperbola),
where $\bs r=\pmatrix{x\cr y}$, $\bs r_0=\pmatrix{x_0\cr y_0}$, and
$A=\pmatrix{a&b\cr b&c}$.  All but $\bs r$ are constant parameters.

\quad c) What is the interpretation of the eigenvectors and eigenvalues of the
matrix $A$?

\quad d) Why doesn't $A$ need an antisymmetric part?

\vspace{.166 in}

Also, Griffiths chapter 1, problems 1, 4, 5, 8, 10, 12, 15, 16, 18.

% 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}
