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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#1, Rev. A, due Wednesday, 2014-01-22 }
\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 \\
    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 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?

{\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 $\bt a = \bs A /( \bs a{\cdot} \bs A)$.  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 projection of $\bs
x$ into a vector parallel to the line plus a vector parallel to the plane.
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 How is this equivalent to the method of finding components by
matrix inversion, as in the
\href{http://www.pa.uky.edu/~crawford/phy416_fa11/notes_01_1.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.

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

% \newpage

% {\bf 2. Conic sections} can be represented as {\it quadratic forms}
% conveniently with matrices.

% {\bf \quad a)} Show that the equation $(\bs r - \bs a)\cdot\bs r = \bs 0$
% defines a sphere, where $\bs r=(x,y,z)$ and $\bs a=(a_x,a_y,a_z)$.  What is
% the interpretation of $\bs a$?
% % Thale's theorem (corollary of inscribed or central angle theorem)

% {\bf \quad b)} Show that the 2-d matrix equation $\bs r^T A \bs r + \bs b^T\bs r
% +f=0$ defines a conic section (an elipse, circle, parabola, or hyperbola),
% where $\bs r=\pmatrix{x\cr y}$, and %$\bs r_0=\pmatrix{x_0\cr y_0}$,
% $A=\pmatrix{a&b\cr b&c}$, $\bs b=\pmatrix{d\cr e}$, and $f$ are constant
% parameters.  Why is the antisymmetric part of $A$ irrevelant?  Rewrite a) in
% this form.

% {\bf \quad c)} What is the interpretation of the eigenvectors and eigenvalues of the
% matrix $A$?  How does this generalize to three dimensions?

% {\bf \quad d)} Bonus: derive formulas to discriminate between elipses, circles,
% \mbox{parabolae, and hyperbolae.}

% {3. \bf Clifford algebra}.  The complete 3-vector algebra including dot and
% cross products can be implented using the identity ($I$) for the unit scalar and Pauli
% matrices ($\bs\sigma$) for the 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}
% Note the difference between the imaginary $i$ and the index $_i$.  Often $I$
% is implicit, not written.

% {\bf \quad a)} Show that any 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.  Thus, the diagonal elements of an antisymmetric matrix are zero.

% {\bf \quad b)} Verify that $\{\bs\sigma_i\bs\sigma_j\}=I
% \bs\sigma_i\cdot\bs\sigma_j = I \delta_{ij}$, so $\{\bs\sigma_x, \bs\sigma_y,
% \bs\sigma_z\}$ is an orthonormal basis.

% {\bf \quad c)} Verify that $\langle \bs\sigma_i\bs\sigma_j \rangle = i \bs\sigma_i
% \times \bs\sigma_j = i \epsilon_{ij}{}^k\bs\sigma_k$, so $\bs\sigma_i \times
% \bs\sigma_j$ is the ordinary cross product.

% {\bf \quad d)} Identify the dot and cross products as symmetric and
% antisymmetric parts of the matrix product (identify terms of
% $\bs\sigma_i\bs\sigma_j = \{\bs\sigma_i\bs\sigma_j\} + \langle
% \bs\sigma_i\bs\sigma_j \rangle$ with Eq.~\ref{eq:dotcross}).  Which terms are
% zero if $i=j$? $i\neq j$?  Thus the product $\bs\sigma_i\bs\sigma_j$ is
% already either symmetric or antisymmetric depending \mbox{on $i$ and $j$.}


% {\bf \quad e)} The imaginary $i$ in the above formula is not present in the ordinary
% cross product.  This distinguishes [axial] pseudovectors from [polar] vectors.
% Calculate the value of the pseudoscalar $\bs\sigma_i\bs\sigma_j\bs\sigma_k$
% for all values of $i,j,k$ using the results above to find the distinction
% between pseudoscalars and scalars.  What is the analog of this tripple product
% in terms of vector products?

% {\bf \quad f)} 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, 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}
