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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#2, Rev. B, due Wednesday, 2014-01-29 }
\end{center}

\vspace{.2 in}

{\bf 1. Coordinate basis}

The {\it coordinate basis} $\bs b_i=\partial \bs r/\partial q^i$, {\it
  reciprocal basis} $\bs b^i=\bs\nabla q^i=\partial q^i/\partial \bs r$, and
{\it unit basis} $\hat e_i = \bs b_i / h_i$ (where the {\it scale factor} is
$h_i=|\bs b_i|$), are the natural bases to describe components of a vector
field in a curvilinear coordinate system $(q^1,q^2,q^3)$.  Note that these
basis vectors change from one point to the next.  Calculate each of the
following in both cylindrical and spherical coordinates.

{\bf \quad a)} Calculate $(\bs b_s, \bs b_\phi, \bs b_z)$ as a functions of
$q^i=(s,\phi,z)$ and $(\bs b_r, \bs b_\theta, \bs b_\phi)$ as functions of
$(r,\theta,\phi)$.

{\bf \quad b)} Calculate the resulting scale factors $h_i$ to get unit vectors.
Write out the basis transformation matrix in terms of these vectors,
i.e. $(\hat{\bs s}, \hat{\bs \phi}, \hat{\bs z}) = (\hat{\bs x}, \hat{\bs y},
\hat{\bs z}) R$.

{\bf \quad c)} Construct the transformation matrices between unit bases, by
considering rotations $R_{\hat{\bs z}}(\phi)$ (rotation about an angle $\phi$
about the $z$-axis) and $R_{\hat{\bs \phi}}(\theta)$ and compare with part b.

{\bf \quad d)} Calculate the reciprocal vectors $(\bs b^s, \bs b^\phi, \bs b^z)$ and
$(\bs b^r, \bs b^\theta, \bs b^\phi)$. \mbox{Show that $\bs b_i\cdot \bs b^j =
  \delta_i{}^j$ in general.}

{\bf \quad e)} Calculate the line element $\bs{dl} = \bs b_i dq^i$, area
element $\bs{da} = \frac12 \bs{dl}\times\bs{dl}$, and volume element $d\tau =
\frac16\bs{dl}\cdot\bs{dl}\times\bs{dl}$.  Note that since differentials
anticommute, their cross products are not zero.

{\bf \quad f)} Bonus: Calculate the metric $g_{ij} = \bs b_i \cdot \bs b_j$ and
$g^{ij} = \bs b^i \cdot \bs b^j$ in terms of $h_i$.  Show they
\mbox{are reciprocal}.

{\bf \quad g)} Bonus: Calculate the derivatives of the basis vectors along
coordinate lines, $\bs\Gamma_{\ij} = \Gamma_{\ij}{}^k \bs b_k = \partial \bs
b_i / \partial q^j = \partial^2 \bs r / \partial q^i \,\partial q^j$.  These
{\it Christoffel symbols} and are needed to calculate derivatives of vectors
in curvilinear coordinates.

\vspace{.166 in}

Also, Griffiths 3ed[4ed] chapter 1, problems \\ \#%
13[13], %p15, sep vec \qquad
19[20], %p20, curl,div=0 \qquad
21[22], %p22, a.del b \qquad
25[26], %p24, laplacian \qquad
38[39], %p42, div thm, pt field \qquad
39[40], %p43, div sph \qquad
40[41], %p43, grad sph field \qquad
42[43]. %p45, div cyl \qquad


\end{document}


\begin{center}
  {\bf Extra problems NOT assigned for credit }
\end{center}


{\bf 2. Affine transformation}.  The affine space (points not vectors) is
distinguished its lack of significance of the origin.  Thus the only two
meaningful operations on points are weighed ({\it affine}) combinations
$w_1x_1 + w_2x_2 + \ldots$ of points, where the total weight
$w=w_1+w_2+\ldots$ equals to 0 (vector difference of points) or 1 (weighted
average of points).  This can be formalized by adding a fourth dimension
(weight) which equals 0 for vectors and 1 for points.
                         
{\bf \quad a)} Show that matrix transformation $T={\scriptsize\pmatrix{R & \bs
    t \cr \bs 0 & 1}}$ transforms a point ${\scriptsize\pmatrix{\bs r\cr 1}}$
by the rotation $R$ about the origin and the translation $\bs t$.  Which
operation is done first, in the active sense?  What operations are done by the
same transformation on vectors ${\scriptsize\pmatrix{\bs v \cr 0}}$?

{\bf \quad b)} A robot actuation can be described by the Denavit-Hartenberg
parameters $(d,\theta,r,\alpha)$, which are described in the video clip
\href{http://tekkotsu.no-ip.org/movie/dh-sd.mp4}{http://tekkotsu.no-ip.org/movie/dh-sd.mp4},
as a translation $d$ along the z-axis, a rotation $\theta$ about the $z$-axis,
a translation $r$ along the new $x$-axis, and a rotation $\alpha$ about the
new $x$-axis.  Build a matrix for each of the above transformations and
compose them into a single transformation matrix.  Be careful to use the
correct order of composition.

{\bf \quad c)} Given $n$ measurements $x_i\pm \delta x_i$ of a quantity $x$ with
gaussian (rms) error $\delta x$, the most accurate estimate of the average
$\bar x$ is calculated by weighting each point by $w_i = \delta^{-2}x_i =
1/(\delta x_i)^2$.  The uncertainty in the average $\delta \bar x$ can be
obtained from the sum of weights, $\delta^{-2}\bar x = w = \sum_i w_i$.  Show
how these can be calculated by adding 2-d weighted vectors $w_i(x_i,1)$ for
each measurement.

{\bf \quad d)} Likewise show how weighted (4-d) vectors $m_i(\bs r_i,1)$ can be
added or integrated to calculate the center of mass of an extended object.


%\newpage


{\bf 3.} The {\bf complex plane} is a vector space consisting of the real and
imaginary components $\bs z=(x,y)$ of complex numbers $z=x + iy$.  In this
exercise we will compare complex and vector rotations.

{\bf \quad a)} Which two complex numbers form the basis of the complex plane for the
\mbox{above components}?

{\bf \quad b)} Show that the dot and cross product of two points $\bs z_1=(x_1,y_1)$
and $\bs z_2=(x_2,y_2)$ can be represented by the real and imaginary parts of
the complex product $z_1^* z_2 = \bs z_1\cdot \bs z_2 + i (\bs z_1 \times \bs
z_2)_z$, where the $z^*=x-iy$ is the {\it complex conjugate} of $z$.  Identify
the symmetric and antisymmetric parts of this product.  Note 1: this shows
that the vector and complex magnitudes coincide, $|z|^2 = z^*z = \bs z \cdot
\bs z$.  Note 2: for vectors in the $xy$-plane, only the $z$-component of the
cross product is nontrivial, thus the cross-product is not actually a vector
in the original space!

{\bf \quad c)} Show graphically (by examining the coordinates) that the operation of
multiplying by $i$ rotates a point $z$ by the angle 90$^\circ$ CCW about the
origin to become the point $iz$.

{\bf \quad d)} Show graphically that the operation $z \to (1+i\, d\theta) z = z +i
z\, d\theta$ produces a new point with the same magnitude (assuming
$d\theta\,^2=0$), but rotated CCW by the angle $d\theta$.

{\bf \quad e)} Finite rotations can be obtained by infinite composition of
these infinitessimal rotations, which can be solved as follows: Formally
integrate the equation $dz = i z\, d\theta$ with the initial condition
$z(\theta=0)=z_0$ to obtain the rotation formula $z(\theta)=R_\theta z_0 = z_0
e^{i\theta}$.  Note that the solution is expressed in the identity
{$\displaystyle e^x=\lim_{n\to\infty}\textstyle(1+\frac{x}{n})^n$}.

{\bf \quad f)} Collect the real and imaginary terms of the Taylor expansion of
$e^{i\theta}$ to show \mbox{$e^{i\theta}=\cos\theta + i \sin\theta$}.

{\bf \quad g)} Show that complex multiplication by $i$ is equivalent to the vector
operation $\hat {\bs z} \times$.

{\bf \quad h)} Determine the matrix representation $M_z$ of the operator $\hat {\bs
  z} \times$, such that $M_z \bs r = \hat {\bs z} \times \bs r$.  Do the same
for $M_x$ and $M_y$.  For any vector $\bs v$, show that $\bs v\times = \bs v
\cdot \bs M = v_x M_x + v_y M_y + v_z M_z$ and derive the matrix
representation of $\bs v \times$.  Note that each of these matrices is
antisymmetric, and the stack of three matrices $\bs M$ (cross product tensor)
is completely antisymmetric, \mbox{with components $\varepsilon_{ijk}$.}

{\bf \quad i)} Calculate $M_x^2$, $M_y^2$, $M_z^2$, and $M_xM_y$.  Note that
$(M_x,M_y,M_z)$ are similar to Hamilton's qua\-ter\-nions $(i,j,k)$, which
extend the rotational properties of complex numbers to 3 dimensions.
%%% MxMy-MyMx   or   zhat x + zhat zhat cdot

{\bf \quad j)} Using identifications above, show that matrix for a vector rotation
of angle $\theta$ CCW in the $xy$-plane can be written $R_\theta = e^{M_z
  \theta} = I \cos\theta + M_z \sin\theta = {\scriptsize\pmatrix{\cos\theta &
    -\sin\theta\cr \sin\theta & \cos\theta}}$, neglecting $z$-components.  In
general, the matrix for a CCW rotation of angle $v=|\bs v|$ about $\hat{\bs
  v}$ can be written $R_{\bs v} = %(e^{\bs M \cdot \bs v} =
I \cos v + \bs M{\cdot} \hat{\bs v} \sin v + \hat{\bs v} \hat{\bs v}^T (1-\cos
v)$.  The third term handles the non-rotating parallel component.

{\bf \quad k)} Calculate the eigenvalues and eigenvectors of
$M_z={\scriptsize \pmatrix{0&-1\cr 1&0}}$ to show that $M_z = VWV^\dagger =
\frac12 {\scriptsize\pmatrix{1&1 \cr i&-i}} {\scriptsize\pmatrix{-i&0\cr0&i}}
{\scriptsize\pmatrix{1&-i\cr1&i}}$ and $e^{M_z \theta} = V
e^{W\theta}V^\dagger = \frac12 {\scriptsize\pmatrix{1&1 \cr i&-i}}
{\scriptsize\pmatrix{e^{-i\theta}&0\cr0&e^{i\theta}}}
{\scriptsize\pmatrix{1&-i\cr1&i}}$.  Multiply this out to verify part j).
Thus real symmetric matrices have real eigenvalues while antisymmetric
matrices have Tr=0 and imaginary eigenvalues.  The exponential of a symmetric
matrix remains symmetric, while the exponential of an antisymmetric matrix is
an orthogonal rotation with Det=1 and unit length eigenvalues.  The
symmetric/antisymmetric decomposition of matrices is frightfully similar to
the real/imaginary decomposition of complex numbers.


\end{document}
