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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#2, Rev. A, due Wednesday, 2014-09-17 }
\end{center}

\vspace{.2 in}

{\bf 1. Curvilinear coordinates}: the {\it coordinate basis} $\bs b_i
\equiv \partial \bs r/\partial q^i$ and {\it reciprocal basis} $\bs b^i \equiv
\bs\nabla q^i=\partial q^i/\partial \bs r$ are the two natural dual bases to
describe components of a vector field in curvilinear coordinates
$(q^1,q^2,q^3)$.  For orthogonal systems, we use the single {\it orthonormal
  basis} $\bh e_i = \bs b_i / h_i = \bs b^i h_i$, where $h_i=|\bs b_i| =
1/|\bs b^i|$ is a {\it scale factor}.  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 $\bs b_i = (\bs b_s, \bs b_\phi, \bs b_z)$ as
functions of $q^i=(s,\phi,z)$ and $(\bs b_r, \bs b_\theta, \bs b_\phi)$
\mbox{as functions of $(r,\theta,\phi)$.}

{\bf \quad b)} Calculate the resulting scale factors $h_i$ to get unit vectors
in both coordinate systems.  Write out the basis transformation matrix $R$ for
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 $R_z(\phi)$ (rotation by an angle $\phi$ about the
$z$-axis) and $R_\phi(\theta)$ 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 metric $g_{ij} = \bs b_i \cdot \bs b_j$ and
reciprocal $g^{ij} = \bs b^i \cdot \bs b^j$ in terms of $h_i$.

{\bf \quad f)} Calculate the general line element $\bs{dl} = \bs b_i dq^i$,
area element $\bs{da} = \frac12 \bs{dl}\times\bs{dl}$, and volume element
$d\tau = \frac16\bs{dl}\cdot\bs{dl}\times\bs{dl}$ in cylindrical and spherical
coorindates.  These products are not zero but have 2 [or 6] identical terms,
because differentials anticommute.

{\bf \quad g)} Apply the formulas $\nabla f = \frac{\bh
  e_i}{h_i}\frac{\partial}{\partial q^i}f$, $\nabla\times\bs A = \frac{\bh
  e_i}{h_jh_k}\frac{\partial}{\partial q^j} h_k A_k$, and $\nabla\cdot\bs B =
\frac{1}{h_1h_2h_3} \frac{\partial}{\partial q^k} h_ih_jB_k$, for $i,j,k$
cyclic, to cylindrical and spherical coordinates.  Compare with
\mbox{Griffiths, inside front cover.}

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 derivative of vector fields in
curvilinear coordinates.


{\bf 2.} The {\bf Hodge dual operator} $*$ converts between scalar/vector and
pseudo-scalar/vector differentials according to the definitions \mbox{$* 1 =
  dx\,dy\,dz = d\tau$},\quad $*d\tau = *dx\,dy\,dz = 1$,\quad
$*dx^i={\textstyle\frac12}\epsilon_{ijk}dx^j dx^k$, and $*dx^i dx^j =
\epsilon_{ijk} dx^k$.  The factor $\frac12$ removes identical terms.

{\bf \quad a)} Calculate $(*dx, *dy, *dz)$ and $(*dy\,dz, *dz\,dx, *dx\,dy)$
to show that $*\bs{dl}=\bs{da}$ and $*\bs{da}=\bs{dl}$.  What is the
equivalent operation in HW1 \#2?

{\bf \quad b)} The {\it codifferential operator} $\delta$ is defined by
$\delta \equiv (-1)^p\,{*}\,d\,*$, where $p$ is the dimension of the
differential acted upon ($p=0$ for scalars, $p=1$ for $\bs{dl}$, $p=2$ for
$\bs{da}$, and $p=3$ for $d\tau$).  Show \\
\begin{tabular}{rlrl}
 \qquad i)   & $\delta f=0$ for all scalar functions $f$, &
 \qquad iii) & $\delta (\bs{A}\cdot\bs{da}) = (\nabla\times\bs{A})\cdot\bs{dl}$, and \\
 \qquad ii)  & $\delta (\bs{A}\cdot\bs{dl})=-\nabla\cdot\bs{A}$ for a vector field $\bs{A}$, &
 \qquad iv)  & $\delta( f\, d\tau) = -(\nabla f)\cdot\bs{da}$.
\end{tabular} \\
Note that $\delta$ is the {\it adjoint} of $d$ in that it switches between
divergence and curl, etc.

{\bf \quad c)} Show that $-\nabla^2 f = -\delta d f = -(d\delta+\delta d)f$
for the Laplacian of a scalar function $f$, and likewise, $(\nabla^2 \bs
A)\cdot\bs{dl} = -(d\delta+\delta d) (\bs A \cdot \bs{dl})$ for the Laplacian
of a vector field.  Note that by mixing differentials and codifferentials, it
possible to generate nontrivial second derivatives.  Interpret each
[co]differential $(d, \delta)$ in terms of its corresponding vector derivative
($\nabla$).

\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}
