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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#3 (Rev. E) due Friday, 2011-09-23 }
\end{center}

\vspace{.1 in}

1. a) Derive formulas for $\nabla f$, $\nabla\times\bs A$, and $\nabla\cdot\bs
B$ in an orthogonal curvilinear coordinate system from the definitions
$df=\nabla f\cdot\bs{dl}$, $d(\bs A\cdot \bs{dl})=(\nabla\times\bs
A)\cdot\bs{da}$, and $d(\bs B\cdot \bs{da})=(\nabla\cdot\bs B)\cdot d\tau$
(see \href{http://www.pa.uky.edu/~crawford/phy416_fa11/notes_01_2.pdf}{class
  notes}), using the differential operator $d=dq^i \partial_i$ and
differential elements $\bs{dl} = \bh{e}_ih_idq^i$, $\bs{da} = \frac12
\bs{dl}\times\bs{dl}$, and $d\tau = \frac13\bs{dl}\cdot\bs{da}$.  The results
are similar to the cartesian formulas, but with extra scale factors.

\vspace{-1.2mm}

\quad b) Expand the formulas from part a) in cylindrical and spherical
coordinates to obtain the formulas in the front cover of Griffiths.

\vspace{-1.2mm}

\quad c) Show that the trival 2$^{\rm{nd}}$ derivatives $\nabla\times\nabla f$
and $\nabla\cdot\nabla\times \bs A$ are special cases \mbox{of the $d^2=0$
rule.}

\vspace{-1.2mm}

\quad d) The Hodge $*$ operator is defined to act on differentials in 3-space
by the equations \mbox{$* 1 = dx\,dy\,dz$},\quad
$*dx^i={\textstyle\frac12}\epsilon_{ijk}dx^j dx^k$,\quad $*dx^i dx^j =
\epsilon_{ijk} dx^k$, and \quad $*dx\,dy\,dz = 1$.  It converts between
differentials for scalars $\leftrightarrow$ pseudoscalars and vectors
$\leftrightarrow$ pseudovectors.  Calculate $(*dx, *dy, *dz)$, and $(*dy\,dz,
*dz\,dx, *dx\,dy)$ to show that $*1=d\tau$, $*\bs{dl}=\bs{da}$,
$*\bs{da}=\bs{dl}$, and $*d\tau=1$. What was the equivalent operation in HW1
\#2?

\vspace{-1.2mm}

\quad e) The {\it codifferential operator} $\delta$ is defined by $\delta
\equiv (-1)^p\,{*}\,d\,*$, where $p$ is the dimension of the differential it acts on
($p=0$ 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 a scalar function $f$, &
 \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}\\
Notice that $\delta$ is the {\it adjoint} of $d$ in that it swaps divergence and
curl, etc.

\vspace{-1.2mm}

\quad f) Show that $\nabla^2 f = -\delta d f = -(d\delta+\delta d)f$ for a
scalar function $f$. Note that by mixing differentials and codifferentials, it
possible to generate nontrivial second derivatives.

\vspace{-1.2mm}

\quad g) Show that the Laplacian of a vector field can be also written as
$(\nabla^2 \bs A)\cdot\bs{dl} = -(d\delta+\delta d) (\bs A \cdot \bs{dl})$.
Interpret each term in terms of vector derivatives involving $\nabla$.

2. a) Plot a 2-D graph of the function $h(x,y)=2xy$ by drawing level curves in
the $xy$-plane.

\vspace{-1.2mm}

\quad b) Plot a 3-D graph of the 2-D function $h(x,y)$, ie. the surface $\{
(x,y,z) ~|~ z=h(x,y) \}$.

\vspace{-1.2mm}

\quad c) Calculate the gradient of $h$ at all points, and the divergence and
curl of the gradient.  

\vspace{-1.2mm}

\quad d) By constructing a 3-D function $g(x,y,z)$ of which $z=h(x,y)$ is a
level surface, calculate a normal vector to the 3-D graph $z=h(x,y)$ at all
points on the surface.  Compare your answer to the normal of the surface
calculated with the cross product (calculate $\bs{da}$ on the surface).

\vspace{-1.2mm}

\quad e) Show that there is a linear coordinate transformation
$(x,y)\to(x',y')$ that transforms the function $h'(x',y')=y'^2-x'^2$ into
$h(x,y)$.  We will find that both of these are \mbox{$m=2$ cylindrical multipoles.}

\vspace{-1.2mm}

\quad f) Show that $h_1(x,y)=2xy$ and $h_2(x,y)=y^2-x^2$ form a basis for all
functions which can be generated from $h_1$ and $h_2$ by rotating the
coordinates $(x,y)$ by an angle $\theta$ via the transformation $\bs r' =
R_\theta \,\bs r$, i.e. $\pmatrix{x'\cr y'} = \pmatrix {c_\phi &-s_\phi\cr
  s_\phi & c_\phi}\pmatrix{x\cr y}$, where $\matrix{ c_\phi=\cos\phi \cr
  s_\phi = \sin\phi}$.

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3. a) Integrate $\int_{\mathcal{S}} \bs v \cdot \bs{da}$ where $\bs v = \bh
x\, x^2 + \bh y\, 2yz + \bh z\, xy$, and $\mathcal{S}$ is the parallelogram in
the figure to the right.

% \vspace{-1.2mm}

\quad b) Integrate $\oint_{\partial\mathcal{S}} \bs v \cdot \bs{dl}$ along the
boundary of $\mathcal{S}$.

% \vspace{-1.2mm}

\quad c) Verify Stoke's theorem for the integral in part b.


\vspace{.166 in}

Also, Griffiths Ch.~1, \#\mbox{%
%14, %p16, del vector \qquad
%17, %p18, div scalar \qquad
28, %p28, 3 line integrals \qquad
29, %p29, div thm \qquad
31, %p31, FTC \qquad
32, %p33, div thm \qquad
33, %p36, stokes thm \qquad
34, %p36, corollary \qquad
35, %p38, int parts \qquad
38, %p42, div thm, pt field \qquad
39, %p43, stokes sph \qquad
40, %p43, grad sph field \qquad
42, %p45, div cyl \qquad
43, %p49, delta integral \qquad
44, %p49, delta scale integral \qquad
45, %p49, delta derivative theta \qquad
46, %p52, pt charge \qquad
47, %p52, int 3d delta \qquad
48, %p52, int parts \qquad
60, %p56, thms by parts \qquad
61, %p57, area integral \qquad
62. %p57, div central power law \qquad
}
\end{document}
