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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#10, Rev. A, due Monday, 2025-04-14 }
\end{center}

\vspace{.3 in}

% ~/hw/phy416_fa11/ps_03.tex #1a, ~/hw/phy416_fa14/ps_02.tex #1g
{\bf 1.} The {\bf gradient}, {\bf curl}, and {\bf divergence} are all applications of $d$ in different dimensions.  They are defined as\  $df=\bs\nabla f\cdot\bs{dl}$,\ \  $d(\bs A\cdot \bs{dl})=(\bs\nabla\times\bs A)\cdot\bs{da}$,\ \   $d(\bs B\cdot \bs{da})=(\bs\nabla\cdot\bs B)\cdot d\tau$, respectively, in terms of the differential operator $d=dq^{i}\partial_{i}$ and elements\  $\bs{dl} = \bh{e}_ih_idq^i$,\ \  $\bs{da} = \frac12 \bs{dl}\times\bs{dl}$,\ \  $d\tau = \frac13\bs{dl}\cdot\bs{da}$.

{\bf \quad a)} [{\it bonus}: Apply these definitions to scalar $f$, polar vector $\bs A$, and axial vector $\bs B$ fields to obtain

$\bs\nabla f 
= \frac{df}{\bs{dr}}
= \frac{\bh e_i}{h_i}\frac{\partial}{\partial q^i}f$,\quad
$\bs\nabla\times\bs A 
= \frac{d\, \bs{dr}\cdot}{\bs{da}}\bs A
= \frac{\bh e_i}{h_jh_k}\frac{\partial}{\partial q^j} h_k A_k$,\quad $\bs\nabla\cdot\bs B 
= \frac{d\, \bs{da}\cdot}{\bs{d\tau}}\bs B
= \frac{1}{h_1h_2h_3} \frac{\partial}{\partial q^k} h_ih_jB_k$,
where $i,j,k$ are cyclic and $\frac{1}{\bs{dr}}$ is the inverse transformation of $\bs{dr}\cdot$ from $\{\bh x, \bh y, \bh z\}$ to $\{dx,dy,dz\}$, etc.]

{\bf \quad b)} Expand these definitions in Cartesian, cylindrical, and spherical coordinates (compare with \href{https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates}{this table}).

{\bf \quad c)} Calculate i) the gradient of $f(r,\theta,\phi)= r^\ell P_\ell(\cos(\theta))$, \href{https://en.wikipedia.org/wiki/Legendre_polynomials#Recurrence_relations}{using} $\frac{x^{2}-1}{\ell}P'_{\ell}(x)=xP_{\ell}(x)-P_{\ell-1}(x)$,

ii) the curl of $\bs A(x,y,z) = (\bh y x - \bh x y)/(x^{2}+y^{2})^n$ in Cartesian and cylindrical coordinates, and iii) the
% \hat\phi/\rho^..
divergence of $\bs B(x,y,z) = (\bh x x+\bh y y+\bh z z)/(x^2+y^2+z^2)^n$ in Cartesian and spherical coordinates.
% \hat\r/r^..

{\bf \quad d)} Derive the formulas for $d^{2} f$ and $d^{2} \bs A \cdot \bs{dl}$ in terms of their partial derivatives to show that the trivial 2$^{\rm{nd}}$ derivatives $\bs\nabla\times\bs\nabla f$ and $\bs\nabla\cdot\bs\nabla\times \bs A$ are special cases of $d^2=0$.


[{\it bonus}: {\bf 2. Potential theory}---the {\it Fundamental Theorem of Differentials (FTD)} $d \int_r \omega +
\int_r d \omega = \omega$ implies that if $d\omega=0$, then the
{\it potential} $\alpha \equiv \int_r\omega \equiv \left[ \int_{r'=0}^r
  \omega_{r\Omega}(r',\Omega) dr' \right ] d\Omega$, integrated along
radial coordinate lines, is its {\it antiderivative}: $d\alpha = \omega$.  We use this formula to derive the antiderivative of the following special vector fields, generalizing the \mbox{\it Fundamental Theorem of Calculus (FTC i)}.

{\bf \quad a)} Show that if $\bs\nabla\times\bs E =\bs 0$ (ie. $\bs E$ is {\it irrotational}), then $\bs E=-\bs\nabla V$ (ie. $\bs E$ is {\it conservative}), with the potential function $V(\bs r)=-\int_r\bs E\cdot \bs{dl}+C$, where $C$ is any constant.  Show that for a radial path in spherical coordinates, this reduces to $\bs V = -\bs r \cdot \int_0^1 \bs E(\lambda\bs r)d\lambda$.  Confirm $\bs E=-\bs\nabla V$.

{\bf \quad b)} Show that if $\bs\nabla\cdot\bs B=0$ (ie. $\bs B$ is {\it incompressible}), then $\bs B = \bs\nabla\times\bs A$ (ie. $\bs B$ is {\it solenoidal}), with the potential function $\bs A\cdot \bs{dl} = \int_r \bs B\cdot\bs{da} + d\chi$, where the `constant of integration' $d\chi$ is the differential (gradient) of any scalar field.  Show that for a radial path in spherical coordinates, this reduces to $\bs A = -\bs r\times\int_0^1 \bs B(\lambda \bs r)\, \lambda\, d\lambda$. Take the curl to confirm $\bs B = \bs\nabla\times\bs A$.

{\bf \quad c)} Show that since the differential of any density field $\rho(\bs r)d\tau$ is zero, it can be written as the exact differential $\rho=\bs\nabla\cdot (\bs D + \nabla\times \bs h)$ of some flux field $\bs D(\bs r)\cdot \bs{da}$, and a `constant of integration' gauge $\bs h(\bs r)\cdot\bs{dl}$.  Derive the formula for $\bs D$ for a radial path in spherical coordinates.]


% %~/hw/phy417_sp15/ps_03.tex
{\bf 3. Stokes' theorems}---the FTD also generalizes the FTC (ii), integrating the derivative of a field over a region out to the boundary, resulting in an integral of the original field on the boundary.

% %~/hw/phy416_fa11/ps_03.tex  NOTE: v.da missing curl in phy416!!!
\begin{wrapfigure}[6]{r}[-.15in]{0in}
\includegraphics[width=2in, trim=0 0 .5in .2in]{H08_fig1}
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\mbox{\quad \bf a)} Integrate ${}_{\mathcal{S}}\int \bs\nabla\times\bs v \cdot \bs{da}$ 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.  Integrate ${}_{\partial\mathcal{S}}\oint \bs v \cdot \bs{dl}$ along the boundary $\partial S$ of $\mathcal{S}$ to verify Stokes' theorem.

\mbox{\quad \bf b)} Verify Stokes' theorem with the function in H10\#1c\,ii) on the disk $\rho<R, z=0$ in the $xy$-plane centered at the origin.

\mbox{\quad \bf c)} Verify Gauss' theorem with the function in H10\#1c\,iii) inside the sphere $r<R$ of radius $R$ centered at the origin.


{\bf 4. Electrostatic and Magnetostatic integrals}

\mbox{\quad \bf a)} Integrate i) $V(\bs r)=\frac{1}{4\pi\epsilon_{0}}\oint \frac{\sigma da'}{\rcurs}$,
% $V=\oint \frac{\sigma da}{4\pi\epsilon_{0}\rcurs}$,
where $\hrcurs\rcurs=\brcurs=\bs r-\bs r'$, and ii) $\bs E(\bs r) = \frac{1}{4\pi\epsilon_{0}}\oint \frac{\sigma da' \hrcurs}{\rcurs^{2}}$ over $\bs r'$ on a sphere of radius $R$ to calculate the electric potential $V$, field $\bs E$ of a spherical shell of uniform surface charge density $\sigma$.  Verify that iii) $\bs E=-\nabla V$ and that you get the same iv) field from Gauss' law $\Phi_{D}=\oint_{\partial V} \epsilon_{0} \bs E\cdot d\bs a=\int_{V}\rho d\tau=Q$ and v) potential $\bs V(\bs r)=-\int_{\bs \infty}^{\bs r}\bs E \cdot \bs{d\ell}$ from the FTVC.

\mbox{\quad \bf b)} Integrate i) the magnetic field $\bs B(\bs r)=\frac{\mu_{0}}{4\pi}\oint \frac{I\bs{d\ell'}\times \hrcurs}{\rcurs^{2}}$ of a infinite wire on the $z$-axis and ii) show that you get the same field from Amp\`ere's law/Stokes' theorem $\mathcal{E}_{H}=\oint_{\partial S} \bs H\cdot \bs{d\ell}=\int_{S}\bs J\cdot\bs{da}=I$, where $\bs B=\mu_{0}\bs H$.  Show that iii) $\bs B=\nabla\times\bs A$, the curl of the vector potential $\bs A=-\frac{\mu_{0}I}{2\pi}\bh z \ln\rho$, and that iv) $\bs H=-\bs\nabla U$, the gradient of the scalar potential $U=-\frac{I}{2\pi}\phi$ with a discontinuity of $\Delta U=I$ at the {\it branch cut} $\phi=\pm \pi$, because $\bs H$ is not conservative---it has curl along the $z$-axis.

\mbox{\quad \bf c)} Expand the formula for the magnetic field $\bs B(\bs r)=\oint \frac{\mu_{0}}{4\pi} \frac{I\bs{d\ell'}\times \hrcurs}{\rcurs^{2}}$ of a ring of radius $R$ centered in the $xy$-plane, and integrate $\bs B(z)$ on the $z$-axis. [{\it bonus}: integrate the field everywhere using elliptic functions, per \mbox{\href{https://www.grant-trebbin.com/2012/04/off-axis-magnetic-field-of-circular.html}{www.grant-trebbin.com/2012/04/off-axis-magnetic-field-of-circular.html}}]


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