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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#8, Rev. A, due Monday, 2023-04-03 }
\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)} 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 a) 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 the gradient of $f(r,\theta,\phi)= r^\ell P_\ell(\cos(\theta))$, \href{https://en.wikipedia.org/wiki/Legendre_polynomials#Recurrence_relations}{using} $\frac{x^{2}-1}{\ell}P'_{\ell}(x)=xP_{\ell}(x)-P_{\ell-1}(x)$,

{\bf \quad d)} the curl of $\bs A(x,y,z) = (\bh y x - \bh x y)/(x^{2}+y^{2})^n$ in Cartesian and \mbox{cylindrical coordinates, and the}
% \hat\phi/\rho^..

{\bf \quad e)} 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 \mbox{spherical coordinates.}
% \hat\r/r^..

{\bf \quad f)} 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 $d^2=0$.


% %~/hw/phy417_sp15/ps_03.tex
{\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 will use this formula to derive the antiderivative of the following special vector fields, generalizing the {\it Fundamental Theorem of Calculus (FTC)}:

{\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 $\bs V = -\bs r \cdot \int_0^1 \bs E(\lambda\bs r)d\lambda$.

{\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 of any scalar field.  Show that for a radial path in spherical coordinates, this reduces to $\bs A = -\bs r\times\int_0^1 \lambda B(\lambda \bs r)d\lambda$.

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

{\bf 3. Stokes' theorems}--the FTD also generalizes the FTC, 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!!!
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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 \#1d) 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 \#1e) inside the sphere $r<R$ of radius $R$ centered at the origin.

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