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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#9, Rev. C, due Wednesday, 2022-04-13 }
\end{center}

\vspace{.3 in}

{\bf 1. Vector chain and product rules}

{\bf \quad a)} Prove that i) $\nabla (f\circ g)(\bs r) = \nabla f(g(\bs r)) = f'(g(\bs r))\nabla g(\bs r)$;  ii) $\nabla fg =  g\nabla f+f \nabla g$;  and iii) $\nabla \cdot f \bs A = \bs A \cdot \nabla f + f \nabla\cdot A$.  [bonus: classify and prove all product rules]

{\bf \quad b)} Calculate i) $\nabla r^{2}$, ii) $\nabla r$, iii) $\nabla\times\bs r$, and iv) $\nabla \cdot \bs r$.

{\bf \quad c)} Use the results of a) and b) to calculate i) $\nabla r^{m}$; ii) $\nabla\; \bs k\cdot \bs r$, iii) $\nabla \times (\bs k \times \bh r r^{m})$ (compare with H08\#1d), iv) $\nabla \cdot (k\bh r r^{m})$, v) $\nabla\cdot \bh r/r^{n}$ (compare with H08\#1e), where $\bs k$ is a constant vector.

{\bf 2. Scalar Laplacian and inverse: Green's function}

{\bf \quad a)} Combine the formulas for divergence and gradient to obtain the formula for $\nabla^{2}f(\bs r)$, called the {\it scalar Laplacian}, in orthogonal curvilinear coordinates $(q^{1},q^{2},q^{3})$ with scale factors $h_{1},h_{2},h_{3}$.

{\bf \quad b)} Calculate $\nabla(1/r)$ and then $\nabla\cdot\nabla(1/r)$ using \#1, and compare with $\nabla^{2}(1/r)$ from \#2a).  Where are the gradient and Laplacian singular?

{\bf \quad c)} Use Gauss' theorem to show that $\int_{r<a}\bh r/r^{2}=4\pi$, and thus $-\nabla^{2}(1/4\pi r)=\delta^{3}(\bs r)$.  We call $G(\brcurs)\equiv -\nabla^{-2}\delta^{3}(\brcurs) = 1/4\pi\rcurs$ the {\it Green's function}. $\brcurs=\bs r-\bs r'$ shifts the {\it pole} from \mbox{$\bv r=\bv 0$ to $\bs r'$.}

{\bf \quad d)} Solve for a {\it particular solution} of the second order partial differential equation $-\nabla^{2}V(\bs r)=\rho(\bs r)$ by expanding $\rho(\bs r)=\int d^{3}\bs r' \rho(\bs r') \delta^{3}(\brcurs)$, and applying c) to each basis function $\delta^{3}(\brcurs)=\delta^{3}(\bs r-\bs r')$.  Thus the Green's function is use to invert the Laplacian operator!

{\bf 3. Vector Laplacian and decomposition: Helmholtz theorem}

{\bf \quad a)} Write down all possible combinations of gradient, curl, and divergence to form second vector derivatives of both scalar and vector fields.  Which `natural' second derivatives are zero?  Show that $\nabla^{2}\bs F = \nabla \nabla \cdot \bs F - \nabla \times (\nabla \times \bs F)$, where $\nabla^{2}=\nabla\cdot\nabla=\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}$ (in Cartesian coordinates only) is called the {\it vector Laplacian}.  Thus the Laplacian with its {\it longitudinal} $\nabla\nabla\cdot\bs F$ and {\it transverse} $\nabla\times(\nabla\times \bs F)$ components is {\it the} unique second derivative of a vector field.  In other coordinates, the Laplacian is defined in terms of curvilinear gradient, divergence and curls by the above equation. [bonus: expand the vector Laplacian in terms of scale factors in an orthogonal curvilinear coordinate system]

{\bf \quad b)} Use the technique of \#2d) to solve the above equation for $\bs F$, and thus prove that any vector field $\bs F(\bs r)$ can be decomposed into $\bs F=\bs E + \bs B$, an {\it irrotational} field $\nabla\times\bs E=\bs 0$ (the {\it longitudinal} part), and an {\it incompressible} field $\nabla\cdot \bs B=0$, (the {\it transverse} part) with {\it sources} $\rho=\nabla\cdot\bs E$ and $\bs J=\nabla\times\bs B$, respectively.  Show that the vector source must satisfy $\nabla\cdot\bs J=\bs 0$.  Show that $\bs E = -\nabla V$ (it is {\it conservative}) with the potential $V=-\nabla^{-2}\rho$ and that $\bs B=\nabla\times\bs A$ (it is {\it solenoidal}) with potential $\bs A=-\nabla^{-2}\bs J$.  Thus any vector field $\bs F$ with sources that vanishes fast enough as $\bs r\to\infty$ is completely determined by its two sources $\rho=\nabla\cdot\bs F$ and $\bs J=\nabla\times\bs F$.

%{\bf \quad c)} Given constant sources $\rho$ and $\bs J$ of a field $\bs F$, calculate its two potentials $V$ and $\bs A$, and then its longitudinal and transverse components $\bs E$ and $\bs B$.  Note this field diverges as $\bs r\to\infty$, but it is easier to calculate than other fields.

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