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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#9, Rev. A, due Monday, 2022-04-11 }
\end{center}

\vspace{.3 in}

{\bf 1. Vector 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) $\nabla fg \nabla f\, g+f \nabla g$;  iii) $\nabla \cdot f \bs A = \nabla f \cdot \bs A + f \nabla\cdot A$.  [bonus: classify and prove all product rules]

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

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

{\bf 2. Scalar Green's function}

{\bf \quad a)} Combine the formulas for divergence and gradient to obtain the formula for $\nabla^{2}f(\bs r)$ in an orthogonal curivlinear coordinate system $(q^{1},q^{2},q^{3})$ 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 an 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 $G(\rcurs)\equiv 1/4\pi\rcurs$ a Green's function.

{\bf \quad d)} Solve for a particular solution of the second order equation $-\nabla^{2}V(\bs r)=\rho(\bs r)$ by expanding $\rho(\bs r)=\int d^{3}\bs r' \rho(r') \delta^{3}(\bs r-\bs r')$, and applying part c) to each basis function.

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