Math 6410 • ODE

According to Maxwell's Equations, the static magnetic field
\(H=(H_1,H_2,H_3)^T\) in an inhomogeneous medium satisfies
\[
  \nabla\cdot(\mu H)=0,\qquad \nabla\times H=0,
\]
where \(\mu\) is a space-dependent permeability. Suppose the interface \(z=0\) separates
a medium in which \(\mu=\mu_+\) from one in which \(\mu=\mu_-\) with \(\mu_\pm\) constants.
\begin{enumerate}
\item[(a)] Define the weak solution: for every given simply connected, smooth domain
\(\Omega_3\subset\R^3\) and arbitrary embedded smooth two dimensional surface with
boundary \(\Omega_2\subset\R^3\), assuming enough regularity, integrate by parts the
equations
\[
  \int_{\Omega_3}\psi\nabla\cdot(\mu H)\dd V=0,\qquad
  \int_{\Omega_2}\psi\nabla\times H\cdot \dd S=0,
\]
where \(\psi\) is an arbitrary, suitable test function.
\item[(b)] Taking \(\Omega_3\) to be a cylindrical domain that cuts the interface
vertically, use Green's Theorem to show that
\[
  [\mu H_3]_-^+=0.
\]
\item[(c)] Taking \(\Omega_2\) to be a rectangular surface that cuts the interface
vertically, use Stokes' Theorem to show that
\[
  [H_1]_-^+=0=[H_2]_-^+.
\]
\end{enumerate}