An anisotropic crystal has diagonal resistivity matrix referred to its principal axes

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Conditions of Validity, Ch The four subspaces and the Fredholm alternative

Problem

An anisotropic crystal has diagonal resistivity matrix A(t)=(t00010001/t) Ω m\mathsf{A}(t)=\begin{pmatrix}t&0&0\\0&1&0\\0&0&1/t\end{pmatrix}\ \Omega\,\mathrm{m} referred to its principal axes, with dimensionless t≥1t\ge1. It gives the electric field b\mathbf{b} (in V m−1\mathrm{V\,m^{-1}}) that drives a current density x\mathbf{x} (in A m−2\mathrm{A\,m^{-2}}) through A(t)x=b\mathsf{A}(t)\mathbf{x}=\mathbf{b}. ((a)) Find det⁡A(t)\det\mathsf{A}(t) for general tt, and show it does not depend on tt. ((b)) Find κ(A(t))\kappa(\mathsf{A}(t)) for general tt, and evaluate it at t=100t=100. ((c)) At t=100t=100, a measured field carries a relative uncertainty ∥δb∥/∥b∥=0.10%\|\delta\mathbf{b}\|/\|\mathbf{b}\|=0.10\%. Bound the relative uncertainty in the computed current density. ((d)) Explain why det⁡A(t)\det\mathsf{A}(t) gives no warning, however large tt becomes, of the growing ill-conditioning found in part (c).

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