Let be a differentiable

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

Let r:Rn→Rn\mathbf r:\mathbb R^n\to\mathbb R^n be a differentiable, invertible change of coordinates with Jacobian matrix Dr\mathrm D\mathbf r and metric g=(Dr)TDrg=(\mathrm D\mathbf r)^{\mathrm T}\mathrm D\mathbf r, so that det⁡g=(det⁡Dr)2\det g=(\det\mathrm D\mathbf r)^2. ((a)) Explain, from this identity alone, why det⁡g\sqrt{\det g} is always ≥0\ge0, whatever the sign of the Jacobian determinant J=det⁡DrJ=\det\mathrm D\mathbf r. ((b)) The plane chart u1=yu^1=y, u2=xu^2=x swaps the two Cartesian coordinates. Write x,yx,y as functions of u1,u2u^1,u^2, compute Dr\mathrm D\mathbf r, JJ, and gg, and verify explicitly that det⁡g=∣J∣\sqrt{\det g}=|J| although J<0J<0. ((c)) State in one sentence why the area element dA=det⁡g du1 du2\mathrm dA=\sqrt{\det g}\,\mathrm du^1\,\mathrm du^2 must be built from ∣J∣|J| and not from JJ itself.

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