Let , , be a differentiable, invertible change of coordinates, with Cartesian…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential 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, r=r(u1,…,un)\mathbf r=\mathbf r(u^1,\dots,u^n), be a differentiable, invertible change of coordinates, with Cartesian components x1,…,xnx^1,\dots,x^n. Let Dr\mathrm D\mathbf r be the n×nn\times n matrix with entries (Dr)ki=∂xk/∂ui(\mathrm D\mathbf r)_{ki}=\partial x^k/\partial u^i, and let gij=ei⋅ejg_{ij}=\mathbf e_i\cdot\mathbf e_j be the metric of the coordinate basis ei=∂r/∂ui\mathbf e_i=\partial\mathbf r/\partial u^i. ((a)) Show that the matrix gg with entries gijg_{ij} satisfies g=(Dr)T Drg=(\mathrm D\mathbf r)^{\mathrm T}\,\mathrm D\mathbf r. ((b)) Hence prove that det⁡g=(det⁡Dr)2\det g=(\det \mathrm D\mathbf r)^2 for any nn, and deduce det⁡g=∣J∣\sqrt{\det g}=|J|, where J=det⁡DrJ=\det\mathrm D\mathbf r is the Jacobian determinant.

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