A change of coordinates on

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

A change of coordinates on R2\mathbb{R}^2, dimensionless, is defined by u=excos⁡y,v=exsin⁡y.u = e^{x}\cos y, \qquad v = e^{x}\sin y. ((a)) Compute the Jacobian determinant ∂(u,v)/∂(x,y)\partial(u,v)/\partial(x,y). ((b)) Using the inverse function theorem, decide whether this map is locally invertible at (x,y)=(0,0)(x,y)=(0,0), and give the value of the Jacobian there.

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