The map , dimensionless, is continuously differentiable and one-to-one on the whole…
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
The map F(x,y)=(x3,y), dimensionless, is continuously differentiable and one-to-one on the whole plane. Someone argues: "F has an inverse everywhere, so by the inverse function theorem its inverse is differentiable at the origin, with derivative (DF(0))−1."
((a)) Compute DF and its determinant, and find every point where the determinant vanishes.
((b)) Write the inverse G explicitly and decide whether it is differentiable at the origin.
((c)) Find the error in the argument. Then use the chain rule to show that if a differentiable map has detDF(a)=0, no inverse of it can be differentiable at F(a).