The bookUnit 1 · Functions of Several Variables13 / 83

Inverse and implicit function theorems, extrema

Why this is asked: Both theorems need a non-vanishing Jacobian (∂F/∂y ≠ 0 for implicit). Where it vanishes, anything can happen — that is exactly what the questions probe.

Inverse and implicit function theorems

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Implicit function theorem: what to differentiate, and when it says nothinginteractive

A worked implicit derivative, plus the point where the hypothesis fails and the curve really does misbehave.

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The trap here

“ $A C^{1}$ map with everywhere non-zero Jacobian is injective” — false

f(x,y) = (eˣ cos y, eˣ sin y) on $\mathbb{R}^{2}$

The Jacobian determinant is $e^{2x}$ ≠ 0, but $f(x, y) = f(x, y + 2\pi)$ . Invertibility is only local.