# Insights The Pantheon of Derivatives - Part V - Comments

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1. Mar 25, 2017

### Staff: Mentor

2. Mar 25, 2017

### zwierz

1) regarding the Implicit Function Theorem:
there is no need to demand $f$ to be totally differentiable in $(x,y)$ Actually $f$ must be differentiable just in the second argument while the first one can belong to a topological space. By the way, $\mathbb{R}^m$ can also be replaced with a Banach space.

2) Formula (15) holds for any Riemann manifold and follows from (14) with the help of results from previous part.

I think that theCauchy-Goursat Theorem can also be obtained from the Stokes formula without a considerable loss of generality.

3. Mar 25, 2017