# The Pantheon of Derivatives - Part V - Comments

• Insights

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.

What an incredible resource! Thank you Fresh!