Analysis on Manifolds — Munkres

James R. Munkres

Rigorous multivariable analysis: the derivative as a linear map, the inverse and implicit function theorems, the change-of-variables formula, differential forms, and integration on manifolds culminating in Stokes’ theorem. The rigorous foundation for the Jacobians, gradients, and multidimensional integrals used throughout optimization, machine learning, and physics.

Chapters

Exercises added as I work through each chapter.