Calculus on Manifolds

About This Book

Spivak’s “Calculus on Manifolds” bridges the gap between undergraduate calculus and graduate differential geometry. Its treatment of differential forms and the generalized Stokes’ theorem provides essential foundations for modern physics and mathematics.

Structure

1. Functions on Euclidean Space: Review of analysis in $\mathbb{R}^n$
2. Differentiation: Derivatives as linear maps, inverse function theorem
3. Integration: Riemann integral in $\mathbb{R}^n$, Fubini’s theorem
4. Integration on Chains: Differential forms, Stokes’ theorem
5. Integration on Manifolds: Manifolds, tangent spaces, orientation

Prerequisites

• Single-variable calculus
• Linear algebra (vector spaces, linear maps, determinants)
• Basic point-set topology