The Differential of a Map

Differential Geometry · Axiom Academy

LESSON The Differential of a Map Understanding how smooth maps transform tangent vectors between manifolds 1. Definition of the Differential Let F: M → N be a smooth map between manifolds, and let p ∈ M. The differential (or pushforward ) of F at p is a linear map: This linear map takes tangent vectors at p in M and produces tangent vectors at F(p) in N. It's also denoted as F *,p or (dF) p . 2. Pushforward of Tangent Vectors If v ∈ T p M is a tangent vector (viewed as a derivation), then dF p (v) acts on smooth functions g: N → ℝ by the rule: This says: to see how the pushed-forward vector acts on g, compose g with F first, then apply v. The differential "pushes" the directional derivative operator from M to N. 3. Matrix Representation: The Jacobian In local coordinates (x¹, ..., xᵐ) on M and (y¹, ..., yⁿ) on N, the differential is represented by the Jacobian matrix : Each entry ∂F^i/∂x^j represents how the i-th component of F changes with respect to the j-th coordinate on M. One of the most important properties of the differential is that it respects composition. If F: M → N and G: N → P are smooth maps, then: This is the functorial property of the differential. It generalizes the familiar chain rule from calculus to arbitrary manifolds. The rank of a smooth map F: M → N at a point p is defined as the rank of the linear map dF p : This is the dimension of the image of dF p , or equivalently, the rank of the Jacobian matrix at p.

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