Differential Maps and Pushforwards

Central Question

How does a smooth map transport first-order motion from one space to another?

Why This Matters

The derivative becomes a map between tangent spaces, revealing the coordinate-free meaning behind Jacobian matrices and the chain rule.

Learning Prompt

Rebuild the derivative of a smooth map $f:M\to N$ as a linear map $df_p:T_pM\to T_{f(p)}N$. Start by following a curve through $p$, then derive the pushforward from its image curve. Show how an ordinary Jacobian represents this map in chosen charts and derive the chain rule intrinsically. Explore an inclusion, a projection and a map whose derivative loses rank. Ask me to track the source and target spaces carefully, and distinguish pushing a tangent vector from pushing a whole vector field.

Ideas

Examples

Questions

Connections

• Charts and Atlases
• Tangent Spaces
• Cotangent Spaces and Differential Forms
• Geometry Meets Physics
• Week overview

One Thing That Surprised Me

Further Rabbit Holes