Tangent Spaces

Central Question

What is a tangent vector when there is no preferred ambient space in which to draw an arrow?

Why This Matters

An intrinsic tangent space turns directional change into a local linear structure, allowing derivatives to be defined independently of an embedding.

Learning Prompt

I understand tangent planes to surfaces in $\mathbb R^3$, but I want an intrinsic definition of tangent vectors. Begin with velocities of curves through a point, ask when two curves represent the same first-order motion, and construct the tangent space. Then connect this with derivations on smooth functions. Work through $\mathbb R^n$, a circle and a sphere, distinguishing a tangent vector from its coordinate components. Make me explain why changing a chart changes the components but not the vector, and why the resulting space is linear.

Ideas

Examples

Questions

Connections

• From Surfaces to Manifolds
• Charts and Atlases
• Differential Maps and Pushforwards
• Cotangent Spaces and Differential Forms
• Week overview

One Thing That Surprised Me

Further Rabbit Holes