Cotangent Spaces and Differential Forms

Central Question

Why should measurements of tangent vectors be treated as geometric objects in their own right?

Why This Matters

Dual spaces encode directional measurements. Cotangent vectors and alternating products prepare the natural language for integration on manifolds.

Learning Prompt

I know linear algebra and dual spaces. Motivate the cotangent space by asking what object measures the first-order change of a function in a direction. Develop $df_p$ as a linear functional on $T_pM$, and compare a covector with a vector without identifying them prematurely. Explain the coordinate roles of $dx^i$ and why a metric is extra structure for converting between vectors and covectors. Then motivate differential forms through alternating multilinear measurements. Make me discover why forms pull back naturally along smooth maps.

Ideas

Examples

Questions

Connections

• Tangent Spaces
• Differential Maps and Pushforwards
• Differential Forms and Integration
• Exterior Derivative
• Week overview

One Thing That Surprised Me

Further Rabbit Holes