Chain Rules: Higher Dimensional Versions

This lecture covers the chain rule for compositions of differential functions in higher dimensions. It explains the standard chain rule and its application to functions from open sets to real numbers. The lecture also generalizes the chain rule for higher dimensional functions, providing theorems and proofs for differentiable functions. Emphasis is placed on understanding the gradient and the impact of directional changes on the functions. The lecture concludes with practical examples and applications of the chain rule in various scenarios.