Differentiability: Functions and Matrices

This lecture delves into the concept of differentiability for functions and matrices, exploring the conditions for partial differentiability and the Jacobean matrix. It covers the definition of differentiability, the Jacobean matrix for vector-valued functions, and the chain rule for differentiable functions. The lecture also discusses the invertibility of vector fields and provides insights into the relationship between differentiability and continuity. Various theorems and generalizations related to differentiability are presented, emphasizing the importance of understanding the properties of functions and matrices in the context of analysis.