Partial Derivatives: Gradient and Theorem of Mean Value

This lecture covers the concept of gradient for functions defined on a non-empty open set, introducing the notation Df or gradf. It also explains the Theorem of Mean Value for functions in several variables, showing the existence of a point where the function's derivative equals the function's increment. The lecture further explores the convexity of sets and functions, demonstrating the connection between convex subsets and continuity. Theorem proofs and practical applications are presented, emphasizing the importance of these concepts in mathematical analysis.