Lecture 4. Gradient. Higher-Order Partial Derivatives

This lecture covers two fundamental topics in multivariable differential calculus: the gradient and higher-order partial derivatives. The material is intended for students studying mathematical analysis.

The first part introduces the gradient as the vector of partial derivatives. Its connection with the directional derivative is examined, revealing its geometric meaning: the gradient points in the direction of the steepest increase of a function, and its length equals the rate of increase in that direction. The orthogonality of the gradient to level lines is proved, and its main properties are discussed.

The second part focuses on second-order and higher-order derivatives. Special attention is given to mixed derivatives. Schwarz's theorem on the equality of mixed derivatives under continuity is formulated and proved, accompanied by a counterexample. The lecture concludes with an examination of the Hessian matrix and its geometric interpretation.