Lecture 2. Differentiability of Functions of Several Variables and Partial Derivatives

This lecture explores differentiability for functions of several variables. It begins by defining first-order partial derivatives, explaining that to compute them, all other variables are treated as constants. The geometric meaning is illustrated: a partial derivative represents the slope of the tangent line to a curve created by slicing the surface with a plane. The core of the lecture introduces the total differential. A function is differentiable if its total increment can be broken down into a linear part plus a negligible error term. This linear part, called the differential, has coefficients that are precisely the partial derivatives. The relationship between diff…