Lecture 2. Differentiability of Functions of Several Variables and Partial Derivatives (S.V. Podkletnova)

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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 differentiability, continuity, and partial derivatives is examined. While differentiability guarantees both continuity and the existence of partial derivatives, the reverse is not true. The lecture concludes with a sufficient condition: continuous partial derivatives imply differentiability.