Differentiability of Functions of Several Variables

This lecture discusses the differentiability of functions of multiple variables, focusing on the existence of directional derivatives and their relationship to partial derivatives. The instructor explains that a function can have directional derivatives at a point without being differentiable at that point. The lecture introduces theorems that establish conditions under which a function is differentiable, emphasizing the importance of continuity of partial derivatives. The instructor illustrates these concepts with examples, including the use of gradients and tangent planes. The discussion also covers the implications of differentiability in terms of continuity and the geometric interpretation of gradients as normal vectors to level curves. The lecture concludes with a focus on the significance of these concepts in higher dimensions and their applications in various fields, including physics and engineering.