Differentiability and Tangent Planes in Multivariable Functions

This lecture discusses the concept of differentiability in multivariable functions and the existence of tangent planes. It begins by establishing the conditions under which a function is differentiable, specifically focusing on the relationship between differentiability and the existence of a non-vertical tangent plane at a point on the graph of the function. The instructor illustrates this with examples, emphasizing the geometric interpretation of tangent planes and the angles involved. The lecture further explores the implications of the implicit function theorem, demonstrating how to derive the equation of the tangent plane for surfaces defined by level sets. The discussion includes practical applications, such as the creation of parabolas, and highlights the importance of gradients in determining the behavior of functions near critical points. Throughout the lecture, the instructor provides detailed explanations and visual aids to clarify complex concepts, ensuring a comprehensive understanding of the material.