Limits and Derivatives in Multivariable Functions

This lecture focuses on the fundamental concepts of limits and derivatives in the context of multivariable functions. The instructor begins by revisiting the definition of limits, emphasizing the transition from single-variable to multivariable analysis. The discussion highlights the challenges of approaching a point from different trajectories and the importance of continuity in determining limits. The instructor introduces polar coordinates as a method to simplify the analysis of limits. The lecture also covers the concept of partial derivatives, explaining how to compute them by fixing one variable while varying another. The instructor illustrates these concepts with examples, demonstrating how to derive functions and calculate limits effectively. The importance of understanding the behavior of functions near singularities and the implications of removable discontinuities are also discussed. The lecture concludes with a discussion on the gradient and its significance in multivariable calculus, setting the stage for further exploration of these concepts in future sessions.