What Is Jacobian - The Right Way of Thinking Derivatives and Integrals

Overview

Explore the concept of Jacobian in multivariable calculus through this 27-minute video lecture. Gain a deeper understanding of derivatives and integrals by rethinking their fundamental meanings. Learn about the Jacobian matrix and determinant, their significance in changing variables, and why they are natural extensions of calculus concepts. Discover how to approach derivatives beyond the traditional slope interpretation, especially in higher dimensions. Follow along as the lecture covers linear maps, derivatives in 1D and 2D, integration concepts, and variable changes in both one and two dimensions. Conclude with an application of these concepts in converting from Cartesian to polar coordinates. While the video focuses on 2D transformations, learn how to generalize the Jacobian to higher dimensions.

Syllabus

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Introduction.
Chapter 1: Linear maps.
Chapter 2: Derivatives in 1D.
Chapter 3: Derivatives in 2D.
Chapter 4: What is integration?.
Chapter 5: Changing variables in integration (1D).
Chapter 6: Changing variables in integration (2D).
Chapter 7: Cartesian to polar.