Implicit Function Theorem with examples, Real Analysis II
In this video, I present the Implicit Function Theorem by focusing on its motivation, the hypotheses, the conclusions, and how to apply it. We will do several detailed examples, beginning with basic example of x^2 + y^2 = 1. The idea of the theorem is this: starting from an equation that defines a relationship between variables, can we locally describe one variable as a function of the others? This information comes from taking the determinant of a certain square submatrix inside of a larger wide Jacobian matrix.
We will also look at a surface in three variables and a curve of intersection between two surfaces in R3. We will use the formula for the Jacobian of the implicit function to do some fun estimates.
The video is example-driven and intended to build intuition for when and how to use the theorem. A separate video will contain a formal proof based on the inverse function theorem.
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