University · Mathematics · Analysis II (Multivariable Calculus)
Implicit and Inverse Function Theorems
Two of the most powerful existence theorems in multivariable calculus: the Implicit Function Theorem tells us when a curve or surface defined by an equation F(x,y)=0 can locally be described as the graph of a function, and the Inverse Function Theorem tells us when a smooth map has a smooth local inverse. Both rest on a single condition — a Jacobian being nonzero or invertible — and both underlie optimization theory, differential geometry, and the study of manifolds.
Inhaltsübersicht
- Motivation and the Implicit Function Theorem in Two Variables
- The General Implicit Function Theorem and a Worked Example
- The Inverse Function Theorem
- Worked Example: Polar Coordinates and Practical Applications
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