MathLiveImplicit function theorem
If a continuously differentiable map f(x,y) from R^m times R^n to R^m vanishes at (a,b) and its Jacobian with respect to x is invertible there, then near b the equation f=0 can be solved for x as a continuous function g(y) with g(b)=a. It is used to guarantee that a system of equations defines a function locally, even when it cannot be solved explicitly.
f\in\mathcal{C}^1(\mathbb{R}^m\times\mathbb{R}^n,\mathbb{R}^m)\land f(a,b)=0\land \det\left.\frac{\partial f(x,y)}{\partial x}\right|_{(a,b)}\neq 0 \implies \exists\, U\subset\mathbb{R}^n \text{ open},\ b\in U,\ \exists\, g\colon U\to\mathbb{R}^m:\ g(b)=a \land \forall y\in U:\ f(g(y),y)=0What the symbols mean
| continuously differentiable map \mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^m, with f(a,b)=0 | |
| variable in \mathbb{R}^m, the unknowns solved for | |
| variable in \mathbb{R}^n, the parameters | |
| point in \mathbb{R}^m with f(a,b)=0 | |
| point in \mathbb{R}^n, the parameter value at which the solution is taken | |
| Jacobian matrix of f with respect to x | |
| open neighbourhood of b in \mathbb{R}^n | |
| implicit function U\to\mathbb{R}^m defined near b |
About this formula
- FieldMathematics
- TopicMultivariable calculus
- LevelUniversity
- Also calledimplicit function theorem formula, implicit function existence, local graph of implicit function, implicit differentiation theorem, Dini's implicit function theorem, solving equations locally for a function
- SourceWikidata Q848375
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