Math 2.0

Real functions Corollary

Implicit function theorem for a surjective derivative

Builds on Implicit function theoremLevel setJacobian matrix

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let m≤nm \le n, and let F ⁣:U→ℝmF \colon U \to \mathbb{R}^m be of class C1C^1. Let p∈Up \in U with F(p)=0F(p) = 0, and suppose that DF(p) ⁣:ℝn→ℝmDF(p) \colon \mathbb{R}^n \to \mathbb{R}^m is surjective. Then there are mm coordinates xi1,…,ximx_{i_1}, \dots, x_{i_m} and an open neighbourhood WW of pp with this property. The set {x∈W:F(x)=0}\{x \in W : F(x) = 0\} is the graph of a C1C^1 function of the other n−mn - m coordinates.

Remarks

The derivative is surjective if and only if the Jacobian matrix JF(p)J_F(p) has rank mm. Then some mm columns of JF(p)J_F(p) are linearly independent, and these columns choose the coordinates. After a renumbering of coordinates, the statement is the implicit function theorem with y=(xi1,…,xim)y = (x_{i_1}, \dots, x_{i_m}). If DF(x)DF(x) is surjective at each point of the level set F−1(0)F^{-1}(0), this set is locally a graph near each of its points. So it is a C1C^1 submanifold of dimension n−mn - m. Example: for F(x,y,z)=x2+y2+z2−1F(x, y, z) = x^2 + y^2 + z^2 - 1, DFDF is surjective on the unit sphere. So the sphere is locally the graph of a function of two coordinates.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptPreimage of a set, concept, no evidence levelPreimage of a setconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptLevel set, definition, no evidence levelLevel setdefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementImplicit function theorem, statement, no evidence levelImplicit function the…statementImplicit function theorem for a surjective derivative, statement, no evidence levelImplicit function the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Preimage of a set concept
  5. Fréchet derivative definition
  6. Function of several variables concept
  7. Limit of a function of several variables concept
  8. Partial derivative concept
  9. Level set definition
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Chain rule in several variables statement
  15. Higher partial derivatives definition
  16. Iterated derivative definition
  17. Jacobian determinant definition
  18. Derivative of the inverse function statement
  19. C^k function characterised by continuous derivatives statement
  20. Inverse function theorem statement
  21. Implicit function theorem statement
  22. Implicit function theorem for a surjective derivative statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function_theorem, Several variables
  2. Web https://en.wikipedia.org/wiki/Implicit_function_theorem, Statement of the theorem
About this page
Tag0NE2W2XPQJ
URIm2://0NE2W2XPQJ
Aliasesregular-value-theorem-euclidean
Kindstatement, corollary
WritingW0 stub
MSC26B10
StatusThis copy of the site has no build data. Run m2 status 0NE2W2XPQJ.
UsesImplicit function theorem, Level set, Jacobian matrix

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.