Math 2.0

Real functions Theorem

Gradient theorem

Builds on Line integral of a vector fieldGradientLine integral as an integral over the parameter intervalChain rule for a composition with a curve

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let ϕ ⁣:U→ℝ\phi \colon U \to \mathbb{R} be of class C1C^1, and let γ ⁣:[a,b]→U\gamma \colon [a, b] \to U be a piecewise C1C^1 curve. Then ∫γ∇ϕ⋅d𝐫=ϕ(γ(b))−ϕ(γ(a)).\int_\gamma \nabla \phi \cdot d\mathbf{r} = \phi(\gamma(b)) - \phi(\gamma(a)).

Remarks

Proof for a C1C^1 curve: by the chain rule along a curve, (ϕ∘γ)′(t)=∇ϕ(γ(t))⋅γ′(t)(\phi \circ \gamma)'(t) = \nabla \phi(\gamma(t)) \cdot \gamma'(t). Then the parametric formula and the fundamental theorem of calculus give the result. For a piecewise C1C^1 curve, add the pieces. So the line integral of a gradient field depends only on the end points. Around a closed curve it is 00. Example: for ϕ(x,y)=x2y\phi(x, y) = x^2 y and any path from (0,0)(0, 0) to (1,2)(1, 2), ∫γ(2xy,x2)⋅d𝐫=2\int_\gamma (2 x y, x^2) \cdot d\mathbf{r} = 2. This is the case of curves in the generalised Stokes theorem: the boundary of a curve consists of its two end points.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptParametrised curve, concept, no evidence levelParametrised curveconceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionGradient, concept, no evidence levelGradientconceptChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionChain rule for a composition with a curve, statement, no evidence levelChain rule for a comp…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionVector field, definition, no evidence levelVector fielddefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionLine integral of a scalar field, definition, no evidence levelLine integral of a sc…definitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionRegular curve, definition, no evidence levelRegular curvedefinitionArc length parametrisation, definition, no evidence levelArc length parametris…definitionLine integral as an integral over the parameter interval, statement, no evidence levelLine integral as an i…statementGradient theorem, statement, no evidence levelGradient theoremstatement
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. Fréchet derivative definition
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Limit of a function of several variables concept
  9. Differentiable function of several variables concept
  10. Parametrised curve concept
  11. Partial derivative concept
  12. Continuity in R^n definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Gradient concept
  16. Chain rule in several variables statement
  17. Differentiable on an open set definition
  18. Gradient definition
  19. Higher partial derivatives definition
  20. Iterated derivative definition
  21. Chain rule for a composition with a curve statement
  22. C^k function characterised by continuous derivatives statement
  23. Parametrised curve definition
  24. Vector field definition
  25. Rectifiable curve definition
  26. Tangent vector of a curve definition
  27. Line integral of a scalar field definition
  28. Line integral of a vector field definition
  29. Regular curve definition
  30. Arc length parametrisation definition
  31. Line integral as an integral over the parameter interval statement
  32. Gradient theorem statement

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gradient_theorem, Proof
  2. Web https://en.wikipedia.org/wiki/Gradient_theorem, lead section
About this page
Tag0SD237FEJD
URIm2://0SD237FEJD
Aliasesfundamental-theorem-of-line-integrals, fundamental-theorem-of-calculus-for-line-integrals
Kindstatement, theorem
WritingW1 stated
MSC26B20
StatusThis copy of the site has no build data. Run m2 status 0SD237FEJD.
UsesLine integral of a vector field, Gradient, Line integral as an integral over the parameter interval, Chain rule for a composition with a curve
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