Math 2.0

Real functions Corollary

Chain rule for a composition with a curve

Builds on Chain rule in several variablesGradientParametrised curve

Statement

Let I⊆ℝI \subseteq \mathbb{R} be an open interval, let U⊆ℝnU \subseteq \mathbb{R}^n be open, let γ ⁣:I→U\gamma \colon I \to U, let f ⁣:U→ℝmf \colon U \to \mathbb{R}^m, and let t0∈It_0 \in I. Suppose that γ\gamma is differentiable at t0t_0 and ff is differentiable at γ(t0)\gamma(t_0). Then f∘γf \circ \gamma is differentiable at t0t_0, and (f∘γ)′(t0)=Df(γ(t0))(γ′(t0)).(f \circ \gamma)'(t_0) = Df(\gamma(t_0))\big(\gamma'(t_0)\big).

Remarks

For m=1m = 1, this reads (f∘γ)′(t0)=∇f(γ(t0))⋅γ′(t0)=∑i=1n∂f∂xi(γ(t0)) γi′(t0)(f \circ \gamma)'(t_0) = \nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = \sum_{i=1}^n \dfrac{\partial f}{\partial x_i}(\gamma(t_0)) \, \gamma_i'(t_0), with the gradient ∇f\nabla f. It is the case n=1n = 1 of the chain rule in several variables, since a linear map ℝ→ℝn\mathbb{R} \to \mathbb{R}^n is s↦sγ′(t0)s \mapsto s \gamma'(t_0). For γ(t)=a+tv\gamma(t) = a + t v it gives the directional derivative Dvf(a)=Df(a)(v)D_v f(a) = Df(a)(v).

Reading path

To understand this page from the ground up, read these 15 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 derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionGradient, concept, no evidence levelGradientconceptChain rule in several variables, statement, no evidence levelChain rule in several…statementChain rule for a composition with a curve, statement, no evidence levelChain rule for a comp…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.
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  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. Differentiable function of several variables definition
  13. Gradient concept
  14. Chain rule in several variables statement
  15. Chain rule for a composition with a curve statement

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chain_rule, Case of scalar-valued multivariate functions
About this page
Tag0A8TJ0VHC7
URIm2://0A8TJ0VHC7
Kindstatement, corollary
WritingW0 stub
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 0A8TJ0VHC7.
UsesChain rule in several variables, Gradient, Parametrised curve

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