Math 2.0

Real functions Proposition

Gradient is orthogonal to level sets

Builds on GradientLevel setChain rule for a composition with a curve

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝf \colon U \to \mathbb{R}, and let c∈ℝc \in \mathbb{R}. Let I⊆ℝI \subseteq \mathbb{R} be an open interval and let γ ⁣:I→U\gamma \colon I \to U be a curve that lies in the level set Lc(f)L_c(f), so that f(γ(t))=cf(\gamma(t)) = c for each t∈It \in I. Let t0∈It_0 \in I, and suppose that γ\gamma is differentiable at t0t_0 and ff is differentiable at γ(t0)\gamma(t_0). Then ∇f(γ(t0))⋅γ′(t0)=0\nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = 0.

Remarks

The proof is one step: f∘γf \circ \gamma is constant, so its derivative is 00, and by the chain rule for a curve that derivative is ∇f(γ(t0))⋅γ′(t0)\nabla f(\gamma(t_0)) \cdot \gamma'(t_0). So the gradient is orthogonal to each tangent vector of the level set at the point. Let ∇f≠0\nabla f \ne 0 and let ff be of class C1C^1. Then the level set near the point is a smooth hypersurface, and ∇f\nabla f is a normal vector to it. For f(x,y,z)=x2+y2+z2f(x, y, z) = x^2 + y^2 + z^2, the level set at c=1c = 1 is the unit sphere, and ∇f(p)=2p\nabla f(p) = 2p is normal to the sphere at pp.

Reading path

To understand this page from the ground up, read these 19 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 compositionconceptPreimage of a set, concept, no evidence levelPreimage of a setconceptFréchet 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 derivativeconceptLevel set, definition, no evidence levelLevel setdefinitionDifferentiable 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…statementGradient, definition, no evidence levelGradientdefinitionChain rule for a composition with a curve, statement, no evidence levelChain rule for a comp…statementGradient is orthogonal to level sets, statement, no evidence levelGradient is orthogona…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. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Limit of a function of several variables concept
  10. Differentiable function of several variables concept
  11. Parametrised curve concept
  12. Partial derivative concept
  13. Level set definition
  14. Differentiable function of several variables definition
  15. Gradient concept
  16. Chain rule in several variables statement
  17. Gradient definition
  18. Chain rule for a composition with a curve statement
  19. Gradient is orthogonal to level sets statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gradient, Level sets
  2. Web https://en.wikipedia.org/wiki/Level_set, Level sets versus the gradient
About this page
Tag0VXYV3NQQ1
URIm2://0VXYV3NQQ1
Kindstatement, proposition
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0VXYV3NQQ1.
UsesGradient, Level set, Chain rule for a composition with a curve

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