Math 2.0

Real functions Corollary

Second-order Taylor expansion

Builds on Taylor's theorem in several variablesHessian matrixGradientC^k function characterised by continuous derivatives

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝf \colon U \to \mathbb{R} be of class C^2 on UU, and let a∈Ua \in U. Let ∇f(a)\nabla f(a) be the gradient of ff at aa, and let Hf(a)H_f(a) be the Hessian matrix of ff at aa. Then f(a+h)=f(a)+∇f(a)⋅h+12 h𝖳Hf(a) h+R(h),lim⁡h→0R(h)∥h∥2=0.f(a + h) = f(a) + \nabla f(a) \cdot h + \tfrac{1}{2} \, h^{\mathsf{T}} H_f(a) \, h + R(h), \qquad \lim_{h \to 0} \frac{R(h)}{\|h\|^2} = 0.

Remarks

This is the case k=2k = 2 of Taylor's theorem in several variables, written with matrices. The quadratic term collects the terms with ∣α∣=2|\alpha| = 2. For n=2n = 2 and h=(u,v)h = (u, v), the quadratic term is 12(fxx(a)u2+2fxy(a)uv+fyy(a)v2)\tfrac{1}{2} \left( f_{xx}(a) u^2 + 2 f_{xy}(a) u v + f_{yy}(a) v^2 \right). At a critical point the linear term is zero, so the sign of h𝖳Hf(a)hh^{\mathsf{T}} H_f(a) h controls the behaviour of ff near aa. This is the basis of the second derivative test.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet 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 derivativeconceptDifferentiable 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 …definitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionHessian matrix, definition, no evidence levelHessian matrixdefinitionMulti-index notation, concept, no evidence levelMulti-index notationconceptC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementTaylor's theorem in several variables, statement, no evidence levelTaylor's theorem in s…statementSecond-order Taylor expansion, statement, no evidence levelSecond-order Taylor e…statement
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Limit of a function of several variables concept
  6. Partial derivative concept
  7. Differentiable function of several variables definition
  8. Partial derivative definition
  9. Differentiable on an open set definition
  10. Gradient definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. Hessian matrix definition
  14. Multi-index notation concept
  15. C^k function characterised by continuous derivatives statement
  16. Taylor's theorem in several variables statement
  17. Second-order Taylor expansion statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Taylor's theorem for multivariate functions
  2. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Example in two dimensions
About this page
Tag0P30YYZN85
URIm2://0P30YYZN85
Aliasessecond-order-taylor-formula
Kindstatement, corollary
WritingW0 stub
MSC26B12
StatusThis copy of the site has no build data. Run m2 status 0P30YYZN85.
UsesTaylor's theorem in several variables, Hessian matrix, Gradient, C^k function characterised by continuous derivatives

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