Math 2.0

Partial differential equations Proposition

Laplacian as a sum of second partial derivatives

Builds on LaplacianDivergenceGradientHigher partial derivativesHessian matrix

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let f ⁣:U→ℝf \colon U \to \mathbb{R} be twice differentiable on UU. Then Δf=∂2f∂x12+∂2f∂x22+⋯+∂2f∂xn2.\Delta f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \dots + \frac{\partial^2 f}{\partial x_n^2}.

Remarks

Proof: ∇f=(∂f/∂x1,…,∂f/∂xn)\nabla f = (\partial f / \partial x_1, \dots, \partial f / \partial x_n), and the divergence adds the derivative of the ii-th component with respect to xix_i. So Δf\Delta f is the trace of the Hessian matrix of ff. The trace does not change under a rotation of coordinates, so neither does Δf\Delta f. Examples: Δ(x2+y2)=4\Delta (x^2 + y^2) = 4, Δ(x2−y2)=0\Delta (x^2 - y^2) = 0 and Δ∥x∥2=2n\Delta \|x\|^2 = 2n on ℝn\mathbb{R}^n. In polar coordinates in the plane, Δf=frr+1rfr+1r2fθθ\Delta f = f_{rr} + \tfrac{1}{r} f_r + \tfrac{1}{r^2} f_{\theta\theta} for r>0r > 0.

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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…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial 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, definition, no evidence levelGradientdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionHessian matrix, definition, no evidence levelHessian matrixdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementVector field, definition, no evidence levelVector fielddefinitionDivergence, definition, no evidence levelDivergencedefinitionLaplacian, definition, no evidence levelLaplaciandefinitionLaplacian as a sum of second partial derivatives, statement, no evidence levelLaplacian as a sum of…statement
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Gradient definition
  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Higher partial derivatives definition
  15. Iterated derivative definition
  16. Hessian matrix definition
  17. C^k function characterised by continuous derivatives statement
  18. Vector field definition
  19. Divergence definition
  20. Laplacian definition
  21. Laplacian as a sum of second partial derivatives statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace_operator, Definition
  2. Web https://en.wikipedia.org/wiki/Laplace_operator, N dimensions
  3. Web https://en.wikipedia.org/wiki/Laplace_operator, Two dimensions
About this page
Tag05RCT1522R
URIm2://05RCT1522R
Aliasescartesian-formula-for-the-laplacian
Kindstatement, proposition
WritingW0 stub
MSC35J05
StatusThis copy of the site has no build data. Run m2 status 05RCT1522R.
UsesLaplacian, Divergence, Gradient, Higher partial derivatives, Hessian matrix

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