Math 2.0

Partial differential equations Definition

Laplacian

Builds on DivergenceGradientC^k function characterised by continuous derivatives

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let f ⁣:U→ℝf \colon U \to \mathbb{R} be twice differentiable on UU. The Laplacian of ff is the function Δf ⁣:U→ℝ\Delta f \colon U \to \mathbb{R}, Δf=div⁡(∇f)=∇⋅∇f,\Delta f = \operatorname{div}(\nabla f) = \nabla \cdot \nabla f, the divergence of the gradient of ff. It is also written ∇2f\nabla^2 f.

Remarks

In Cartesian coordinates, Δf\Delta f is the sum of the unmixed second partial derivatives of ff. The operator Δ\Delta is linear, and it maps functions of class C^k to functions of class Ck−2C^{k-2} for k≥2k \ge 2. Let ff be of class C2C^2, and let fˉr\bar{f}_r be the average of ff on the sphere of radius rr about xx. Then Δf(x)\Delta f(x) is 2n2n times the limit of (fˉr−f(x))/r2(\bar{f}_r - f(x)) / r^2 as r→0r \to 0. So Δf(x)>0\Delta f(x) > 0 means that f(x)f(x) is below its nearby averages. The Laplacian appears in the heat equation ∂tu=Δu\partial_t u = \Delta u, the wave equation and Laplace's equation Δu=0\Delta u = 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 derivativedefinitionC^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 levelLaplaciandefinition
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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
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  10. Partial derivative definition
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  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Higher partial derivatives definition
  15. Iterated derivative definition
  16. C^k function characterised by continuous derivatives statement
  17. Vector field definition
  18. Divergence definition
  19. Laplacian definition

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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, Averages
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DefinesLaplacian
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