Math 2.0

Potential theory Definition

Harmonic function

Builds on LaplacianLaplacian as a sum of second partial derivativesC^k function characterised by continuous derivatives

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open. A function u ⁣:U→ℝu \colon U \to \mathbb{R} is harmonic on UU if uu is of class C^2 on UU and its Laplacian vanishes on UU: Δu=∂2u∂x12+⋯+∂2u∂xn2=0.\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \dots + \frac{\partial^2 u}{\partial x_n^2} = 0.

Remarks

Each affine function is harmonic. In the plane, x2−y2x^2 - y^2, xyx y and exsin⁡ye^x \sin y are harmonic, and so is ln⁡(x2+y2)\ln(x^2 + y^2) on ℝ2∖{0}\mathbb{R}^2 \setminus \{0\}. On ℝ3∖{0}\mathbb{R}^3 \setminus \{0\}, 1/∥x∥1 / \|x\| is harmonic. It is the potential of a unit point charge at the origin. The real and imaginary parts of a holomorphic function of z=x+iyz = x + i y are harmonic functions of (x,y)(x, y). Mean value property: let the closed ball of centre aa and radius rr lie in UU. Then u(a)u(a) is the average of uu on the sphere of radius rr about aa, and also on the ball.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
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…statementHarmonic function, definition, no evidence levelHarmonic functiondefinition
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. 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
  22. Harmonic function definition

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_function, lead section
  2. Web https://en.wikipedia.org/wiki/Harmonic_function, Examples
  3. Web https://en.wikipedia.org/wiki/Harmonic_function, The mean value property
About this page
Tag0W5NJQ8MKY
URIm2://0W5NJQ8MKY
Kinddefinition
WritingW1 stated
MSC31B05
DefinesHarmonic function
UsesLaplacian, Laplacian as a sum of second partial derivatives, C^k function characterised by continuous derivatives

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by