Math 2.0

Real functions Definition

Divergence

Builds on Vector fieldPartial derivativeJacobian matrix

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let F=(F1,…,Fn) ⁣:U→ℝnF = (F_1, \dots, F_n) \colon U \to \mathbb{R}^n be a vector field whose partial derivatives ∂Fi/∂xi\partial F_i / \partial x_i exist on UU. The divergence of FF is the function div⁡F ⁣:U→ℝ\operatorname{div} F \colon U \to \mathbb{R}, div⁡F=∇⋅F=∂F1∂x1+∂F2∂x2+⋯+∂Fn∂xn.\operatorname{div} F = \nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \dots + \frac{\partial F_n}{\partial x_n}.

Remarks

The divergence is the trace of the Jacobian matrix of FF. For F(x)=xF(x) = x on ℝn\mathbb{R}^n, div⁡F=n\operatorname{div} F = n. For the rotation field F(x,y)=(−y,x)F(x, y) = (-y, x), div⁡F=0\operatorname{div} F = 0. The divergence is linear, and div⁡(ϕF)=∇ϕ⋅F+ϕdiv⁡F\operatorname{div}(\phi F) = \nabla \phi \cdot F + \phi \operatorname{div} F for a differentiable function ϕ\phi. Take a C1C^1 field and a small region around x0x_0. The divergence at x0x_0 is the limit of the outward flux through the boundary of the region, divided by its volume. This follows from the divergence theorem.

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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 derivativedefinitionDifferentiable 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 levelDivergencedefinition
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  1. Euclidean space object
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  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
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  12. Jacobian matrix definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Vector field definition
  17. Divergence definition

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38 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Divergence, Cartesian coordinates
  2. Web https://en.wikipedia.org/wiki/Divergence, Properties
About this page
Tag02NSMQVQWK
URIm2://02NSMQVQWK
Kinddefinition
WritingW1 stated
MSC26B12
DefinesDivergence
UsesVector field, Partial derivative, Jacobian matrix

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