Math 2.0

statement proposition

Divergence of a curl is zero

Statement

Let U⊆R3U \subseteq \mathbb{R}^3 be open, and let A ⁣:U→R3A \colon U \to \mathbb{R}^3 be a vector field of class C2C^2. Then ∇⋅(∇×A)=0on U.\nabla \cdot (\nabla \times A) = 0 \quad \text{on } U.

Remarks

Proof: with A=(A1,A2,A3)A = (A_1, A_2, A_3), the divergence of the curl is ∂x(∂yA3−∂zA2)+∂y(∂zA1−∂xA3)+∂z(∂xA2−∂yA1)\partial_x (\partial_y A_3 - \partial_z A_2) + \partial_y (\partial_z A_1 - \partial_x A_3) + \partial_z (\partial_x A_2 - \partial_y A_1). The six terms cancel in pairs by Schwarz's theorem. So a field that has a vector potential is divergence-free. On an open ball, the converse is also true: a C1C^1 divergence-free field is the curl of some field. The identity is a special case of the rule d d = 0 for the exterior derivative.

Explanations

L1 human

The curl of a field describes rotation, and pure rotation does not create or destroy anything. So the curl of any smooth field has no sources or sinks, which means its divergence is zero.

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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…statementSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementVector field, definition, no evidence levelVector fielddefinitionSchwarz's theorem for C^2 functions, statement, no evidence levelSchwarz's theorem for…statementCurl, definition, no evidence levelCurldefinitionDivergence, definition, no evidence levelDivergencedefinitionDivergence of a curl is zero, statement, no evidence levelDivergence of a curl …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. Differentiable on an open set definition
  12. Jacobian matrix definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Symmetry of second derivatives statement
  17. Vector field definition
  18. Schwarz's theorem for C^2 functions statement
  19. Curl definition
  20. Divergence definition
  21. Divergence of a curl is zero statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Vector_calculus_identities, Divergence of curl is zero
  2. Web https://en.wikipedia.org/wiki/Divergence, Properties

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