Math 2.0

Real functions Definition

Curl

Builds on Vector fieldPartial derivative

Definition

Let U⊆ℝ3U \subseteq \mathbb{R}^3 be open, and let F=(P,Q,R) ⁣:U→ℝ3F = (P, Q, R) \colon U \to \mathbb{R}^3 be a vector field whose first partial derivatives exist on UU. The curl of FF is the vector field curl⁡F ⁣:U→ℝ3\operatorname{curl} F \colon U \to \mathbb{R}^3, curl⁡F=∇×F=(∂R∂y−∂Q∂z, ∂P∂z−∂R∂x, ∂Q∂x−∂P∂y).\operatorname{curl} F = \nabla \times F = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right). A field with curl⁡F=0\operatorname{curl} F = 0 on UU is irrotational, or curl-free.

Remarks

As a mnemonic, ∇×F\nabla \times F is the formal determinant of the matrix with rows (𝐢,𝐣,𝐤)(\mathbf{i}, \mathbf{j}, \mathbf{k}), (∂x,∂y,∂z)(\partial_x, \partial_y, \partial_z) and (P,Q,R)(P, Q, R). For F(x,y,z)=(−y,x,0)F(x, y, z) = (-y, x, 0), curl⁡F=(0,0,2)\operatorname{curl} F = (0, 0, 2). For F(x,y,z)=(x,y,z)F(x, y, z) = (x, y, z), curl⁡F=0\operatorname{curl} F = 0. For a plane field (P,Q)(P, Q), one uses the scalar curl ∂Q/∂x−∂P/∂y\partial Q / \partial x - \partial P / \partial y, the third component of the curl of (P,Q,0)(P, Q, 0). It appears in Green's theorem. Take small loops orthogonal to a unit vector nn. The component of curl⁡F\operatorname{curl} F along nn is the limit of the circulation of FF around these loops, divided by the enclosed area. This follows from Stokes' 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…conceptContinuity 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 …definitionHigher 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 fielddefinitionCurl, definition, no evidence levelCurldefinition
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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. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Vector field definition
  15. Curl definition

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Curl_(mathematics), In Cartesian coordinate systems
  2. Web https://en.wikipedia.org/wiki/Curl_(mathematics), Examples
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WritingW1 stated
MSC26B12
DefinesCurl
UsesVector field, Partial derivative

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