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| '''Curl''' measures the spin of a [[Calculus/VectorField|vector field]]. | '''Curl''' measures the circulation density of a [[Calculus/VectorField|vector field]]. |
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| The curl of a [[Calculus/VectorField|vector field]] given as ''F = <P(x,y), Q(x,y)>'' is calculated as: | Curl refers to how much rotation there is in a [[Calculus/VectorField|vector field]]. It is measured using [[Calculus/Derivative|differentiation]]. For a vector field given as ''F = <P(x,y), Q(x,y)>'', it is calculated as: |
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| where ''Q,,x,, = ∂Q/∂x'' and ''P,,y,, = ∂P/∂y''. | where ''k̂'' is the [[Calculus/UnitVector|unit basis vector]]. |
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| For an ''F'' given as ''<P(x,y,z), Q(x,y,z), R(x,y,z)>'', it is calculated as: | For a vector field given as ''F = <P,Q,R>'', it is calculated as: |
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| where ''P,,z,, = ∂P/∂z'', ''Q,,z,, = ∂Q/∂z'', ''R,,x,, = ∂R/∂x'', and ''R,,y,, = ∂R/∂y''. | where ''î'', ''ĵ'', and ''k̂'' are all unit basis vectors. |
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| An '''irrotational''' [[Calculus/VectorField|vector field]] has zero curl. |
Curl
Curl measures the circulation density of a vector field.
Contents
Description
Curl refers to how much rotation there is in a vector field. It is measured using differentiation.
For a vector field given as F = <P(x,y), Q(x,y)>, it is calculated as:
where k̂ is the unit basis vector.
For a vector field given as F = <P,Q,R>, it is calculated as:
where î, ĵ, and k̂ are all unit basis vectors.
Note that curl returns a new vector field. Generally this vector field is then evaluated at a given point.
Properties
An irrotational vector field has zero curl.
The curl of a conservative vector field is always 0.