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IEVref: | 102-05-22 | ID: | |

Language: | en | Status: Standard | |

Term: | rotation | ||

Synonym1: | curl [Preferred] | ||

Synonym2: | |||

Synonym3: | |||

Symbol: | |||

Definition: | vector rot associated at each point of a given space region with a vector U, equal to the limit of the integral over a closed surface S of the vector product of the vector surface element and the vector U, divided by the volume of the interior of the surface, when the surface is contained in a sphere the radius of which tends to zeroU$\mathsf{rot}U=\underset{V\to 0}{\mathrm{lim}}\frac{1}{V}{\displaystyle \underset{\text{S}}{\u222f}{e}_{n}\times U\mathrm{d}A}$ where Note 1 to entry: In orthonormal Cartesian coordinates, the three coordinates of the rotation are: $\frac{\partial \text{\hspace{0.17em}}{U}_{z}}{\partial \text{\hspace{0.17em}}y}-\frac{\partial \text{\hspace{0.17em}}{U}_{y}}{\partial \text{\hspace{0.17em}}z}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}{U}_{x}}{\partial \text{\hspace{0.17em}}z}-\frac{\partial \text{\hspace{0.17em}}{U}_{z}}{\partial \text{\hspace{0.17em}}x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}{U}_{y}}{\partial \text{\hspace{0.17em}}x}-\frac{\partial \text{\hspace{0.17em}}{U}_{x}}{\partial \text{\hspace{0.17em}}y}$ Note 2 to entry: The rotation of a polar vector is an axial vector and the rotation of an axial vector is a polar vector. Note 3 to entry: The rotation of the vector field | ||

Publication date: | 2008-08 | ||

Source: | |||

Replaces: | |||

Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2017-08-24: Added a <b> tag that was missing for first U. LMO | ||

CO remarks: | |||

TC/SC remarks: | |||

VT remarks: | |||

Domain1: | |||

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Domain5: |

$\mathsf{rot}U=\underset{V\to 0}{\mathrm{lim}}\frac{1}{V}{\displaystyle \underset{\text{S}}{\u222f}{e}_{n}\times U\mathrm{d}A}$

where *e*_{n}d*A* is the vector surface element oriented outwards and *V* is the volume

Note 1 to entry: In orthonormal Cartesian coordinates, the three coordinates of the rotation are:

$\frac{\partial \text{\hspace{0.17em}}{U}_{z}}{\partial \text{\hspace{0.17em}}y}-\frac{\partial \text{\hspace{0.17em}}{U}_{y}}{\partial \text{\hspace{0.17em}}z}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}{U}_{x}}{\partial \text{\hspace{0.17em}}z}-\frac{\partial \text{\hspace{0.17em}}{U}_{z}}{\partial \text{\hspace{0.17em}}x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}{U}_{y}}{\partial \text{\hspace{0.17em}}x}-\frac{\partial \text{\hspace{0.17em}}{U}_{x}}{\partial \text{\hspace{0.17em}}y}$

Note 2 to entry: The rotation of a polar vector is an axial vector and the rotation of an axial vector is a polar vector.

Note 3 to entry: The rotation of the vector field ** U** is denoted by $\mathsf{rot}U$ or $\mathbf{\nabla}\times U$. In some English texts, the rotation is denoted by $\mathsf{curl}U$.