IEVref: 102-05-22 ID: Language: en Status: Standard Term: rotation Synonym1: curl [Preferred] Synonym2: Synonym3: Symbol: Definition: vector rot U 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 zero$\mathbsf{rot}U=\underset{V\to 0}{\mathrm{lim}}\frac{1}{V}\underset{\text{S}}{∯}{e}_{n}×U\mathrm{d}A$ where endA is the vector surface element oriented outwards and V is the volumeNote 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 $\mathbsf{rot}U$ or $\mathbf{\nabla }×U$. In some English texts, the rotation is denoted by $\mathbsf{curl}U$. 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 tag that was missing for first U. LMO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: