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$\ointclockwise$

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This template is used to include the oriented integrals around closed surfaces (or hypersurfaces in higher dimensions), usually in a mathematical formula. They are additional symbols to the non-oriented integrals \oiint and \oiiint which are not yet rendered on Wikipedia.

Arguments

preintegral the text or formula immediately before the integral
symbol the integral symbol,

Select one of...	Arrow up, integrals over a closed			Arrow down, integrals over a closed
Select one of...	1-surface	2-surface	3-surface	1-surface	2-surface	3-surface
Clockwise orientation	oint = $\oiint$	oiint = $\oiint$	oiiint = $\oiint$	varoint = $\oiint$	varoiint = $\oiint$	varoiiint = $\oiint$
Counterclockwise orientation	ointctr = $\oiint$	oiintctr = $\oiint$	oiiintctr = $\oiint$	varointctr = $\oiint$	varoiintctr = $\oiint$	varoiiintctr = $\oiint$

The default is $\oiint$

intsubscpt the subscript below the integral
integrand the text or formula immediately after the formula

All parameters are optional.

Examples

The work done in a thermodynamic cycle on an indicator diagram: $W=$ $\varointclockwise$ ${\scriptstyle \Gamma }$ $p\,{\rm {d}}V$

{{intorient
| preintegral = <math>W = </math>
| symbol = varoint
| intsubscpt = <math>{\scriptstyle \Gamma}</math>
| integrand = <math>p \, {\rm d}V</math>
}}

In complex analysis for contour integrals: $\varointclockwise$ ${\scriptstyle \Gamma }$ ${\frac {{\rm {d}}z}{(z+a)^{3}\,z^{1/2}}}$

{{intorient|
| preintegral = 
| symbol = varoint
| intsubscpt = <math>{\scriptstyle \Gamma}</math>
| integrand = <math>\frac{{\rm d}z}{(z+a)^3 \, z^{1/2}}</math>
}}

Line integrals of vector fields: $\ointclockwise$ ${\scriptstyle \partial S}$ $\mathbf {F} \cdot {\rm {d}}\mathbf {r} =-$ $\ointctrclockwise$ ${\scriptstyle \partial S}$ $\mathbf {F} \cdot {\rm {d}}\mathbf {r}$

{{intorient|<!-- You have to nest things this way to insure everything stays in one line. -->
| preintegral = {{intorient|
| preintegral =
| symbol = oint
| intsubscpt = <math>{\scriptstyle \partial S}</math>
| integrand = <math>\mathbf{F} \cdot {\rm d}\mathbf{r} = -</math>
}}
|symbol=ointctr
| intsubscpt = <math>{\scriptstyle \partial S}</math>
| integrand = <math>\mathbf{F} \cdot {\rm d}\mathbf{r}</math>
}}

Other examples: $\oiintclockwise$ ${\scriptstyle \Sigma }$ $(E+H\wedge T)\,{\rm {d}}^{2}\Sigma$

{{Intorient|
| preintegral = 
| symbol = oiiintctr
| intsubscpt = <math>{\scriptstyle \Sigma}</math>
| integrand = <math>(E + H \wedge T) \, {\rm d}^2 \Sigma</math>
}}

$\varoiiintctrclockwise$

{\scriptstyle \Omega }

(E+H\wedge T)\,{\rm {d}}^{4}\Omega

{{Intorient|
| preintegral = 
| symbol = varoiiintctr
| intsubscpt = <math>{\scriptstyle \Omega}</math>
| integrand = <math>(E + H \wedge T) \, {\rm d}^4 \Omega</math>
}}

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Template:Intorient

Arguments

Examples

See also

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