Template:Intorient

$\ointclockwise$

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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[edit]

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[edit]

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>
}}

Arguments[edit]

Examples[edit]

See also[edit]