# Template:Intorient

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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
1-surface 2-surface 3-surface 1-surface 2-surface 3-surface
Clockwise
orientation
oint= oiint= oiiint= varoint= varoiint= varoiiint=
Counterclockwise
orientation
ointctr= oiintctr= oiiintctr= varointctr= varoiintctr= varoiiintctr=
The default is
• 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: ${\displaystyle W=}$ ${\displaystyle {\scriptstyle \Gamma }}$ ${\displaystyle p\,{\rm {d}}V}$
{{intorient
| preintegral = $W =$
| symbol = varoint
| intsubscpt = ${\scriptstyle \Gamma}$
| integrand = $p \, {\rm d}V$
}}

• In complex analysis for contour integrals: ${\displaystyle {\scriptstyle \Gamma }}$ ${\displaystyle {\frac {{\rm {d}}z}{(z+a)^{3}\,z^{1/2}}}}$
{{intorient|
| preintegral =
| symbol = varoint
| intsubscpt = ${\scriptstyle \Gamma}$
| integrand = $\frac{{\rm d}z}{(z+a)^3 \, z^{1/2}}$
}}

• Line integrals of vector fields: ${\displaystyle {\scriptstyle \partial S}}$ ${\displaystyle \mathbf {F} \cdot {\rm {d}}\mathbf {r} =-}$ ${\displaystyle {\scriptstyle \partial S}}$ ${\displaystyle \mathbf {F} \cdot {\rm {d}}\mathbf {r} }$
{{intorient|<!-- You have to nest things this way to insure everything stays in one line. -->
| preintegral = {{intorient|
| preintegral =
| symbol = oint
| intsubscpt = ${\scriptstyle \partial S}$
| integrand = $\mathbf{F} \cdot {\rm d}\mathbf{r} = -$
}}
|symbol=ointctr
| intsubscpt = ${\scriptstyle \partial S}$
| integrand = $\mathbf{F} \cdot {\rm d}\mathbf{r}$
}}

• Other examples: ${\displaystyle {\scriptstyle \Sigma }}$ ${\displaystyle (E+H\wedge T)\,{\rm {d}}^{2}\Sigma }$
{{Intorient|
| preintegral =
| symbol = oiiintctr
| intsubscpt = ${\scriptstyle \Sigma}$
| integrand = $(E + H \wedge T) \, {\rm d}^2 \Sigma$
}}

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