Template:Intorient/doc

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	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$\mathrm{\Gamma }$ $p\mathrm{d}V$

{{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$\mathrm{\Gamma }$ $\frac{\mathrm{d}z}{(z+a{)}^3{z}^{1/2}}$

{{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$\partial S$ $\mathbf{𝐅}\cdot \mathrm{d}\mathbf{𝐫}=-$ \ointctrclockwise$\partial S$ $\mathbf{𝐅}\cdot \mathrm{d}\mathbf{𝐫}$

{{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$\mathrm{\Sigma }$ $(E+H\wedge T){\mathrm{d}}^2\mathrm{\Sigma }$

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

\varoiiintctrclockwise$\mathrm{\Omega }$ $(E+H\wedge T){\mathrm{d}}^4\mathrm{\Omega }$

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

See also

Non-oriented boundary integrals over a 2-surface and 3-surface can be implemented respectively by:

• {{Oiint}}
• {{Oiiint}}

These symbols are currently not supported in MediaWiki's Math mode and this feature request is tracked in phab:T182127.