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: <math>W = </math> \varointclockwise<math>{\scriptstyle \Gamma}</math> <math>p \, {\rm d}V</math>

<source lang="tex"><math>W = </math> \varointclockwise<math>{\scriptstyle \Gamma}</math> <math>p \, {\rm d}V</math> </source>

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

<source lang="tex"> \varointclockwise<math>{\scriptstyle \Gamma}</math> <math>\frac{{\rm d}z}{(z+a)^3 \, z^{1/2}}</math> </source>

• Line integrals of vector fields: \ointclockwise<math>{\scriptstyle \partial S}</math> <math>\mathbf{F} \cdot {\rm d}\mathbf{r} = -</math> \ointctrclockwise<math>{\scriptstyle \partial S}</math> <math>\mathbf{F} \cdot {\rm d}\mathbf{r}</math>

<source lang="tex"> \ointclockwise<math>{\scriptstyle \partial S}</math> <math>\mathbf{F} \cdot {\rm d}\mathbf{r} = -</math> \ointctrclockwise<math>{\scriptstyle \partial S}</math> <math>\mathbf{F} \cdot {\rm d}\mathbf{r}</math> </source>

• Other examples: \oiintclockwise<math>{\scriptstyle \Sigma}</math> <math>(E + H \wedge T) \, {\rm d}^2 \Sigma</math>

<source lang="tex"> \oiiintctrclockwise<math>{\scriptstyle \Sigma}</math> <math>(E + H \wedge T) \, {\rm d}^2 \Sigma</math> </source>

\varoiiintctrclockwise<math>{\scriptstyle \Omega}</math> <math>(E + H \wedge T) \, {\rm d}^4 \Omega</math>

<source lang="tex"> \varoiiintctrclockwise<math>{\scriptstyle \Omega}</math> <math>(E + H \wedge T) \, {\rm d}^4 \Omega</math> </source>

See also

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

• {{Oiint}}
• {{Oiiint}}