Шаблон: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 \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: $W=$${\scriptstyle \Gamma }$$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: ${\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)^3z^{1/2}}</math>
}}

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

{{intorient|
| 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: ${\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>
}}

${\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>
}}

See also

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

• {{Oiint}}
• {{Oiiint}}