Template:Intorient/doc

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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 \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=x45px oiint=x45px oiiint=x45px varoint=x45px varoiint=x45px varoiiint=x45px
Counterclockwise
orientation
ointctr=x45px oiintctr=x45px oiiintctr=x45px varointctr=x45px varoiintctr=x45px varoiiintctr=x45px
The default is x45px
  • intsubscpt the subscript below the integral
  • integrand the text or formula immediately after the formula

All parameters are optional.

Examples

{{intorient
| preintegral=<math>W=</math>
| symbol = varoint
| intsubscpt = <math>{\scriptstyle \Gamma}</math>
| integrand = <math>p{\rm d}V</math>
}}
{{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: x44px<math>{\scriptstyle \partial S}</math> <math>\mathbf{F}\cdot{\rm d}\mathbf{r}=-</math> x44px<math>{\scriptstyle \partial S}</math> <math>\mathbf{F}\cdot{\rm d}\mathbf{r}</math>
{{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: x44px<math>{\scriptstyle \Sigma}</math> <math>(E+H\wedge T) {\rm d}^2 \Sigma</math>
{{Intorient|
| preintegral = 
|symbol=oiiintctr
| intsubscpt = <math>{\scriptstyle \Sigma}</math>
| integrand = <math>(E+H\wedge T) {\rm d}^2 \Sigma</math>
}}
x44px<math>{\scriptstyle \Omega}</math> <math>(E+H\wedge T) {\rm d}^4\Omega</math>
{{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: