Template:Intorient

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\ointclockwise

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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[edit]

  • 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[edit]

{{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: \ointclockwise{\scriptstyle \partial S}\mathbf{F}\cdot{\rm d}\mathbf{r}=-\ointctrclockwise{\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: \oiintclockwise{\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>
}}
\varoiiintctrclockwise{\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[edit]

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