Fyrimynd:Intorient/doc

Hetta er ein dokumentatión undirsíða til Fyrimynd:Intorient.
Hon inniheldur kunning um nýtsluna, bólkar og um annað innihald, ið ikki er við í uppruna fyrimynd síðuni.

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
Select one of...	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: $W=$ $\varointclockwise$ ${\scriptstyle \Gamma }$ $p{\rm {d}}V$

{{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: $\varointclockwise$ ${\scriptstyle \Gamma }$ ${\frac {{\rm {d}}z}{(z+a)^{3}z^{1/2}}}$

{{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: $\ointclockwise$ ${\scriptstyle \partial S}$ $\mathbf {F} \cdot {\rm {d}}\mathbf {r} =-$ $\ointctrclockwise$ ${\scriptstyle \partial S}$ $\mathbf {F} \cdot {\rm {d}}\mathbf {r}$

{{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>
}}

Sí eisini

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

{{Oiint}}
{{Oiiint}}