The SurfaceInt(f, dom) command computes the surface integral of the function f over the surface specified on the right side of dom.  The left side of dom is the list of variables of the function f.

Surfaces are represented by unevaluated function calls.  The possible surfaces are Box(r1, r2, r3), Sphere(cen, rad), and Surface(v).

Box(r1, r2, r3)

Each ri must have type algebraic..algebraic.  These represent the sides of the box, and the surface integral taken over each face of the box.

Sphere(cen, rad)

The first parameter of Sphere, cen, must have type 'Vector'(3, algebraic) and rad must have type algebraic. These represent the center and radius of the sphere, respectively.  If a coordinate system attribute is specified on cen, the center is interpreted in this coordinate system.

Surface(v)

The first argument of v must have type 'Vector'(3, algebraic). The second argument can be of the form list(name) = region where the names of the two parameters are in the list and the region of the second argument is any valid two dimensional region that VectorCalculus:-int accepts, or an equation of the form name=range.  If the region is in the form name=range, the third argument, range is also of the form name=range.  This gives explicit ranges to the two parameters. If the two-dimensional region over which the surface is defined is given by name as Ellipse, Triangle, Circle, Sector, or Rectangle, this geometric region will be in the parameter space, and not necessarily in the space of the variables of integration.

Finally, an optional fourth argument can be coords=name or coordinates=name. It is the coordinate system in which v is interpreted. The variables in the integrand are then also interpreted in this same coordinate system. In the second example below,  setting the coordinate system to cylindrical causes $a,b,c$ to be  interpreted as $r$,$\mathrm{\theta }$,$z$, the variables of the cylindrical system.  The components of v are interpreted as the parametric equations for a surface defined in cylindrical coordinates:  $r\left(u\,v\right)=u,\mathrm{\theta }\left(u\,v\right)=v,z\left(u\,v\right)=1$.  $u=r$ can be inferred from the known value of the surface-area element in cylindrical coordinates.

The SurfaceInt(f, dom, inert) command returns the integral form of the surface integral of f over dom.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{SurfaceInt}\left(1\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Surface}\left(⟨r\&comma;s\&comma;t⟩\&comma;s=0..\mathrm{\pi }\&comma;t=0..2\mathrm{\pi }\&comma;\mathrm{coords}=\mathrm{spherical}\right)\right)\mathbf{assuming}0<r$

$4\mathrm{\pi }{r}^2$

$\mathrm{simplify}\left(\mathrm{SurfaceInt}\left(f\left(a\&comma;b\&comma;c\right)\&comma;\left[a\&comma;b\&comma;c\right]=\mathrm{Surface}\left(⟨u\&comma;v\&comma;1⟩\&comma;u=0..1\&comma;v=0..2\&comma;\mathrm{coords}=\mathrm{cylindrical}\right)\&comma;\mathrm{inert}\right)\right)$

${\int }_0^1{\int }_0^{2}f\left(u\&comma;v\&comma;1\right)\mathrm{csgn}\left(u\right)u\&DifferentialD;v\&DifferentialD;u$

$\mathrm{SurfaceInt}\left(x+y+z\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Surface}\left(⟨s\&comma;t\&comma;4-2s-t⟩\&comma;\left[s\&comma;t\right]=\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨1\&comma;0⟩\&comma;⟨0\&comma;1⟩\right)\right)\right)$

$\frac{11\sqrt{6}}{6}$

$\mathrm{SurfaceInt}\left(y^2\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Sphere}\left(⟨0\&comma;0\&comma;0⟩\&comma;r\right)\right)$

$\frac{4\mathrm{\pi }{r}^4}{3}$

$\mathrm{SurfaceInt}\left(xyz\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Box}\left(1..2\&comma;3..4\&comma;5..6\right)\right)$

$\frac{693}{4}$

$\mathrm{SurfaceInt}\left(xyz\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Box}\left(1..2\&comma;3..4\&comma;5..6\right)\&comma;'\mathrm{inert}'\right)$

${\int }_5^6\left({\int }_3^{4}3st\&DifferentialD;s\right)\&DifferentialD;t+{\int }_5^6\left({\int }_1^{2}7st\&DifferentialD;s\right)\&DifferentialD;t+{\int }_3^4\left({\int }_1^{2}11st\&DifferentialD;s\right)\&DifferentialD;t$