The Flux(f, dom) calling sequence computes the flux of the vector field f through the surface (3-D) or curve (2-D) specified by dom.

Specify the surface or curve dom using unevaluated function calls. The surfaces are Box, Sphere, and Surface. The curves are Circle, Line, LineSegments, and Path.

Box(r1, r2, r3, direction)

Each ri must have type algebraic..algebraic. These represent the sides of the box. The integral is taken over each face of the box.

The optional fourth argument, direction, specifies the direction of the normal vector. It must be inward or outward.  The default value is outward.

Sphere(center, radius, direction)

The first parameter of Sphere, center, must have type 'Vector'(3, algebraic) and radius must have type algebraic.  These represent the center and radius of the sphere, respectively.

The optional third argument, direction, specifies the direction of the normal vector.  It must be inward or outward.  The default value is outward.

Surface(v, range)

The first argument, v, must have type 'Vector'(3, algebraic). The second argument, range, can be:

[name1, name2] = region(arguments) where region' is any two-dimensional region that Student[VectorCalculus][int] accepts: Circle, Ellipse, Rectangle, Region, Sector, or Triangle.

name1=range1, name2=range2  This explicitly specifies the ranges for the two parameters.

The normal vector is the cross-product of the derivatives of v.

Circle(center, radius, direction)

The parameter center is the center of the circle and must have type 'Vector'(algebraic). The parameter radius is the radius of the circle and must have type algebraic.

The optional third argument, direction, specifies the direction of the normal vector.  It must be inward or outward.  The default value is outward.

Line(p1, p2)

The parameters p1 and p2 must have type 'Vector'(algebraic). They represent the endpoints of the directed line segment from p1 to p2.

The normal is taken Pi/2 to the right of the direction of the directed line segment.

LineSegments(p1, p2, ..., pk)

Similar to Line(p1, p2), the parameters pi represent the endpoints of $k-1$ line segments. The path is the collection of line segments directed from p1 to p2, p2 to p3, ..., and p(k-1) to pk.

Path(v, range)

The first parameter, v, is a Vector representing the components of the path. The second parameter, range, must have type {range, name=range}. If no parameter name is specified in range, it is inferred from v.

The normal is taken Pi/2 to the right of the tangent vector that points in the direction of increasing parameter.

The options arguments primarily control plot options.

output = value, plot, or integral

This option controls the return value of the command.

output = value returns the value of the line integral. Plot options are ignored if output = value.  This is the default value.

output = plot returns a plot of the path, normal vectors of the path, and vector field. In 2-D, plots are available for the regions Path, Circle, LineSegment and Line. In 3-D, plots are available for the regions Surface (with two explicit ranges), Sphere and Box.

output = integral returns the inert representation of the line integral. Plot options are ignored if output = integral.

fieldoptions = list

A list of plot options for plotting the Vector field. For more information on plotting options, see plot/options. The default value is []. A list of plot options for plotting the Vector field. The field is plotted using a call to plots[fieldplot] or plots[fieldplot3d], so options valid for these commands can be passed in this option.

pathoptions = list

A list of plot options for plotting the path. For more information on plotting options, see plot/options. The default value is [].

surfaceoptions = list

A list of plot options for plotting the surface. For more information on plotting options, see plot/options. The default value is [].

title = string

Give the plot a title. The default value is the empty string.

vectoroptions = list

A list of plot options for plotting the normal vectors. For more information on plotting options, see plot/options. The default value is []. Free Vectors and rooted Vectors are plotted using plots[arrow].

view = [realcons..realcons, realcons..realcons, realcons..realcons] (3-D) or [realcons..realcons, realcons..realcons] (2-D)

Specify the plot view. For more information, see plot3d/option or plot/options.

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Surface}\left(⟨r\,s\,t⟩\,s=0..\mathrm{\pi }\,t=0..2\mathrm{\pi }\right)\right)$

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Surface}\left(⟨r\,s\,t⟩\,\left[s\,t\right]=\mathrm{Circle}\left(⟨0\,0⟩\,2\right)\right)\right)$

$4r\mathrm{\pi }$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Surface}\left(⟨2\,s\,t⟩\,\left[s\,t\right]=\mathrm{Circle}\left(⟨0\,0⟩\,2\right)\right)\right)$

$8\mathrm{\pi }$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨y\,-x\,0⟩\right)\,\mathrm{Surface}\left(⟨s\,t\,s^2+t^2⟩\,s=0..1\,t=2..3\right)\right)$

$0$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Sphere}\left(⟨0\,0\,0⟩\,r\right)\right)$

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Sphere}\left(⟨0\,0\,0⟩\,r\right)\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_0^{2\mathrm{\pi }}\left({\int }_0^{\mathrm{\pi }}{r}^3\sin \left(\mathrm{\phi }\right)\ⅆ\mathrm{\phi }\right)\ⅆ\mathrm{\theta }$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\,\mathrm{Sphere}\left(⟨0\,0\,0⟩\,r\,'\mathrm{inward}'\right)\right)$

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨y\,-x\,0⟩\right)\,\mathrm{Box}\left(1..2\,3..4\,5..6\right)\,\mathrm{output}=\mathrm{plot}\,\mathrm{surfaceoptions}=\left[\mathrm{color}=\mathrm{yellow}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y⟩\right)\,\mathrm{Circle}\left(⟨0\,0⟩\,r\,'\mathrm{inward}'\right)\right)$

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y⟩\right)\,\mathrm{Path}\left(⟨\cos \left(t\right)\,\sin \left(t\right)⟩\,t=0..2\mathrm{\pi }\right)\right)$

$2\mathrm{\pi }$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y⟩\right)\,\mathrm{Line}\left(⟨0\,0⟩\,⟨1\,1⟩\right)\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_0^{1}0\ⅆt$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨x\,y⟩\right)\,\mathrm{LineSegments}\left(⟨0\,0⟩\,⟨1\,1⟩\,⟨2\,3⟩\right)\,\mathrm{output}=\mathrm{plot}\,\mathrm{pathoptions}=\left[\mathrm{color}=\mathrm{blue}\right]\,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{black}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨\frac{1}{x^2}\,0\,1⟩\right)\,\mathrm{Box}\left(-1..1\,-1..1\,-1..1\right)\,\mathrm{output}=\mathrm{plot}\,\mathrm{scaling}=\mathrm{constrained}\right)$