The Flux(f, dom) command computes the flux of the vector field f through the surface or curve specified by dom.

Surfaces and curves are represented by unevaluated function calls. The possible surfaces are Box(r1, r2, r3, dir), Sphere(cen, rad, dir), and Surface(v, param). The possible curves are Arc(obj, start, finish), Circle(cen, rad, dir), Ellipse(eqn, varx, vary, dir), Line(p1, p2), LineSegments(p1, p2, ..., pk), and Path(v, rng, c).

Box(r1, r2, r3, dir)

Each ri must have type algebraic..algebraic (a range). These represent the sides of the box, and the integral is taken over each face of the box.

If the optional fourth argument dir is specified, it specifies the direction of the normal vector. It must be the word inward or outward.  The default is outward.

Sphere(cen, rad, dir)

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 that coordinate system.

If the optional third argument, dir, is specified, it specifies the direction of the normal vector.  It must be the word inward or outward.  The default is outward.

Surface(v, param)

This construct can be used to define a general two-parameter surface. The first argument v must be a free Vector or a position Vector; it represents the surface through which the flux will be calculated. The second argument, param, provides information about the parameters that occur in v. It must be of the form [x1, x2] = region, where the names x1 and x2 are the two parameter names and region specifies the bounds on those two parameter names. The region argument must either be (1) any valid two-dimensional region structure that VectorCalculus[int] accepts or (2) a sequence of two equations of the form x1 = range1, x2 = range2, where range1 and range2 are explicit ranges that bound x1 and x2, respectively.

Finally, an optional last argument, c, can be given, and must be of the form coords=sys or coordinates=sys. This is the coordinate system in which v is interpreted. Note: if this argument is supplied, any existing coordinate attribute on v is overwritten (and therefore ignored).

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

Arc(obj, start, finish)

The first parameter of Arc, obj, is a Circle or Ellipse structure. The Arc structure defines a segment of the circle or ellipse with endpoints specified by the start and finish angles.

To define precisely how the endpoints are determined from the given start and finish angles, it suffices to discuss only circles and ellipses centered at the origin. For a circle or ellipse centered elsewhere, the start and finish endpoints are determined as if the circle or ellipse were first translated to the origin.

For a Circle centered at the origin, angle is measured counterclockwise from the positive x-axis. Therefore, the angle $\frac{3\mathrm{\pi }}{2}$ specifies the negative y-axis.

To define how angle is measured for an Ellipse centered at the origin, we first define the right semimajor axis of the ellipse to be the semimajor axis in the right half-plane (the first and fourth quadrants of the plane). If the major axis of the ellipse is coincident with the y-axis, then its right semimajor axis is defined to be the one on the negative y-axis. Thus, for an ellipse centered at the origin with its major axis sitting on the line y = x, its right semimajor axis is the one inside the first quadrant.

For an Ellipse centered at the origin, angle is measured counterclockwise from its right semimajor axis. Therefore, in the example ellipse given in the previous paragraph, the angle $\frac{\mathrm{\pi }}{4}$ specifies the positive y-axis.

Once the terminal arm of the angle is determined, that angle specifies the point where the terminal arm (viewed as a ray) intersects the circle or ellipse. The arc is always traversed in its entirety from start to finish; therefore, it is possible to traverse the circle or ellipse for more than one full revolution or to traverse in the opposite (clockwise) direction by specifying appropriate start and finish angles.

Circle(cen, rad)

The parameter cen is the center of the circle and must have type 'Vector'(algebraic) and rad is the radius of the circle and must have type algebraic.  If a coordinate system attribute is specified on cen, it is interpreted in that coordinate system.

If the optional third argument dir is specified, it specifies the direction of the normal vector. It must be the word inward or outward.  The default is outward.

Ellipse(cen, a, b, phi, dir)

The parameter cen is the center of the ellipse and must have type 'Vector'(algebraic). If a coordinate system attribute is specified on cen, it is interpreted in that coordinate system. The parameters a and b are the lengths of the semimajor and semiminor axes, respectively. The resulting ellipse is constructed via the following process: Start with an ellipse centered at the origin having the specified axes lengths, with its major axis initially on the x-axis. It is rotated through an angle of phi in the counterclockwise direction and translated to cen.

If the optional fifth argument dir is specified, it specifies the direction of the normal vector. It must be the word inward or outward.  The default is outward.

Ellipse(eqn, varx, vary, dir)

The parameter eqn is either a Cartesian equation specifying the ellipse or an algebraic expression such that the equation eqn = 0 specifies the ellipse. A Cartesian equation for a general conic section is of the form $A{x}^2+Bxy+C{y}^2+Ex+Fy+G=0$, and specifies the locus of all points $⟨x\,y⟩$ that satisfy the equation; this locus is a non-degenerate, real ellipse if and only if three conditions hold:

The two variable names that appear in eqn can be specified via varx and vary. The variable specified by varx represents the x-axis, and vary the y-axis. Both varx and vary can be omitted, but only if F is in Cartesian coordinates and its coordinate names are the same as the variables that appear in the equation.

If the optional last argument dir is specified, it specifies the direction of the normal vector. It must be the word inward or outward.  The default is outward.

Line(p1, p2)

The parameters p1 and p2 must be of type 'Vector'(algebraic), and they represent the endpoints of the directed line segment from p1 to p2. If coordinate system attributes are specified on the points, they are interpreted in their respective coordinate systems.

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) above, the pi's represent the endpoints of $k-1$ line segments.  The path of integration is the collection of line segments directed from p1 to p2, p2 to p3, ..., p(k-1) to pk. If any coordinate system attributes are specified on these points, they are interpreted in their respective coordinate systems.

Path(v, rng, c)

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

If the optional third argument c is specified, it must be an equation of the form coords=sys or coordinates=sys. This is the coordinate system in which v is interpreted. Note: if this argument is supplied, any existing coordinate attribute on v is overwritten (and therefore ignored).

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

The Flux(f, dom, inert) command returns the integral form of the flux of f over dom.

For some surfaces or curves, the Student[VectorCalculus][Flux] command offers a way to visualize the surface or curve, normal vectors and vector field.

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

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

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

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

$0$

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

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

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

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

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

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

$\mathrm{Flux}\left(\mathrm{VectorField}\left(⟨y\,-x\,0⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)\,\mathrm{Box}\left(1..2\,3..4\,5..6\right)\right)$

$0$

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

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