The PathInt(f, dom) command computes the path integral of the function f over the path specified by dom. The right-hand side of dom is the path of integration. The left-hand side of dom is a list of the variables of integration.

Paths of integration are represented by unevaluated function calls.  The possible paths of integration are Arc(obj, start, finish), Circle(cen, rad), Circle3D(cen, rad, normal), Ellipse(cen, a, b, phi), Ellipse(eqn), Line(p1, p2), LineSegments(p1, p2, ..., pk), and Path(v, rng, c).

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 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.

Circle3D(cen, rad, normal)

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

Ellipse(cen, a, b, phi)

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.

Ellipse(eqn)

The parameter eqn is either an 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 must be the ones specified on the left-hand side of dom.

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.

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, the points 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 PathInt(f, dom, inert) command returns the integral form of the path integral of f over dom.

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

$\mathrm{PathInt}\left(x^2\,\left[x\,y\right]=\mathrm{Line}\left(⟨0\,0⟩\,⟨1\,1⟩\right)\right)$

$\frac{\sqrt{2}}{3}$

$\mathrm{PathInt}\left(y^2\,\left[x\,y\right]=\mathrm{LineSegments}\left(⟨0\,0⟩\,⟨1\,1⟩\,⟨1\,2⟩\right)\right)$

$\frac{\sqrt{2}}{3}+\frac{7}{3}$

$\mathrm{PathInt}\left(y\,\left[x\,y\right]=\mathrm{Path}\left(⟨\cos \left(t\right)\,\sin \left(t\right)⟩\,t=0..\mathrm{\pi }\right)\right)$

$2$

$\mathrm{PathInt}\left(y\,\left[x\,y\right]=\mathrm{Path}\left(⟨1\,t⟩\,0..\mathrm{\pi }\,'\mathrm{coords}'='\mathrm{polar}'\right)\right)$

$2$

$\mathrm{PathInt}\left(x^2+y^2\,\left[x\,y\right]=\mathrm{Circle}\left(⟨0\,0⟩\,3\right)\right)$

$54\mathrm{\pi }$

$\mathrm{PathInt}\left(x^2+y^2\,\left[x\,y\right]=\mathrm{Circle}\left(⟨0\,0⟩\,3\right)\,'\mathrm{inert}'\right)$

${\int }_0^{2\mathrm{\pi }}3\left(9{\cos \left(t\right)}^2+9{\sin \left(t\right)}^2\right)\sqrt{{\cos \left(t\right)}^2+{\sin \left(t\right)}^2}\ⅆt$

$\mathrm{PathInt}\left(x^2\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Circle3D}\left(⟨0\&comma;0\&comma;0⟩\&comma;r\&comma;⟨1\&comma;y\&comma;3⟩\right)\right)\mathbf{assuming}0<r$

$\frac{\mathrm{\pi }{r}^3\left(y^2+9\right)}{y^2+10}$

$\mathrm{PathInt}\left(1\&comma;\left[x\&comma;y\right]=\mathrm{Ellipse}\left(\frac{x^2}{4}+\frac{y^2}{9}-1\right)\right)$

$12\mathrm{EllipticE}\left(\frac{\sqrt{5}}{3}\right)$

$\mathrm{PathInt}\left(x\&comma;\left[x\&comma;y\right]=\mathrm{Arc}\left(\mathrm{Circle}\left(⟨0\&comma;0⟩\&comma;1\right)\&comma;0\&comma;\frac{\mathrm{\pi }}{2}\right)\right)$

$1$