LineInt - Maple Help

Student[VectorCalculus]

 LineInt
 compute the line integral of a vector field in R^2 or R^3

 Calling Sequence LineInt(F, dom, options)

Parameters

 F - vector field or a Vector-valued procedure; specify the vector field to integrate dom - unevaluated function call; specify the path of integration options - (optional) equation(s) of the form option=value where option is one of output, fieldoptions, pathoptions, title, vectoroptions, or view

Description

 • The LineInt(F, dom) calling sequence computes the line integral of the vector field F over the path specified by dom.
 • Specify the path of integration using unevaluated function calls.  The possible paths of integration are Circle, Circle3D, Line, LineSegments, and Path.
 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 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.
 The line integral of the tangential component of the field is computed using Stokes' theorem, converting it to a surface integral. If the inert option is used, a double integral over the disk bounded by the given circle is returned. (Note that the applicability of Stokes' theorem is not verified.)
 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.
 LineSegments(p1, p2, ..., pk)
 Similar to Line(p1, p2), the parameters pi 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, ..., 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 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, tangent vectors of the path, and vector field.
 – 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 [].
 • title = string
 Give the plot a title. The default value is the empty string.
 • vectoroptions = list
 A list of plot options for plotting the tangent 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)
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨x,y⟩\right),\mathrm{Line}\left(⟨1,2⟩,⟨3,-4⟩\right)\right)$
 ${10}$ (1)
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨x,y⟩\right),\mathrm{Line}\left(⟨1,2⟩,⟨3,-4⟩\right),\mathrm{output}=\mathrm{integral}\right)$
 ${{\int }}_{{0}}^{{1}}\left({-}{10}{+}{40}{}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}$ (2)
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨x,y⟩\right),\mathrm{LineSegments}\left(⟨0,0⟩,⟨1,1⟩,⟨1,-1⟩\right)\right)$
 ${1}$ (3)
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨{x}^{2},{y}^{2}⟩\right),\mathrm{Path}\left(⟨t,{t}^{2}⟩,t=0..2\right)\right)$
 ${24}$ (4)
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨y,-x,z⟩\right),\mathrm{Line}\left(⟨0,0,0⟩,⟨1,1,1⟩\right)\right)$
 $\frac{{1}}{{2}}$ (5)

The commands to create the plots from the Plotting Guide are

 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨y,-x⟩\right),\mathrm{Circle}\left(⟨0,0⟩,1\right),\mathrm{output}=\mathrm{plot},\mathrm{pathoptions}=\left[\mathrm{color}=\mathrm{blue}\right],\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{green}\right],\mathrm{fieldoptions}=\left[\mathrm{color}=\mathrm{red}\right],\mathrm{scaling}=\mathrm{constrained}\right)$
 > $\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨y,-x,z⟩\right),\mathrm{Line}\left(⟨0,0,0⟩,⟨1,1,1⟩\right),\mathrm{output}=\mathrm{plot},\mathrm{axes}=\mathrm{boxed},\mathrm{orientation}=\left[170,85,5\right]\right)$