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VectorCalculus

 PlotPositionVector
 plot a curve or surface defined by a position Vector

 Calling Sequence PlotPositionVector(pv) PlotPositionVector(pv, r) PlotPositionVector(pv,r, options1) PlotPositionVector(pv, r1,r2) PlotPositionVector(pv,r1,r2, options2)

Parameters

 pv - 'Vector'(algebraic); the position Vector representing a curve or surface r - range or name=range; the range of the parameter of the curve r1 - name=range; the range of one of the parameters of the surface r2 - name=range; the range of one of the parameters of the surface options1 - (optional) curve options, equation(s) of the form keyword = value, where keyword is either 'points', 'vectorfield', 'vectorfieldoptions', 'pvdiff', 'diffoptions', 'tangent', 'tangentoptions', 'normal', 'normaloptions', 'binormal', 'binormaloptions' or 'curveoptions' options2 - (optional) surface options, equation(s) of the form keyword = value, where keyword is either 'points', 'pointoptions', , 'coordcurve', 'curveoptions', 'pvdiff', 'diffoptions', 'normal', 'normaloptions', 'normalfield', 'normalfieldoptions', 'normalorientation', 'vectorfield', 'vectorfieldoptions', or 'surfaceoptions'

Options

 • 'points'=[a] or 'points'=[[a,b]] If the option 'points'=[a] or 'points'=[[a,b]] is given, the position Vector is evaluated at the values of the parameters and the corresponding point is plotted along the curve or surface.
 – For a position Vector with two parameters the point [a,b] is interpreted as param1=a and param2= b where param1 and param2 are the parameter names given in the ranges r1 and r2 respectively.
 – For the point [a], a must lie in the range r and for the point [a,b], a and b must lie in the ranges r1 and r2 respectively. Otherwise, an error is raised.
 – Use [a1,a2,...,ak] or [[a1,b1],...,[ak,bk]] to specify more than one point.

Curve Options:

 • 'vectorfield' = VField

If the option 'vectorfield' = VField is provided, the vector field VField is evaluated at equally spaced points on the curve and the resulting rooted Vectors are displayed. The number of vectors displayed can be controlled with the option 'vectornum'=posint and the evaluation points on the curve can be provided with the option 'points'.

 – Since the position Vector is a Cartesian Vector, the vector field will be converted to Cartesian coordinates before the evaluation.
 – A list of vector fields can be specified using 'vectorfield'=[VField1,..,VFieldk].
 – For more information on vector fields, see VectorField.
 • 'pvdiff'=[param$k] The option 'pvdiff'=[param$k] plots the rooted Vector corresponding to the $\mathrm{kth}$ derivative of the components of the position Vector along the curve. The number of vectors displayed can be controlled with the option 'vectornum'=posint. The evaluation point(s) on the curve can be provided with the option 'points'.

 – More than one derivative can be plotted using 'pvdiff'=[[d1],[d2]...], where each sublist is a valid second argument to diff.
 • 'tangent'= truefalse

The option 'tangent'= truefalse controls whether or not normalized tangent vectors are plotted along the curve. The number of vectors displayed can be controlled by the option 'vectornum'=posint. The default value is false. The evaluation point(s) on the curve can be provided with the option 'points'.

 • 'normal'= truefalse

The option 'normal'= truefalse controls whether or not normalized principal normal vectors are plotted along the curve. The number of vectors displayed can be controlled by the option 'vectornum'=posint. The default value is false. The evaluation point(s) on the curve can be provided with the option 'points'.

 • 'binormal'= truefalse

The option 'binormal'= truefalse controls whether or not normalized binormal vectors are plotted along a three dimensional curve. The number of vectors displayed can be controlled by the option 'vectornum'=posint. The default value is false. The evaluation point(s) on the curve can be provided with the option 'points'. If pv represents a two dimensional curve, an error is raised.

 • 'curveoptions' =list

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

 • 'pointoptions', 'vectorfieldoptions', 'tangentoptions', 'normaloptions', 'binormaloptions' =list

A list of plot options for plotting the vectorfield, tangent vectors, principal normal vectors, and binormal vectors on the curve. For more information on plotting options, see plot/options. Their default value is [].

Surface Options:

 • 'coordcurve'= [{param1, param2}=a,options1 ]

The option 'coordcurve' plots a coordinate curve on the surface obtained by fixing one of the parameters, param1 or param2 to the value a. Any of the curve options in options1 can be specified for the coordinate curve.

 – If the value a of the parameter is out of range an error is raised.
 – A list of coordinate curves can be specified, using 'coordcurve'=[[...],...,[...]].
 • 'vectorfield' = VField

If the option 'vectorfield' = VField is provided, the vector field VField is evaluated at equally spaced points on the surface and the resulting rooted Vectors are displayed. The number of vectors displayed can be controlled with the option vectorgrid=[posint,posint] and the evaluation points on the surface can be provided with the option 'points'.

 – Since the position Vector is a Cartesian Vector, the vector field will be converted to Cartesian coordinates before the evaluation.
 – This option can take a list of vector fields, using 'vectorfield'=[VField1,..,VFieldk].
 – For more information on vector fields, see VectorField.
 • 'pvdiff'=[{param1,param2}$k] The option 'pvdiff'={param1,param2}$k plots the rooted Vector corresponding to the $\mathrm{kth}$ derivative of the components of the position Vector along the surface. The number of vectors displayed can be controlled with the option 'vectorgrid'=[posint,posint]. The evaluation point(s) on the curve can be provided with the option 'points'.

 – More than one derivative can be plotted using 'pvdiff'=[[param$k1],[param$k2]...].
 • 'normal' = truefalse

The option 'normal'= truefalse controls whether or not a normal vector to the surface is plotted. By default the normal vector is evaluated at the center of the specified ranges of the parameters, however the evaluation point(s) on the surface can be provided with the option 'points'. The default value is false.

 – The orientation of the normal vector can be controlled with the option 'normalorientation'=truefalse.
 – The formula used to compute the normal vector of a position Vector $\mathrm{pv}\left(u,v\right)=x\left(u,v\right)i+y\left(u,v\right)j+z\left(u,v\right)k$ is $\mathbit{N}=\frac{\partial \left(y,z\right)}{\partial \left(u,v\right)}\cdot \mathbit{i}+\frac{\partial \left(z,x\right)}{\partial \left(u,v\right)}\cdot \mathbit{j}+\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}\cdot \mathbit{k}$.
 • 'normalfield' = truefalse

If a vector field is provided with the option vectorfield, the additional option 'normalfield'= truefalse controls whether or not a normal field is evaluated at equally spaced points on the surface. The resulting rooted vectors are displayed. If no vector field is provided, an error is raised. The default value is false.

 – The formula used to compute the normalfield of a vector field is $\left(\mathbit{N}\mathbf{.}\mathbit{F}\right)\mathbf{\cdot }\mathbit{N}$ where $\mathbit{N}\mathbf{.}\mathbit{F}$ is the dot product of the normalized normal vector $\mathbit{N}$ and $\mathbit{F}$.
 • 'surfaceoptions'=list

A list of plot option for plotting the surface. For more information on plotting options, see plot3d/options. The default value is [].

 • 'pointoptions', 'diffoptions', 'normaloptions', 'vectorfieldoptions', 'normalfieldoptions'=list;

Lists of plot options for plotting points, vectors corresponding to derivatives, normal vectors, vectorfields and normal fields on the surface. For more information on plotting options, see plot3d/options. Their default value is [].

Description

 • The PlotPositionVector command plots a curve or surface defined by a two or three dimensional position Vector. For a curve, the plot can be in two or three dimensional space.
 • The first argument pv is a position Vector. The components of the position Vector represent the parametric description of the curve or surface in Cartesian coordinates. For more information about position Vectors, see PositionVector.
 – If the position Vector has one parameter it is assumed to represent a curve.
 – If the position Vector has two parameters and three components it is assumed to represent a surface. If the position Vector has two parameters and two components it is assumed to represent unevaluated curves.
 – If the position Vector has no indeterminates, a single Vector rooted at the origin is plotted.
 • Given a position Vector with one parameter representing a curve, the range r is specified in the form param = a..b or a..b. Given a position Vector with two parameters and three dimensions representing a surface, the ranges r1 and r2 are specified in the form param1=a..b and param2=c..d respectively. Given a position Vector with two parameters and two components, one parameter has to be assigned a specific value and the other must be a range of the form param = a..b.
 • The options1 and options2 arguments control extra structures that can be added to the plot of the curve and surface as well as providing plot options for specific elements of the plot. Additional plot options, as described on the plot/option and plot3d/option help pages, that are applicable may also be provided.

Examples

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

Position Vectors

 > $\mathrm{pv1}≔\mathrm{PositionVector}\left(\left[1,2,3\right],\mathrm{cartesian}\left[x,y,z\right]\right)$
 ${\mathrm{pv1}}{≔}\left[\begin{array}{c}{1}\\ {2}\\ {3}\end{array}\right]$ (1)
 > $\mathrm{PlotPositionVector}\left(\mathrm{pv1}\right)$
 • Specify the range of the parameter.
 > $\mathrm{R1}≔\mathrm{PositionVector}\left(\left[p,{p}^{2}\right],\mathrm{polar}\left[r,t\right]\right)$
 ${\mathrm{R1}}{≔}\left[\begin{array}{c}{p}{}{\mathrm{cos}}{}\left({{p}}^{{2}}\right)\\ {p}{}{\mathrm{sin}}{}\left({{p}}^{{2}}\right)\end{array}\right]$ (2)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R1},p=1..2\right)$
 > $\mathrm{R2}≔\mathrm{PositionVector}\left(\left[v,v\right],\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)$
 ${\mathrm{R2}}{≔}\left[\begin{array}{c}{v}{}{\mathrm{cos}}{}\left({v}\right)\\ {v}{}{\mathrm{sin}}{}\left({v}\right)\end{array}\right]$ (3)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},0..3\mathrm{\pi }\right)$

Display tangent vectors on the curve.

 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{\pi },\mathrm{tangent}=\mathrm{true}\right)$

Evaluate a vector field on the curve.

 > $\mathrm{VF1}≔\mathrm{VectorField}\left(⟨-x,-y⟩,\mathrm{cartesian}\left[x,y\right]\right)$
 ${\mathrm{VF1}}{≔}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{y}\right){\stackrel{{_}}{{e}}}_{{y}}$ (4)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{\pi },\mathrm{vectorfield}=\mathrm{VF1}\right)$

Display the tangent, principal normal and binormal vectors on a curve.

 > $\mathrm{R3}≔\mathrm{PositionVector}\left(\left[1,\frac{\mathrm{\pi }}{2}+\mathrm{arctan}\left(\frac{1}{2}t\right),t\right],\mathrm{spherical}\right)$
 ${\mathrm{R3}}{≔}\left[\begin{array}{c}\frac{{2}{}{\mathrm{cos}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ \frac{{2}{}{\mathrm{sin}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ {-}\frac{{t}}{\sqrt{{{t}}^{{2}}{+}{4}}}\end{array}\right]$ (5)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{\pi }\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{\pi },\mathrm{tangent}=\mathrm{true},\mathrm{normal}=\mathrm{true},\mathrm{binormal}=\mathrm{true}\right)$
 > $\mathrm{R4}≔\mathrm{PositionVector}\left(\left[1,p,p\right],\mathrm{cylindrical}\left[r,p,s\right]\right)$
 ${\mathrm{R4}}{≔}\left[\begin{array}{c}{\mathrm{cos}}{}\left({p}\right)\\ {\mathrm{sin}}{}\left({p}\right)\\ {p}\end{array}\right]$ (6)

Specify points on the curve for plotting other structures.

 > $\mathrm{PlotPositionVector}\left(\mathrm{R4},p=0..4\mathrm{\pi },\mathrm{tangent}=\mathrm{true},\mathrm{points}=\left[\mathrm{\pi }\right]\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{R4},p=0..4\mathrm{\pi },\mathrm{tangent}=\mathrm{true},\mathrm{normal}=\mathrm{true},\mathrm{binormal}=\mathrm{true},\mathrm{points}=\left[\mathrm{\pi },\frac{\mathrm{\pi }}{2}\right]\right)$

Two dimensional position Vectors with two parameters.

 > $\mathrm{pv2}≔\mathrm{PositionVector}\left(\left[p,q\right],\mathrm{polar}\left[r,t\right]\right)$
 ${\mathrm{pv2}}{≔}\left[\begin{array}{c}{p}{}{\mathrm{cos}}{}\left({q}\right)\\ {p}{}{\mathrm{sin}}{}\left({q}\right)\end{array}\right]$ (7)

Provide numeric value of one of the parameters.

 > $\mathrm{PlotPositionVector}\left(\mathrm{pv2},p=2,q=0..\mathrm{\pi }\right)$

The commands to create the plots from the Plotting Guide are

 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{\pi },\mathrm{tangent}=\mathrm{true},\mathrm{pvdiff}=\left[v\right]\right)$
 > $\mathrm{VF2}≔\mathrm{VectorField}\left(⟨r,0,0⟩,\mathrm{spherical}\left[r,p,t\right]\right)$
 ${\mathrm{VF2}}{≔}\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{p}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{t}}$ (8)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{\pi },\mathrm{vectorfield}=\mathrm{VF2},\mathrm{vectornum}=6\right)$

Surfaces

 > $\mathrm{S1}≔\mathrm{PositionVector}\left(\left[t,\frac{v}{\mathrm{sqrt}\left(1+{t}^{2}\right)},\frac{vt}{\mathrm{sqrt}\left(1+{t}^{2}\right)}\right],\mathrm{cartesian}\left[x,y,z\right]\right)$
 ${\mathrm{S1}}{≔}\left[\begin{array}{c}{t}\\ \frac{{v}}{\sqrt{{{t}}^{{2}}{+}{1}}}\\ \frac{{v}{}{t}}{\sqrt{{{t}}^{{2}}{+}{1}}}\end{array}\right]$ (9)
 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3\right)$

Specify points on the surface.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{points}=\left[\left[0,0\right],\left[1,1\right],\left[2,2\right],\left[3,3\right]\right]\right)$

Visualize tangent vectors along coordinate curves.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{pvdiff}=\left[\left[t\right],\left[v\right]\right]\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨x,y,0⟩,\mathrm{cartesian}\left[x,y,z\right]\right),\mathrm{vectorgrid}=\left[3,3\right]\right)$

Visualize multiple coordinate curves and different vectors along those curves

 > $\mathrm{PlotPositionVector}\left(\mathrm{PositionVector}\left(\left[x,y,{x}^{2}{y}^{2}\right]\right),x=-1..1,y=-1..1,\mathrm{coordcurve}=\left[\left[x=\frac{1}{2},\mathrm{tangent}=\mathrm{true},\mathrm{normal}=\mathrm{true},\mathrm{vectornum}=2,\mathrm{tangentoptions}=\left[\mathrm{color}=\mathrm{red}\right]\right],\left[x=-1,\mathrm{tangent}=\mathrm{true},\mathrm{vectornum}=2\right]\right],\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{frame}\right)$

Visualize the normal field of a given vector field.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨x,y,0⟩,\mathrm{cartesian}\left[x,y,z\right]\right),\mathrm{normalfield}=\mathrm{true}\right)$

 > $\mathrm{S2}≔\mathrm{PositionVector}\left(\left[1,p,q\right],\mathrm{toroidal}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$
 ${\mathrm{S2}}{≔}\left[\begin{array}{c}\frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{cos}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{sin}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sin}}{}\left({1}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\end{array}\right]$ (10)

Display a coordinate curve on a surface.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S2},p=0..2\mathrm{\pi },q=0..2\mathrm{\pi },\mathrm{coordcurve}=\left[p=\frac{\mathrm{\pi }}{2},\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨r,0,0⟩,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)\right]\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{PlotPositionVector}\left(\mathrm{S2},p=0..2\mathrm{\pi },q=0..2\mathrm{\pi },\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨r,0,0⟩,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right),\mathrm{surfaceoptions}=\left[\mathrm{color}=\mathrm{red}\right],\mathrm{vectorgrid}=\left[3,3\right]\right)$