PositionVector - Maple Help

VectorCalculus

 PositionVector
 create a position Vector with specified components and a coordinate system

 Calling Sequence PositionVector(comps) PositionVector(comps, c)

Parameters

 comps - list(algebraic); specify the components of the position Vector c - symbol or symbol[name, name, ...]; specify the coordinate system, possibly indexed by the coordinate names

Description

 • The PositionVector procedure constructs a position Vector, one of the principal data structures of the Vector Calculus package.
 • The call PositionVector(comps, c) returns a position Vector in a Cartesian enveloping space with components interpreted using the corresponding transformations from c coordinates to Cartesian coordinates.
 • If no coordinate system argument is present, the components of the position Vector are interpreted in the current coordinate system (see SetCoordinates).
 • The position Vector is a Cartesian Vector rooted at the origin, and has no mathematical meaning in non-Cartesian coordinates.
 • The c parameter specifies the coordinate system in which the components are interpreted; they will be transformed into Cartesian coordinates.
 • If comps has indeterminates representing parameters, the position Vector serves to represent a curve or a surface.
 – To differentiate a curve or a surface specified via a position Vector, use diff.
 – To evaluate a curve or a surface given by a position Vector, use eval.
 – To evaluate a vector field along a curve or a surface given by a position Vector, use evalVF.
 – A curve or surface given by a position Vector can be plotted using PlotPositionVector.
 • The position Vector is displayed in column notation in the same manner as rooted Vectors are, as a position Vector can be interpreted as a Vector that is (always) rooted at the Cartesian origin.
 • A position Vector cannot be mapped to a basis different than Cartesian coordinates.  In order to see how the same position Vector would be described in other coordinate systems, use GetPVDescription.
 • Standard binary operations between position Vectors like +/-,*, Dot Product, Cross Product are defined.
 • Binary operations between position Vectors and vector fields, free Vectors or rooted Vectors are not defined; however, a position Vector can be converted to a free Vector in Cartesian coordinates via ConvertVector.
 • For details on the differences between position Vectors, rooted Vectors and free Vectors, see VectorCalculus,Details.

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)
 > $\mathrm{About}\left(\mathrm{pv1}\right)$
 $\left[\begin{array}{cc}{\mathrm{Type:}}& {\mathrm{Position Vector}}\\ {\mathrm{Components:}}& \left[{1}{,}{2}{,}{3}\right]\\ {\mathrm{Coordinates:}}& {{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\\ {\mathrm{Root Point:}}& \left[{0}{,}{0}{,}{0}\right]\end{array}\right]$ (2)
 > $\mathrm{PositionVector}\left(\left[1,\frac{\mathrm{\pi }}{2}\right],\mathrm{polar}\left[r,t\right]\right)$
 $\left[\begin{array}{c}{0}\\ {1}\end{array}\right]$ (3)
 > $\mathrm{PositionVector}\left(\left[1,3\right],\mathrm{parabolic}\left[u,v\right]\right)$
 $\left[\begin{array}{c}{-4}\\ {3}\end{array}\right]$ (4)

Curves

 > $\mathrm{R1}≔\mathrm{PositionVector}\left(\left[p,{p}^{2}\right],\mathrm{cartesian}\left[x,y\right]\right)$
 ${\mathrm{R1}}{≔}\left[\begin{array}{c}{p}\\ {{p}}^{{2}}\end{array}\right]$ (5)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R1},p=1..2\right)$
 > $\mathrm{R2}≔\mathrm{PositionVector}\left(\left[v,v\right],\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)$
 ${\mathrm{R2}}{≔}\left[\begin{array}{c}{v}{}{\mathrm{cos}}{}\left({v}\right)\\ {v}{}{\mathrm{sin}}{}\left({v}\right)\end{array}\right]$ (6)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{\pi }\right)$
 > $\mathrm{R3}≔\mathrm{PositionVector}\left(\left[1,\frac{\mathrm{\pi }}{2}+\mathrm{arctan}\left(\frac{1}{2}t\right),t\right],\mathrm{spherical}\right)$
 ${\mathrm{R3}}{≔}\left[\begin{array}{c}\frac{{2}{}{\mathrm{cos}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ \frac{{2}{}{\mathrm{sin}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ {-}\frac{{t}}{\sqrt{{{t}}^{{2}}{+}{4}}}\end{array}\right]$ (7)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{\pi }\right)$

Surfaces

 > $\mathrm{S1}≔\mathrm{PositionVector}\left(\left[t,\frac{v}{\mathrm{sqrt}\left(1+{t}^{2}\right)},\frac{vt}{\mathrm{sqrt}\left(1+{t}^{2}\right)}\right],\mathrm{cartesian}\left[x,y,z\right]\right)$
 ${\mathrm{S1}}{≔}\left[\begin{array}{c}{t}\\ \frac{{v}}{\sqrt{{{t}}^{{2}}{+}{1}}}\\ \frac{{v}{}{t}}{\sqrt{{{t}}^{{2}}{+}{1}}}\end{array}\right]$ (8)
 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3\right)$
 > $\mathrm{S2}≔\mathrm{PositionVector}\left(\left[1,p,q\right],\mathrm{toroidal}\left[r,p,t\right]\right)$
 ${\mathrm{S2}}{≔}\left[\begin{array}{c}\frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{cos}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{sin}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sin}}{}\left({1}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\end{array}\right]$ (9)
 > $\mathrm{PlotPositionVector}\left(\mathrm{S2},p=0..2\mathrm{\pi },q=0..2\mathrm{\pi }\right)$