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VectorCalculus

 *
 An overloaded version of Star that deals with scalar multiplication of Vectors

 Calling Sequence s*v

Parameters

 s - algebraic; the scalar to scale the Vector v - Vector(algebraic); the Vector to scale

Description

 • Returns the scalar multiplication of s and v.
 • An overloaded version for the VectorCalculus package that deals with scaling Vectors (scalar multiplication) in different coordinate systems.
 • The following table describes the interaction between different types of Vector objects in different coordinate systems when the Star operator is applied.

 $v$ coord($v$) $s$*$v$ coord($s$*$v$) 1 free Vector cartesian free Vector cartesian free Vector curved error 2 rooted Vector(root) any rooted Vector (root) any 3 vector field any vector field any 4 position Vector cartesian position Vector cartesian

 • Note that in 2-D math, the Star operator appears as a dot.

Examples

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

Only free Vectors in cartesian coordinates can be scaled.

 > $2⟨1,1,2⟩$
 ${2}{{e}}_{{x}}{+}{2}{{e}}_{{y}}{+}{4}{{e}}_{{z}}$ (1)
 > $\mathrm{v1}≔\mathrm{Vector}\left(⟨1,2⟩,\mathrm{coordinates}={\mathrm{cartesian}}_{x,y}\right)$
 ${\mathrm{v1}}{:=}{{e}}_{{x}}{+}{2}{{e}}_{{y}}$ (2)
 > $3\mathrm{v1}$
 ${3}{{e}}_{{x}}{+}{6}{{e}}_{{y}}$ (3)

Rooted Vectors in any coordinate system can be scaled.

 > $\mathrm{v2}≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1,\frac{\mathrm{π}}{2}\right],\left[1,2\right],{\mathrm{polar}}_{r,t}\right)$
 ${\mathrm{v2}}{:=}\left[\begin{array}{r}{1}\\ {2}\end{array}\right]$ (4)
 > $3\mathrm{v2}$
 $\left[\begin{array}{r}{3}\\ {6}\end{array}\right]$ (5)
 > $\mathrm{v3}≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1,2\right],\left[1,1\right],{\mathrm{parabolic}}_{u,v}\right)$
 ${\mathrm{v3}}{:=}\left[\begin{array}{r}{1}\\ {1}\end{array}\right]$ (6)
 > $2\mathrm{v3}$
 $\left[\begin{array}{r}{2}\\ {2}\end{array}\right]$ (7)

Vector Fields in any coordinate system can be scaled.

 > $\mathrm{vf}≔\mathrm{VectorField}\left(⟨x,y⟩,{\mathrm{cartesian}}_{x,y}\right)$
 ${\mathrm{vf}}{:=}\left({x}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({y}\right){\stackrel{{_}}{{e}}}_{{y}}$ (8)
 > $x\mathrm{vf}$
 $\left({{x}}^{{2}}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({x}{}{y}\right){\stackrel{{_}}{{e}}}_{{y}}$ (9)

Position Vectors can be scaled.

 > $\mathrm{pv}≔\mathrm{PositionVector}\left(\left[p,p\right],{\mathrm{polar}}_{r,t}\right)$
 ${\mathrm{pv}}{:=}\left[\begin{array}{c}{p}{}{\mathrm{cos}}{}\left({p}\right)\\ {p}{}{\mathrm{sin}}{}\left({p}\right)\end{array}\right]$ (10)
 > $\frac{1\mathrm{pv}}{2}$
 $\left[\begin{array}{c}\frac{{1}}{{2}}{}{p}{}{\mathrm{cos}}{}\left({p}\right)\\ \frac{{1}}{{2}}{}{p}{}{\mathrm{sin}}{}\left({p}\right)\end{array}\right]$ (11)