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Student[VectorCalculus]

 CrossProduct
 compute the cross product of Vectors and differential operators

 Calling Sequence CrossProduct(v1, v2) v1 &x v2

Parameters

 v1 - Vector, Vector-valued procedure, or differential operator v2 - Vector, Vector-valued procedure, or differential operator

Description

 • The CrossProduct(v1, v2) calling sequence computes the cross product (vector product) of v1 and v2, where v1 and v2 can be Vectors, vector fields, Student[VectorCalculus][Del], or Student[VectorCalculus][Nabla].
 Note: You can enter vector fields as Vector-valued procedures or operators.
 • The Student[VectorCalculus] package has an &x cross product operator that you can use in place of the CrossProduct command. For example, CrossProduct(v1, v2) is equivalent to $\mathrm{v1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{v2}$.
 Also, $\mathrm{Del}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$ is equivalent to Curl(F).

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{CrossProduct}\left(⟨a,b,c⟩,⟨d,e,f⟩\right)$
 $\left({b}{}{f}{-}{c}{}{e}\right){{e}}_{{x}}{+}\left({-}{a}{}{f}{+}{c}{}{d}\right){{e}}_{{y}}{+}\left({a}{}{e}{-}{b}{}{d}\right){{e}}_{{z}}$ (1)
 > $⟨a,b,c⟩\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}⟨d,e,f⟩$
 $\left({b}{}{f}{-}{c}{}{e}\right){{e}}_{{x}}{+}\left({-}{a}{}{f}{+}{c}{}{d}\right){{e}}_{{y}}{+}\left({a}{}{e}{-}{b}{}{d}\right){{e}}_{{z}}$ (2)
 > $F≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
 ${F}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > $\mathrm{Del}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-2}\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
 > $\mathrm{CrossProduct}\left(\nabla ,F\right)$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-2}\right){\stackrel{{_}}{{e}}}_{{z}}$ (5)
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (6)
 > $\mathrm{r1}≔\mathrm{RootedVector}\left(\left[2,2,1\right],\mathrm{root}=⟨1,2,3⟩\right)$
 ${\mathrm{r1}}{≔}\left[\begin{array}{c}{2}\\ {2}\\ {1}\end{array}\right]$ (7)
 > $\mathrm{r2}≔\mathrm{RootedVector}\left(\left[3,-1,0\right],\mathrm{root}=⟨1,2,3⟩\right)$
 ${\mathrm{r2}}{≔}\left[\begin{array}{c}{3}\\ {-1}\\ {0}\end{array}\right]$ (8)
 > $\mathrm{r1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{r2}$
 $\left[\begin{array}{c}{1}\\ {3}\\ {-8}\end{array}\right]$ (9)