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Student[VectorCalculus]

 DotProduct
 compute the dot product of Vectors and differential operators

 Calling Sequence DotProduct(v1, v2) v1 . v2

Parameters

 v1 - Vector or differential operator v2 - Vector or differential operator

Description

 • The DotProduct(v1, v2) calling sequence computes the dot product (scalar product) of v1 and v2, where v1 and v2 can be Vectors, vector fields, Student[VectorCalculus]:-Del, or Student[VectorCalculus]:-Nabla.
 • The Student[VectorCalculus] package has a . operator that you can use in place of the DotProduct command.  For example, DotProduct(v1, v2) is equivalent to $\mathrm{v1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{v2}$.
 Also, $\mathrm{Del}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}F$ is equivalent to Divergence(F).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{VectorCalculus}]\right):$
 > $\mathrm{DotProduct}\left(⟨a,b⟩,⟨c,d⟩\right)$
 ${a}{}{c}{+}{b}{}{d}$ (1)
 > $\mathrm{SetCoordinates}\left({\mathrm{polar}}_{r,\mathrm{θ}}\right)$
 ${{\mathrm{polar}}}_{{r}{,}{\mathrm{θ}}}$ (2)
 > $v≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1,2\right],\left[a,b\right]\right)$
 ${v}{:=}\left[\begin{array}{c}{a}\\ {b}\end{array}\right]$ (3)
 > $w≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1,2\right],\left[c,d\right]\right)$
 ${w}{:=}\left[\begin{array}{c}{c}\\ {d}\end{array}\right]$ (4)
 > $v\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}w$
 ${a}{}{c}{+}{b}{}{d}$ (5)
 > $F≔\mathrm{VectorField}\left(⟨{r}^{2},\mathrm{θ}⟩\right)$
 ${F}{:=}\left({{r}}^{{2}}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({\mathrm{θ}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}$ (6)
 > $\mathrm{Del}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}F$
 $\frac{{3}{}{{r}}^{{2}}{+}{1}}{{r}}$ (7)
 > $\mathrm{Divergence}\left(F\right)$
 $\frac{{3}{}{{r}}^{{2}}{+}{1}}{{r}}$ (8)
 > $\mathrm{SetCoordinates}\left({\mathrm{cartesian}}_{x,y,z}\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (9)
 > $L≔\mathrm{VectorField}\left(⟨x,y,z⟩\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{Del}\left(f\left(x,y,z\right)\right)$
 ${L}{:=}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{y}{}\left(\frac{{\partial }}{{\partial }{y}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{z}{}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right)$ (10)