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 $\mathrm{v1}·\mathrm{v2}$.

Also, $\mathrm{Del}·F$ is equivalent to Divergence(F).

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{DotProduct}\left(⟨a\,b⟩\,⟨c\,d⟩\right)$

$ac+bd$

$\mathrm{SetCoordinates}\left(\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)$

${\mathrm{polar}}_{r,\mathrm{\theta }}$

$v≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1\,2\right]\,\left[a\,b\right]\right)$

$v≔\left[\begin{array}{c}a\\ b\end{array}\right]$

$w≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1\,2\right]\,\left[c\,d\right]\right)$

$w≔\left[\begin{array}{c}c\\ d\end{array}\right]$

$v·w$

$ac+bd$

$F≔\mathrm{VectorField}\left(⟨r^2\,\mathrm{\theta }⟩\right)$

$F≔\left(r^2\right){\stackrel{\_}{e}}_r\+\left(\mathrm{\theta }\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$

$\mathrm{Del}·F$

$\frac{3r^2+1}{r}$

$\mathrm{Divergence}\left(F\right)$

$\frac{3r^2+1}{r}$

$\mathrm{SetCoordinates}\left(\mathrm{cartesian}\left[x,y,z\right]\right)$

${\mathrm{cartesian}}_{x,y,z}$

$L≔\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)·\mathrm{Del}\left(f\left(x\,y\,z\right)\right)$

$L≔x\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)+y\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)+z\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)$