The DirectionalDiff(F,v,c) command computes the directional derivative of an expression F at the location and direction specified by v.  The expression F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system.

The DirectionalDiff(F,p,d,c) command computes the directional derivative of F at the point p in the direction d, where F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system.  The point p can be a list, a free Vector in Cartesian coordinates or a position Vector. The direction d can be a free Vector in Cartesian coordinates, a position Vector or a vector field.  The result is the value of DirectionalDiff(F,d,c) evaluated at the point p.

If c is a list of names, the directional derivative of F is taken with respect to these names in the current coordinate system.

If c is an indexed coordinate system, F is interpreted in the combination of that coordinate system and coordinate names.

If c is not specified, F is interpreted in the current coordinate system, whose coordinate name indices define the function's variables.

If c is not specified, and the current coordinate name is not indexed by any coordinate variable names, then the first n indeterminates of F, sorted alphabetically, will define the function's variables, where n is determined by the dimension of the current coordinate system. If this fails because there are not enough indeterminates in F, then the standard coordinate names will define the function's variables. For example, r and theta will be used for polar coordinates.

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

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

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

$\mathrm{v1}≔⟨1\,2⟩\:$

$\mathrm{DirectionalDiff}\left(r^2\,\mathrm{v1}\,\mathrm{polar}\left[r,t\right]\right)$

$\frac{2r\cos \left(t\right)\sqrt{5}}{5}+\frac{4r\sin \left(t\right)\sqrt{5}}{5}$

$W≔\mathrm{VectorField}\left(⟨u+v\,v⟩\,\mathrm{cartesian}\left[u,v\right]\right)$

$\mathrm{DirectionalDiff}\left(r^2\,\mathrm{point}=\left[1\,\mathrm{\pi }\right]\,W\,\mathrm{polar}\left[r,t\right]\right)$

$2$

$\mathrm{dd}≔\mathrm{DirectionalDiff}\left(r^2\,W\,\mathrm{polar}\left[r,t\right]\right)\:$

$\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{dd}\,\left[r=1\,t=\mathrm{\pi }\right]\right)\right)$

$2$

$\mathrm{DirectionalDiff}\left(pq\,⟨1\,2⟩\,\left[p\,q\right]\right)$

$\frac{q\sqrt{5}}{5}+\frac{2p\sqrt{5}}{5}$

$\mathrm{v2}≔⟨1\,0⟩\:$

$\mathrm{SetCoordinates}\left(\mathrm{polar}\right)$

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

$\mathrm{dd}≔\mathrm{DirectionalDiff}\left(r\cos \left(\mathrm{\theta }\right)\,\mathrm{v2}\,\left[r\,\mathrm{\theta }\right]\right)\:$

$\mathrm{simplify}\left(\mathrm{dd}\right)$

$1$

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

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

$\mathrm{vs}≔\mathrm{VectorSpace}\left(\left[1\,\frac{\mathrm{\pi }}{2}\right]\,\mathrm{polar}\left[r,t\right]\right)\:$

$\mathrm{v3}≔\mathrm{vs}:-\mathrm{Vector}\left(\left[1\,1\right]\right)$

$\mathrm{v3}≔\left[\begin{array}{c}1\\ 1\end{array}\right]$

$\mathrm{v4}≔\mathrm{vs}:-\mathrm{Vector}\left(\left[0\,1\right]\right)$

$\mathrm{v4}≔\left[\begin{array}{c}0\\ 1\end{array}\right]$

$\mathrm{DirectionalDiff}\left(r^2\,\mathrm{v3}\right)$

$\sqrt{2}$

$\mathrm{DirectionalDiff}\left(r^2\,\mathrm{v4}\right)$

$0$

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

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

$\mathrm{DirectionalDiff}\left(y^2{x}^2\,\mathrm{point}=\left[1\,2\right]\,⟨0\,1⟩\,\mathrm{cartesian}\left[x,y\right]\right)$

$4$

$\mathrm{DirectionalDiff}\left(y^2{x}^2\,\mathrm{RootedVector}\left(\mathrm{root}=\left[1\,2\right]\,\left[0\,1\right]\right)\,\mathrm{cartesian}\left[x,y\right]\right)$

$4$

$\mathrm{DirectionalDiff}\left(y^2{x}^2\,\mathrm{RootedVector}\left(\mathrm{root}=\left[1\,\frac{\mathrm{\pi }}{2}\right]\,\left[1\,1\right]\,\mathrm{polar}\left[r,t\right]\right)\,\mathrm{cartesian}\left[x,y\right]\right)$

$0$

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

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

$V≔\mathrm{VectorField}\left(⟨yz\,xz\,xy⟩\right)$

$\mathrm{normal}\left(\left(V·\mathrm{Del}\right)\left(xyz\right)\right)$

$\sqrt{y^2{x}^2+x^2{z}^2+y^2{z}^2}$

$\left(V·\mathrm{Del}\right)\left(\mathrm{VectorField}\left(⟨\frac{1}{x}\,\frac{1}{y}\,\frac{1}{z}⟩\right)\right)$