The DirectionalDiff(F,v,c) command, where F is a scalar function, computes the directional derivative of 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,v,c) command, where F is a VectorField, computes the VectorField of directional derivatives of each component of F with respect to v.

The argument v can be a free Vector in Cartesian coordinates, a position Vector, a vector field or a rooted Vector.  If v is one of the first three, the result will be a scalar field of all directional derivatives in $R^n$ in the directions specified by v; this scalar field will be given in the same coordinate system as is used to interpret expression F.  If v is a rooted Vector, the result is the value of the directional derivative of F in the direction of v taken at the root point of v.

If F is a scalar function, the Vector v is normalized. If F is a VectorField, the Vector v is not normalized.

The DirectionalDiff(F,p,dir,c) command computes the directional derivative of F at the point p in the direction dir, 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 dir can be a free Vector in Cartesian coordinates, a position Vector or a vector field.  The result is the value of DirectionalDiff(F,dir,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.

An operator implementing the directional derivative with respect to a VectorField can be obtained using the dot operator with Del, as in $V·\mathrm{Del}$.

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

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

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

$\mathrm{v1}≔⟨1\,2⟩\:$

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

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

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

$W≔\left(u+v\right){\stackrel{\_}{e}}_u\+\left(v\right){\stackrel{\_}{e}}_v$

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

$2$

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

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

$2$

$\mathrm{dd}≔\mathrm{DirectionalDiff}\left(\mathrm{VectorField}\left(⟨\frac{x}{y}\,xy⟩\right)\,W\right)$

$\mathrm{dd}≔\left(\frac{x+y}{y}-\frac{x}{y}\right){\stackrel{\_}{e}}_x\+\left(\left(x+y\right)y+yx\right){\stackrel{\_}{e}}_y$

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

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

$\mathrm{v2}≔⟨1\,0⟩\:$

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

$\mathrm{polar}$

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

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

$1$

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

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

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

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

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

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

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

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

$\sqrt{2}$

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

$0$

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

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

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

$4$

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

$4$

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

$0$

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

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

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

$V≔\left(yz\right){\stackrel{\_}{e}}_x\+\left(xz\right){\stackrel{\_}{e}}_y\+\left(yx\right){\stackrel{\_}{e}}_z$

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

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

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

$\left(-\frac{yz}{x^2}\right){\stackrel{\_}{e}}_x\+\left(-\frac{xz}{y^2}\right){\stackrel{\_}{e}}_y\+\left(-\frac{yx}{z^2}\right){\stackrel{\_}{e}}_z$