The function directional_diff(f, V, coord) computes the directional derivative of the scalar field f with respect to the coordinates coord in the direction of the contravariant vector V.

f must be of type algebraic, and V must be a tensor_type with index character [1] (a contravariant vector field).  The result is an algebraic expression.

Simplification:  This routine uses the `tensor/directional_diff/simp` routine for simplification purposes.  The simplification routine is applied to each component of the gradient of f (an intermediate calculation) and once to the contraction of this gradient with V (the result).  By default, `tensor/directional_diff/simp` is initialized to the `tensor/simp` routine. It is recommended that the `tensor/directional_diff/simp` routine be customized to suit the needs of the particular problem.

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

$\mathrm{coord}≔\left[x\,y\,z\right]\:$

$f≔\frac{3x}{y+z}$

$f≔\frac{3x}{y+z}$

$V≔\mathrm{create}\left(\left[1\right]\,\mathrm{array}\left(\left[xy\,yz\,zx\right]\right)\right)$

$V≔table\left(\left[\mathrm{index\_char}=\left[1\right]\,\mathrm{compts}=\left[\begin{array}{ccc}xy& yz& zx\end{array}\right]\right]\right)$

$\mathrm{directional\_diff}\left(f\,V\,\mathrm{coord}\right)$

$-\frac{3x\left(zx-y^2\right)}{{\left(y+z\right)}^2}$

$U≔\mathrm{create}\left(\left[1\right]\,\mathrm{array}\left(\left[\mathrm{u1}\,\mathrm{u2}\,\mathrm{u3}\right]\right)\right)$

$U≔table\left(\left[\mathrm{index\_char}=\left[1\right]\,\mathrm{compts}=\left[\begin{array}{ccc}\mathrm{u1}& \mathrm{u2}& \mathrm{u3}\end{array}\right]\right]\right)$

$\mathrm{directional\_diff}\left(g\left(x\,y\,z\right)\,U\,\mathrm{coord}\right)$

$\left(\frac{\partial }{\partial x}g\left(x\,y\,z\right)\right)\mathrm{u1}+\left(\frac{\partial }{\partial y}g\left(x\,y\,z\right)\right)\mathrm{u2}+\left(\frac{\partial }{\partial z}g\left(x\,y\,z\right)\right)\mathrm{u3}$