Important: The linalg package has been deprecated. Use the superseding command VectorCalculus[Gradient], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The function grad computes the gradient of expr with respect to v.

It computes the following vector of partial derivatives:

vector( [diff(expr, v[1]), diff(expr, v[2]), ...] ).

In the case of three dimensions, where expr is a scalar expression of three variables and v is a list or a vector of three variables:

If the optional third argument co is of the form coords = coords_name or coords = coords_name({[const]}), grad will operate on commonly used orthogonally curvilinear coordinate systems. See ?coords for the list of the different coordinate systems known to Maple.

If the optional third argument co is a list of three elements which specify the scale factors, grad will operate on orthogonally curvilinear coordinate systems.

To compute the gradient in other orthogonally curvilinear coordinate systems, use the addcoords routine.

The two dimensional case is similar to the three dimensional one.

The command with(linalg,grad) allows the use of the abbreviated form of this command.

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

$\mathrm{grad}\left(3x^2+2yz\,\mathrm{vector}\left(\left[x\,y\,z\right]\right)\right)$

$\left[\begin{array}{ccc}6x& 2z& 2y\end{array}\right]$

$f≔r\sin \left(\mathrm{\theta }\right)z^2\:$$v≔\left[r\,\mathrm{\theta }\,z\right]\:$

$\mathrm{grad}\left(f\,v\,\mathrm{coords}=\mathrm{cylindrical}\right)$

$\left[\begin{array}{ccc}\sin \left(\mathrm{\theta }\right)z^2& \cos \left(\mathrm{\theta }\right)z^2& 2r\sin \left(\mathrm{\theta }\right)z\end{array}\right]$

$g≔r^2\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)\:$$v≔\left[r\,\mathrm{\theta }\,\mathrm{\phi }\right]\:$

$\mathrm{grad}\left(g\,v\,\mathrm{coords}=\mathrm{spherical}\right)$

$\left[\begin{array}{ccc}2r\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)& r\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)& -r\sin \left(\mathrm{\phi }\right)\end{array}\right]$

$h≔\left[1\,r\,r\sin \left(\mathrm{\theta }\right)\right]\:$

$\mathrm{grad}\left(g\,v\,h\right)$

$\left[\begin{array}{ccc}2r\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)& r\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)& -r\sin \left(\mathrm{\phi }\right)\end{array}\right]$

$l≔\cosh \left(\mathrm{\xi }\right)\cos \left(\mathrm{\eta }\right)\cos \left(\mathrm{\phi }\right)\:$$v≔\left[\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right]\:$

$\mathrm{grad}\left(l\,v\,\mathrm{coords}=\mathrm{prolatespheroidal}\left(1\right)\right)$

$\left[\begin{array}{ccc}\frac{\sinh \left(\mathrm{\xi }\right)\cos \left(\mathrm{\eta }\right)\cos \left(\mathrm{\phi }\right)}{\sqrt{{\sinh \left(\mathrm{\xi }\right)}^2+{\sin \left(\mathrm{\eta }\right)}^2}}& -\frac{\cosh \left(\mathrm{\xi }\right)\sin \left(\mathrm{\eta }\right)\cos \left(\mathrm{\phi }\right)}{\sqrt{{\sinh \left(\mathrm{\xi }\right)}^2+{\sin \left(\mathrm{\eta }\right)}^2}}& -\frac{\cosh \left(\mathrm{\xi }\right)\cos \left(\mathrm{\eta }\right)\sin \left(\mathrm{\phi }\right)}{\sinh \left(\mathrm{\xi }\right)\sin \left(\mathrm{\eta }\right)}\end{array}\right]$