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

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

laplacian(f, v) computes the Laplacian of f with respect to v.

The Laplacian is defined to be the sum of the second derivatives $\frac{{\ⅆ}^{2}f}{\ⅆx^2}$ for x in v.

In the case of three dimensions, where f 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]}), laplacian will operate on commonly used orthogonally curvilinear coordinate systems. See ?coords for the list of coordinate systems supported by Maple.

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

To compute the Laplacian 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,laplacian) allows the use of the abbreviated form of this command.

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

$\mathrm{laplacian}\left(x^{2}yz\,\left[x\,y\,z\right]\right)$

$2yz$

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

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

$2r\sin \left(\mathrm{\theta }\right)$

$h≔\left[1\,r\,1\right]\:$

$\mathrm{laplacian}\left(f\,v\,h\right)$

$2r\sin \left(\mathrm{\theta }\right)$

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

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

$\frac{5r^2{\sin \left(\mathrm{\theta }\right)}^2\cos \left(\mathrm{\phi }\right)+r^2{\cos \left(\mathrm{\theta }\right)}^2\cos \left(\mathrm{\phi }\right)-r^2\cos \left(\mathrm{\phi }\right)}{r^2\sin \left(\mathrm{\theta }\right)}$