# Laplacian

Laplacian[f,{x1,,xn}]

gives the Laplacian .

Laplacian[f,{x1,,xn},chart]

gives the Laplacian in the given coordinates chart.

# Details • Laplacian[f,x] can be input as f. The character can be typed as del or \[Del]. The list of variables x and the 2 are entered as a subscript and superscript, respectively.
• An empty template can be entered as del2 , and moves the cursor from the subscript to the main body.
• All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
• Laplacian[f,{x1,x2,}] yields a result with the same dimensions as f.
• In Laplacian[f,{x1,,xn},chart], if f is an array, it must have dimensions {n,,n}. The components of f are interpreted as being in the orthonormal basis associated to chart.
• Coordinate charts in the third argument of Laplacian can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
• Laplacian works with SparseArray and StructuredArray objects.

# Examples

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## Basic Examples(4)

The Laplacian in three-dimensional Cartesian coordinates:

 In:= Out= The Laplacian in three-dimensional cylindrical coordinates:

 In:= Out= The Laplacian in two-dimensional polar coordinates:

 In:= Out= Use del to enter , for the list of subscripted variables, and to enter the 2:

 In:= Out= Use del2 to enter the template , fill in the variables, press , and fill in the function:

 In:= Out= ## Interactive Examples(1)

Introduced in 2012
(9.0)
|
Updated in 2014
(10.0)