Laplacian

Laplacian[f,{x1,,xn}]
gives the Laplacian .

Laplacian[f,{x1,,xn},chart]
gives the Laplacian in the given coordinates chart.

DetailsDetails

  • Laplacian[f,x] can be input as . The character can be typed as EscdelEsc or \[Del]. The list of variables and the are entered as a subscript and superscript, respectively.
  • An empty template can be entered as Escdel2Esc, and Tab 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 . 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 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.

ExamplesExamplesopen allclose all

Basic Examples  (4)Basic Examples  (4)

The Laplacian in three-dimensional Cartesian coordinates:

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The Laplacian in three-dimensional cylindrical coordinates:

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The Laplacian in two-dimensional polar coordinates:

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Use EscdelEsc to enter , Ctrl+- for the list of subscripted variables, and Ctrl+% to enter the 2:

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Use Escdel2Esc to enter the template , fill in the variables, press Tab, and fill in the function:

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