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 f. The character can be typed as EscdelEsc 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 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 {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.

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)