BUILT-IN MATHEMATICA SYMBOL

# Laplacian

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

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

## DetailsDetails

• 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 (3)Basic Examples (3)

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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