Student[VectorCalculus][Laplacian] - compute the Laplacian of a function or a vector field
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Calling Sequence
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Laplacian(f, c)
Laplacian(F)
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Parameters
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f
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algebraic expression
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c
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(optional) specify the coordinate system
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F
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(optional) vector field or Vector-valued procedure; a vector field
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Description
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The Laplacian(f) calling sequence computes the Laplacian of the function f in the current coordinate system. If no coordinate system has been explicitly specified, the command will assume a cartesian system with coordinates the variables which appear in the expression f. This is equivalent to and .
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The Laplacian(f, c) calling sequence computes the Laplacian of the function f in the coordinate system specified by the parameter c, which can be given as:
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* an indexed name, e.g.,
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* a name, e.g., spherical; default coordinate names will be used
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* a list of names, e.g., ; the current coordinate system will be used, with these as the coordinate names
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The Laplacian(c) calling sequence returns the differential form of the Laplacian operator in the coordinate system specified by the parameter c.
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The Laplacian() calling sequence returns the differential form of the Laplacian operator in the current coordinate system. If no coordinate system has been set (by a call to SetCoordinates), cartesian coordinates are assumed.
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The Laplacian(F) calling sequence, where F is either a vector field or a Vector-valued procedure (which is interpreted as a vector field), computes the Laplacian of the vector field as follows.
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* If the coordinate system of F is cartesian, the command maps the algebraic Laplacian onto the component functions.
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* If F is a 3-D vector field, use the formula .
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* Otherwise, map F to cartesian coordinates, apply the algebraic Laplacian to the component functions, and then map the result back to the original coordinate system of F.
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Examples
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The Laplacian in cylindrical coordinates:
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