“Laplacian”的概念、定义、习题解题方法及翻译-数学辞典

单词

Laplacian

释义

Laplacian

The Laplacian operator for a Scalar function is defined by

(1)

in Vector notation, where the

are the Scale Factorsof the coordinate system. In Tensor notation, the Laplacian is written


			(2)

where

is a Covariant Derivative and

(3)

The finite difference form is



	(4)

For a pure radial function



			(5)

Using the Vector Derivative identity

(6)


			(7)

Therefore, for a radial Power law,


			(8)

A Vector Laplacian can also be defined for a Vector A by

(9)

in vector notation. In tensor notation, A is written

, and the identity becomes


			(10)

Similarly, a Tensor Laplacian can be given by

(11)

An identity satisfied by the Laplacian is

(12)

where

is the Hilbert-Schmidt Norm,

is a row Vector, and

is theMatrix Transpose of A.

To compute the Laplacian of the inverse distance function , where , and integrate theLaplacian over a volume,

(13)

This is equal to



			(14)

where the integration is over a small Sphere of Radius

. Now, for

and

, the integralbecomes 0. Similarly, for

and

, the integral becomes

. Therefore,

(15)

where

is the Delta Function.

See also Antilaplacian

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