The Laplacian operator for a Scalar function is defined by

(1) |

(2) |

where is a Covariant Derivative and

(3) |

(4) |

(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) |

(10) |

Similarly, a Tensor Laplacian can be given by

(11) |

An identity satisfied by the Laplacian is

(12) |

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

(13) |

(14) |

where the integration is over a small Sphere of Radius . Now, for and , the integral becomes 0. Similarly, for and , the integral becomes . Therefore,

(15) |

© 1996-9

1999-05-26