Discussion of “the meaning of the vector Laplacian”

✍ Scribed by Harold K. Farr

• Publisher: Elsevier Science
• Year: 1954
• Tongue: English
• Weight: 99 KB
• Volume: 258
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(54)90777-2

✦ Synopsis

A proposal has been made recently that a special notation be adopted for the Laplacian of a vector. It is shown that such a notation is unjustified since the usual scalar Laplacian gives the correct result when operating on a vector.

In a recent article Moon and Spencer (1) 2 develop expressions for the Laplacian of a vector in rectangular, cylindrical and spherical coordinate systems. It is indeed surprising that these expressions~are not listed in most standard texts on field theory. In making them easily available, these authors have satisfied an important need.

They point out correctly that some confusion may arise from the fact that the expression for a scalar component of the Laplacian 'of a vector does not in general have the same form as the Laplacian of a scalar. They propose to avoid the confusion by using two different symbols for the Laplacian, depending on whether operation is on a scalar or a vector.

This dual notation is unjustified and would very likely only add to the confusion. Such a notation implies that the usual scalar expression for the Laplacian is not necessarily correct when operating on a vector. This belief seems to be rather widespread (2, 3). Now, vector operations are properly defined to be independent of the choice of coordinate system and this is in fact true of the Laplacian. The confusion apparently results from overlooking an important point in the computation of the derivative of a vector. It must be remembered that the unit vectors d,, do, a, used in cylindrical or spherical coordinates are functions of position. For example, whereas in rectangular coordinates