Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/52

If the region for which $\nabla \times \omega =0$ is unlimited, these functions will be angle-valued. If the region is limited, but acyclic,^[1] the functions will still be single-valued and satisfy the condition $\nabla u=\omega$ within the same region. If the region is cyclic, we may determine functions satisfying the condition $\nabla u=\omega$ within the region, but they will not necessarily be single-valued. 68. If $\omega$ is any vector function of position in space, $\nabla .\nabla \times \omega =0.$ This may be deduced directly from the definitions of No. 54.

The converse of this proposition will be proved hereafter.

69. If $u$ is any scalar function of position in space, we have by Nos. 52 and 54

$\nabla .\nabla u=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)u.$

70. Def.—If $\omega$ is any vector function of position in space, we may define $\nabla .\nabla \omega$ by the equation

$\nabla .\nabla \omega =\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)\omega ,$

the expression $\nabla .\nabla$ being regarded, for the present at least, as a single operator when appKed to a vector. (It will be remembered that no meaning has been attributed to $\nabla$ before a vector.) It should be noticed that if

$\omega =i{\text{X}}+j{\text{Y}}+k{\text{Z}},$ $\nabla .\nabla \omega =i\nabla .\nabla {\text{X}}+j\nabla .\nabla {\text{Y}}+k\nabla .\nabla {\text{Z}},$

that is, the operator $\nabla .\nabla$ applied to a vector affects separately its scalar components.

71. From the above definition with those of Nos, 52 and 54 we may easily obtain

$\nabla .\nabla \omega =\nabla \nabla .\omega -\nabla \times \nabla \times \omega .$

The effect of the operator $\nabla .\nabla$ is therefore independent of the directions of the axes used in its definition.

72. The expression $-{\tfrac {1}{6}}a^{2}\nabla .\nabla u$, where $a$ is any infinitesimal scalar, evidently represents the excess of the value of the scalar function $u$

1. If every closed line within a given region can oontraot to a single point, without breaking its continuity, or passing out of the region, the region is called acyclic, otherwise cyclic.
   A cyclic region may be made acyclic by diaphragms, which must then be regarded as forming part of the surface bounding the region, each diaphragm contributing its own area twice to that surface. This process may be used to reduce many-valued functions of position in space, having single-valued derivatives, to single-valued functions.
   When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. Diaphragms may be applied to bring functions naturally many-valued under the application of some of the following theorems, as Nos. 74 ff.