36 VECTOR ANALYSIS. 



If the region for which Vxet> = is unlimited, these functions will 

 be single-valued. If the region is limited, but acyclic,* the functions 

 will still be single- valued and satisfy the condition Vu = w within the 

 same region. If the region is cyclic, we may determine functions 

 satisfying the condition Vu = c within the region, but they will not 

 necessarily be single- valued. 



68. If o) is any vector function of position in space, V.Vx = 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 



__ (d* d* d*\ 

 V. vu = ( -T-O 4- -j-3 + -j-o ) u. 

 \dx z dy 2 dz 2 / 



70. Def. If fc> is any vector function of position in space, we may 

 define V.Vw by the equation 



^ _/d 2 d* 



= \d^ + d^ + d 



the expression V.V being regarded, for the present at least, as a single 

 operator when applied to a vector. (It will be remembered that no 

 meaning has been attributed to V before a vector.) It should be 

 noticed that if 



V. Vo> = iV. VX + j V. VY + jfeV. VZ, 



that is, the operator V.V 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 



The effect of the operator V.V is therefore independent of the direc- 

 tions of the axes used in its definition. 



72. The expression %a?V.Vu, where a is any infinitesimal scalar, 

 evidently represents the excess of the value of the scalar function u 



*If every closed line within a given region can contract to a single point without 

 breaking its continuity, or passing out of the region, the region is called acyclic, other- 

 wise 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, imless 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. 



