32 VECTOR ANALYSIS. 



55. Surface-integrals. The integral ffto.da; in which da- represents 

 an element of some surface, is called the surface-integral of o> for that 

 surface. It is understood here and elsewhere, when a vector is said 

 to represent a plane surface (or an element of surface which may be 

 regarded as plane), that the magnitude of the vector represents the 

 area of the surface, and that the direction of the vector represents 

 that of the normal drawn toward the positive side of the surface. 

 When the surface is defined as the boundary of a certain space, the 

 outside of the surface is regarded as positive. 



The surface-integral of any given space (i.e., the surface-integral 

 of the surface bounding that space) is evidently equal to the sum of 

 the surface-integrals of all the parts into which the original space 

 may be divided. For the integrals relating to the surfaces dividing 

 the parts will evidently cancel in such a sum. 



The surface-integral of o> for a closed surface bounding a space dv 

 infinitely small in all its dimensions is 



V.wdv. 



This follows immediately from the definition of Vo>, when the space 

 is a parallelepiped bounded by planes perpendicular to i, j, k. In 

 other cases, we may imagine the space or rather a space nearly 

 coincident with the given space and of the same volume dv to be 

 divided up into such parallelepipeds. The surface-integral for the 

 space made up of the parallelepipeds will be the sum of the surface- 

 integrals of all the parallelepipeds, and will therefore be expressed by 

 V.wdv. The surface-integral of the original space will have sensibly 

 the same value, and will therefore be represented by the same 

 formula. It follows that the value of V. w does not depend upon the 

 system of unit vectors employed in its definition. 



It is possible to attribute such a physical signification to the 

 quantities concerned in the above proposition, as shall make it evident 

 almost without demonstration. Let us suppose co to represent a flux 

 of any substance. The rate of decrease of the density of that sub- 

 stance at any point will be obtained by dividing the surface-integral 

 of the flux for any infinitely small closed surface about the point by 

 the volume enclosed. This quotient must therefore be independent of 

 the form of the surface. We may define V.w as representing that 

 quotient, and then obtain equation (1) of No. 54 by applying the 

 general principle to the case of the rectangular parallelepiped. 



56. Skew surface-integrals. The integral ffdcrxw may be called 

 the skew surface-integral of o>. It is evidently a vector. For a 

 closed surface bounding a space dv infinitely small in all dimensions 

 this integral reduces to Vxwdv, as is easily shown by reasoning like 

 that of No. 55. 



