34, 35] DEFINITE INTEGRALS. 67 



any closed surface S within which the function is uniform and 

 continuous, multiplied by the area of the surface, is equal to the 

 mean value of the divergence of the vector in the space within S 

 multiplied by its volume. The theorem was proved for a vector 

 which was the parameter of a scalar point-function V, but it is 

 evident that it may be proved directly by partial integration 

 whether this is the case or not. 



Let us consider the geometrical nature of a vector point- 

 function R whose divergence vanishes in a certain region. In the 

 neighbourhood of any point, the vector will at some points be 

 directed toward the point and at others away. We may then 

 draw curves of such a nature that at every point of any curve the 

 tangent is in the direction of the vector point-function R at that 

 point. Such curves will be called lines of the vector function. 

 Suppose that such lines be drawn through all points of a closed 

 curve, they will generate a tubular surface, which will be called 

 a tube of the vector function. Let* us now construct any two 

 surfaces S and $ 2 cutting across the vector 

 tube and apply the divergence theorem to the 

 portion of space inclosed by the tube and the 

 two surfaces or caps $j and $ 2 . Since at every 

 point on the surface of the tube, R is tangent 

 to the tube, the normal component vanishes. 

 The only parts contributing anything to the 

 surface integral are accordingly the caps, and since the divergence 

 everywhere vanishes in r, we have 



Rcos(Rn 1 )dS l +jl R cos (Rn,) dS 2 = 0. 



If we draw the normal to S 2 in the other direction, so that as 

 we move the cap along the tube the direction of the normal is 

 continuous, the above formula becomes 



Rcos(Rn l )dS 1 -jj 



or the surface integral of the normal component of R over any cap 

 cutting the same vector tube is constant. 



Such a vector will be termed solenoidal, or tubular, and the 

 o -y f\v ^\y 



condition -= h ~ H ^-=0 will be termed the solenoidal condition 

 ox dy dz 



(Maxwell). We may abbreviate it, div. R = 0. If a vector point- 



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