VECTOR ANALYSIS. 33 



57. Integration. If dv represents an element of any space, and dcr 

 an element of the bounding surface, 



For the first member of this equation represents the sum of the 

 surface-integrals of all the elements of the given space. We may 

 regard this principle as affording a means of integration, since we 

 may use it to reduce a triple integral (of a certain form) to a double 

 integral. 



The principle may also be expressed as follows : 



The surface-integral of any vector function of position in space for 

 a closed surface is equal to the volume-integral of the divergence of 

 that function for the space enclosed. 



58. Line-integrals. The integral fw.dp, in which dp denotes the 

 element of a line, is called the line-integral of o> for that line. It is 

 implied that one of the directions of the line is distinguished as 

 positive. When the line is regarded as bounding a surface, that side 

 of the surface will always be regarded as positive, on which the 

 surface appears to be circumscribed counter-clockwise. 



59. Integration. From No. 51 we obtain directly 



f Vu . dp = u" u', 



where the single and double accents distinguish the values relating 

 to the beginning and end of the line. 



In other words, The line-integral of the derivative of any (con- 

 tinuous and single-valued) scalar function of position in space is equal 

 to the difference of the values of the function at the extremities of 

 the line. For a closed line the integral vanishes. 



60. Integration. The following principle may be used to reduce 

 double integrals of a certain form to simple integrals. 



If da- represents an element of any surface, and dp an element of 

 the bounding line, 



In other words, The line-integral of any vector function of position 

 in space for a closed line is equal to the surface-integral of the curl of 

 that function for any surface bounded by the line. 



To prove this principle, we will consider the variation of the 

 line-integral which is due to a variation in the closed line for which 

 the integral is taken. We have, in the first place, 



<5/w . dp =f$& . dp +Jco . 8 dp. 

 But h).Sdp = d((*).Sp) du).8p. 



Therefore, since fd(w.Sp) = for a closed line, 



<5/fc> . dp =jSu> . dp jdw . Sp. 

 G. n. c 





