216 E. B. Wilson — Divergence and Curl. 



To evaluate this expression several methods are available. 

 In the first place one may apply the definition (2) directly to 

 an infinitesimal cube oriented with its edges parallel to the 

 axes. Consider the two faces perpendicular to the JT-axis and 

 at a distance dx apart. If the flow across the left face be 

 V(x\ y, z) that across the right face is V(x+dx, y, z). The sum 

 of the integrals for these two faces is therefore 



Jj L v( x + dx > y> z ) - v( x > y, z ) J *i d y (h =/J ^r dx d y dz = ^ (h 



With similar operations for the other faces, it appears that 

 ,. Tr bX bY bZ 



div r=—- + — + --. (4) 



bx by bz 



A second, and in many ways, a far better method will be 

 given in § 4. 



3. If there is in hydrodynamics any one theorem which 

 deserves to rank above all others it is Kelvin's on the constancy 

 of circulation ; or, as Heaviside calls it, circnitation. The 

 circulation in a curve would naturally be taken to be the 

 integral around the curve of the flux. This, however, is not 

 the case. The curvilinear integral of the velocity 



fv.dv instead of fpv.dr =ifV,dv 

 c be 



is defined as the circulation (or circnitation). To maintain an 

 analytical analogy with the former sections we may set p equal 

 to unity, and then the circulation will be numerically (though 

 not dimensionally) equal to the curvilinear integral of the 

 flux V which becomes the equivalent of the velocity. 



To study the simplest case consider the velocity as due to 

 an angular velocitv co about an axis along the unit vector n. 

 Then' 



V = «>n X r ; fV.dr =f o>n x r.dr = © / n.r X dr. 



CO c 



Now rxdr is the double area of the triangle formed by 

 r, f+dr, and dr. Let this area, denoted as a vector, be dA, 

 and let the total area of the curve c be A. Then 



fV.dr — '2om.fdA = 2am. A. 



G C 



If in particular this relation be applied to a closed plane 

 curve or to an infinitesimal closed curve (which may be 

 regarded as plane) of area da, 



f V.dp = 2wn.da. (5) 



