122 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



expressed as the normal component to the surface of the vector (a), (curl A). 

 Thus 



(b) J A s ds = J' (curl A) da 



or in words : 



The line-integral of the tangential component of any vector taken around a 

 closed curve is equal to the surface-integral of the normal component of 

 the curl of the vector taken over any surface which has the given closed 

 curve as contour. (Stokes's theorem.) 



As long as we deal with two-dimensional vectors only, the vector-nature of the 

 curl does not become apparent, as we have then to deal only with the component 

 of the vector normal to the surface which contains the two-dimensional vector-field. 

 In this respect the case is analogous to that of the vector-product. 



The general theorem allows us to demonstrate an important property of every 

 vector which is the curl of another vector. If the surface a is closed, the con- 

 tour 5 will disappear, and thus the line-integral around this be zero. We then get 

 the equation 



(c) J* (curl A) n d<x = o 



where the integral is extended to the closed surface. But this equation indicates 

 that the vector curl A is a solenoidal vector. This result can also be verified if we 

 substitute the expressions of the components (a) of the curl into the solenoidal 

 condition in its differential form. This leads to the identity 



(c') div curl A = o 



Thus : The curl of a vector is a solenoidal vector. 



174. Complex Differential Operations. Divergence and curl may be considered 

 as the intrinsic derivatives of a vector-field. The intrinsic structure of a field is 

 known when we know curl and divergence. 



Besides the differential operations leading to these intrinsic derivatives, we 

 shall have to consider also a differential operation of a more complex nature. A 

 and B being two vectors, we shall consider a vector F which has the three components 



2 A 2 A PA 



F.-B.-'+B.-'+B.-' 



S A c A c A 



CiA 0/1 oJ 



F z = 5^+5^+5.' 

 Sx ?y 9z 



Remembering the definitions of the scalar product and of the ascendant, we see that 

 the expression of each component may be written as the scalar product of the vector 

 B and the three ascendants VA X , \/A y and VA Z , thus 



F X = B.VA X F y = B.VA y F z = B.VA t 



