the curve by the value of q at that element and add the products, thus forming the line inte- 

 gral 



§ q^ ds 

 where ^ indicates that the integration extends around the whole curve. This integral is 

 called the circulation of the fluid around the curve C, It may be regarded as a measure of the 

 extent to which the fluid is moving in a rotational manner along this particular path. 



The integral can also be written (^ q^ds= ^(u dx + vdy + w dz) since the direction 

 cosines of ^s are dx/ds, dy/ds, dz/ds and hence, Dy Equation [lb], 



//t dy dz 



q =u— + v — + w — 



ds ds ds 



If the fluid is actually moving like a rotating rigid body, the circulation about any closed 

 curve lying in a plane perpendicular to the axis of rotation is equal to twice the angular 

 velocity of rotation u> multiplied by the area enclosed by the curve. This is easily seen in the 

 special case of a circle whose axis is the axis of rotation, as in Figure 5b; here q - q = cur, 

 where r denotes distance from the axis, and the circulation around the circle is 

 jiq^ ds = ^ CO rds = cor^ ds = coT {2n r) = 2 {co) (rrr^). The same result is obtained, by evaluation 

 of the integral, for a circle centered anywhere. 



Motion of a fluid in which the circulation is zero around any continuously collapsible 

 curve is called irrotational motion or flow. The significance of the collapsibility of the curve 

 can be illustrated in the space occupied by the body of a doughnut. Any closed curve in this 

 space that is not linked with the central hole can be shrunk down continuously onto a point, 

 or can be deformed in continuous fashion into any other curve of the same type. Curves that 

 link with the hole, on the other hand, although continuously deformable into each other, can 

 never be shrunk below a certain minimum size. A region in which closed curves fall into two 

 classes with respect to collapsibility is called doubly connected; a region in which all curves 

 are completely collapsible is called singly connected. In some cases there are more than two 

 such classes of curves. Regions in which there are at least two classes are called multiply 

 connected. In irrotational motion in a multiply connected region, the circulation is required to 

 vanish only about the closed curves that are continuously collapsible down to a point. 



The great importance of irrotational flow arises from the following dynamical theorem, 

 which is proved in Sec. 33 of Lamb's Hydrodynamics.^ 



Suppose that the fluid is frictionless, and that its density, if not uniform and constant, 

 is at least a definite, fixed function of the pressure. Let the external forces be conservative, 

 as are those due to gravity; that is, the total work done by these forces on a given mass 

 vanishes when the mass is carried around any closed curve. Then the circulation around any 

 closed curve that is allowed to move with the fluid is constant in time. 



It follows from this theorem that, if a mass of frictionless fluid acted on only by con- 

 servative forces happens to be moving irrotationally at any instant, it will continue to move 

 irtotationally thereafter. Closed curves moving with the fluid may change their shape, but they 



