30, 31] 



STOKES'S THEOREM. CURL. 



87 



The normal must be drawn toward that side of the surface that 

 shall make the rotation of a right-handed screw advancing along the 

 normal agree with the direction of traversing the closed contour of 

 integration. 



37) IE cos (B y ds) ds = I Xdx + Ydy + Zdz = I la cos (on) dS 



= I I { % cos (nx) + ri cos (ny) + g cos (ni)} dS. 



The vector co related to the vector point -function E by the differ- 

 ential equations above is called the rotation, spin (Clifford), or curl 

 (Maxwell and Heaviside) of JR. Such 

 vectors are of frequent occurrence 

 in mathematical physics. (See 

 Part III.) 



The significance of the geo- 

 metrical term curl can be seen 

 from the physical example in 

 which the vector E represents 

 the velocity of a point instant- 

 aneously occupying the position 

 x, y, z in a rigid body turning 

 about the ^"-axis with an angular 



velocity co. Then the vector E= OQ is perpendicular to the radius 

 and its components are (Fig. 22), 







rig 



X = B cos (Ex) = E sin 



= BGOS(QX) = 



= XG), 



where co is constant, and 



2co. 



_ 



dx dy 



So that the - component of the^curl of the linear velocity is twice 

 the angular velocity about the ^-axis. Further examples are presented 

 to us in the theory of fluid motion. 



31. Lamellar Vectors. In finding the variation of the integral I 

 in the previous section, since the variations dx, dy, 82 are perfectly 

 arbitrary functions of s, if the integral is to be independent of the 

 path, dl must vanish, which can happen for all possible choices of 

 dx, dy, dz, only if 



dZ dY_dX dZ_3Y dX _ n 



~5i~'9i l -~**Ji ~~ d*~ z>* 3y ~ 



that is if the curl of E vanishes everywhere. In case this condition 

 is satisfied, I depends only on the positions of the limiting points A 



