Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 325Figure 5.35Meaning of curl.where A, is the area (nr2) of S, and (x*, y*, z*) is a suitably chosen point of S,. Onecan now writecurI,u(x*, y*, z*) =In the case of a fluid motion with velocity u the integral jcr u~ ds is termed thecirculation around C,; it measures the extent to which the corresponding fluid motionis a rotation around the circle C, in the given direction. If r is now allowed to approach0, we findc,that is, the component of curl u at (x,, y,, zl) in the direction of n is the limitingratio of circulation to area for a circle about (xi, yl, zl) with n as normal. Briefly,curl equals circulation per unit area. In the limit process the circular disk can bereplaced by a more general surface with normal n, provided that the shrinking tozero is properly carried out. If n is taken as i, j, and k successively, one obtains thethree components of curl u along the axes.Since (5.98) has a significance independent of the coordinate system chosen,this equation proves that the curl of a vector field has a meaning independent of theparticular (right-handed) coordinate system chosen in space. One could in fact use(5.98) to define the curl. [If the orientation of space is reversed, the direction of thecurl will also be reversed (cf. also Section 3.8).]5.13 INTEGRALS INDEPENDENT OF PATH w IRROTATIONALAND SOLENOIDAL FIELDS;I" - i?Since the generalization of Green's theorem to space takes two different forms, theDivergence theorem and Stokes's theorem, one can generalize the discussions ofSections 5.6 and 5.7 in two directions: to surface integrals and to line integrals.In the case of line integrals the results for two dimensions carry over with minormodifications. The surface integrals require a somewhat different treatment. .Line integrals independent of path in space are defined just as in the plane. Thefollowing theorems then hold.