Advanced Calculus fi..

270 Advanced Calculus, Fifth EditionThere are corresponding component integrals//f(x,~.z)dxdy. //f(x,y,z)dydz, . .Sand a vector surface integral/I F . d~ = J/(F. n) d ~ ,SSwhere do = nda is the "area element vector," n being - a unit normal vector to thesurface.It will be seen that the basic theorems-those of Green, Gauss, and Stokesconcernthe relations between line, surface, and volume (triple) integrals. These correspondto fundamental physical relations between such quantities as flux, circulation,divergence, and curl. The applications will be considered at the end of thechapter.S5.2 LINE INTEGRALS IN THE PLANE8 b , ~ < * $ ,We now state in precise form the definitions outlined in the preceding section.By a smooth curve C in the xy-plane will be meant a curve representable in theform:*4where x and y are continuous and have continuous derivatives for h 5 t 5 k. Thecurve C can be assigned a direction, which will usually be that of increasing t. If Adenotes the point [#(h), I,b(h)] and B denotes the point [#(k), I,b(k)], then C can bethought of as the path of a point moving continuously from A to B. This path maycross itself, as for the curve C1 of Fig. 5.3. If the initial point A and terminal point Bcoincide, C is termed a closed curve; if, in addition, (x, y) moves from A to B = Awithout retracing any other point, C is called a simple closed curve (curve C2 ofFig. 5.3).Figure 5.3Paths of integration.