By Brian H. Chirgwin

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**Sample text**

10). T h e conditions of smoothness, continuity and absence of double points ensure that the transformation is continuous and reversible (see V o l . I I , § 5:10). Therefore W e now apply Green's theorem for a plane to the r. h. side of this equation and obtain 34 A C O U R S E O F M A T H E M A T I C S But etc. 39). 35) for a plane is the t w o dimensional form of b o t h the Divergence theorem and Stokes's Theorem. 11. note that the vector element of arc (dx 1} dx2) can be replaced b y an element of 'area' η as directed normal t o the curve, (Fig.

Side of this equation and obtain 34 A C O U R S E O F M A T H E M A T I C S But etc. 39). 35) for a plane is the t w o dimensional form of b o t h the Divergence theorem and Stokes's Theorem. 11. note that the vector element of arc (dx 1} dx2) can be replaced b y an element of 'area' η as directed normal t o the curve, (Fig. 35) n o w becomes : 2 2 E x a m p l e . U s e Green's l e m m a t o e v a l u a t e φ (x y formed b y y dx -f- y dy), < where (J is t h e l o o p = x, y — χ (Fig. 1 2 ) . Therefore FIG.

Side of the last equation into an integral involving u, v, the parameters of the surface χ χ (Η , ν), y — y(u, v). where R{L2) denotes R evaluated at L2 in the upper element of area. Since the elementary column enters the surface at Lx, ^ , as shown in the figure, makes an obtuse angle with the ζ -axis, so that at Lx, and therefore Therefore # x being the lower portion and # 2 the upper portion of S. 38). If the surface S is n o t c o n v e x , we divide the v o l u m e into a finite number of sections each b o u n d e d b y a c o n v e x surface, and add the various contributions.