A Course in Mathematical Analysis (Volume: 1)
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 Excerpt: ...toward the observer. To the boundary T of S will correspond a closed contour C in the xy plane; and these two curves are described simultaneously in the sense indicated by the arrows. Let 2 =/(i V) be the equation of the given surface, and let P(x, y, z) be a function which is continuous in a region of space which contains S. Then the line integral jr)P(i, y, z)dx is identical with the line integral taken along the plane curve C. Let us apply Green's theorem ( 126) to this latter integral. Setting dP(x, y) _ gP dP a _ SP dP cosfl dy By dz dy dy dz cosy where a, /3, 7 are the direction angles of the normal to the upper side of S. Hence, by Green's theorem, / P)dx = f f cny), JlO JJ(A)e!l /COS 7 where the double integral is to be taken over the region A of the xy plane bounded by the contour C. But the right-hand side is simply the surface integral The sense in which r is described and the side of the surface over which the double integral is taken correspond according to the convention made above. IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS 137. Volumes. Let us consider, as above, a region of space bounded by the xy plane, a surface S above that plane, and a cylinder whose generators are parallel to the z axis. We shall suppose that the section of the cylinder by the plane z = 0 is a contour similar to that drawn in Fig. 25, composed of two parallels to the y axis and two curvilinear arcs A PB and A'QB'. If = fix, y) is the equation of the surface S, the volume in question is given, by 124, by the integral The volume of a solid bounded in any way whatever is equal to the algebraic sum of several volumes bounded as above. For instance, to find the volume of a solid bounded by a convex closed surface we should circumscribe the solid by a cyl...
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