Surface Integral

Surface Integral - 832 CHAPTER 15 Vector Fields 23 Review...

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832 CHAPTER 15 Vector Fields 23. Review Example 5 in Section 15.2 in which it was shown that y ± y x 2 + y 2 ² = x ± x x 2 + y 2 ² , for all ( x , y ) 6= ( 0 , 0 ) . Why does this result, together with that of Exercise 22, not contradict the ±nal assertion in the remark following Theorem 1? 24. 3 (Winding number) Let C be a piecewise smooth curve in the xy -plane which does not pass through the origin. Let θ = θ( x , y ) be the polar angle coordinate of the point P = ( x , y ) on C , not restricted to an interval of length 2 π , but varying continuously as P moves from one end of C to the other. As in Example 5 of Section 15.2, it happens that θ =− y x 2 + y 2 i + x x 2 + y 2 j . If, in addition, C is a closed curve, show that w( C ) = 1 2 π I C xdy ydx x 2 + y 2 has an integer value. w is called the winding number of C about the origin. 15.5 Surfaces and Surface Integrals This section and the next are devoted to integrals of functions de±ned over surfaces in 3-space. Before we can begin, it is necessary to make more precise just what is meant by the term “surface.” Until now we have been treating surfaces in an intuitive way, either as the graphs of functions f ( x , y ) or as the graphs of equations f ( x , y , z ) = 0. A smooth curve is a one-dimensional object because points on it can be located by giving one coordinate (for instance, the distance from an endpoint). Therefore, the curve can be de±ned as the range of a vector-valued function of one real variable. A surface is a two-dimensional object;points on it can be located by using two coordinates , and it can be de±ned as the range of a vector-valued function of two real variables. We will call certain such functions parametric surfaces. Parametric Surfaces DEFINITION 4 A parametric surface in 3-space is a continuous function r de±ned on some rectangle R given by a u b , c v d in the u v -plane and having values in 3-space: r ( u ,v) = x ( u i + y ( u j + z ( u k ,( u in R . Figure 15.16 A parametric surface S de±ned on parameter region R .The contour curves on S correspond to the rulings of R x y z a b c d v u r ( u ( u R S Actually, we think of the range of the function r ( u as being the parametric surface. It is a set S of points ( x , y , z ) in 3-space whose position vectors are the vectors r ( u for ( u in R . (See Figure 15.16.) If r is one-to-one,then the surface does not intersect itself. In this case r maps the boundary of the rectangle R (the four edges) onto a curve in 3-space, which we call the boundary of the parametric surface . The requirement
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SECTION 15.5: Surfaces and Surface Integrals 833 that R be a rectangle is made only to simplify the discussion. Any connected, closed, bounded set in the u v -plane, having well-defined area and consisting of an open set together with its boundary points, would do as well. Thus, we will from time to time consider parametric surfaces over closed disks, triangles, or other such domains in the u v -plane. Being the range of a continuous function defined on a closed, bounded set, a parametric surface is always bounded in 3-space.
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This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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Surface Integral - 832 CHAPTER 15 Vector Fields 23 Review...

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