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Unformatted text preview: Stokes theorem 1 Chapter 13 Stokes theorem In the present chapter we shall discuss R 3 only. We shall use a right-handed coordinate system and the standard unit coordinate vectors , , k . We shall also name the coordinates x , y , z in the usual way. The basic theorem relating the fundamental theorem of calculus to multidimensional in- tegration will still be that of Green. In this chapter, as well as the next one, we shall see how to generalize this result in two directions. In this chapter we generalize it to surfaces in R 3 , whereas in the next chapter we generalize to regions contained in R n . But in all of these procedures it is still Greens theorem that is fundamental. A. Orientable surfaces We shall be dealing with a two-dimensional manifold M R 3 . Well just use the word surface to describe M . There are two features of M that we need to discuss first. The first is the idea of a normal vector for M . We assume that M is of class C 1 , so that at each point p M there is a vector of unit norm which is orthogonal to M , in the sense that it is orthogonal to the tangent space T p M . There are of course two choices of such a normal vector, and we now need to make a choice. DEFINITION. The surface M is said to be orientable if there exists a unit normal vector b N ( p ) at each point p M which is a continuous function of p . The continuity of b N ( p ) is all-important. For instance, one can construct a Mobius strip and obtain a surface which is not orientable: image created with Mathematica In case a surface is described implicitly by an equation g ( x,y,z ) = 0 such that g is never 0 at any point of the surface, and if g is of class C 1 , then g is continuous and we have two choices for b N : b N = g k g k or b N =- g k g k . 2 Chapter 13 For example, for the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 we may take b N = ( x a 2 , y b 2 , z c 2 ) q x 2 a 4 + y 2 b 4 + z 2 c 4 . For the sphere S (0 ,a ) we have in particular either N b N = ( x,y,z ) a or b N =- ( x,y,z ) a . REMARK. An orientable surface is also said to be two-sided . The reason for this is that the continuous normal vector b N serves to define a direction of up at points of M . Thus at points of M there is a definite sense of two sides of M , an up side and a down side. A Mobius strip for example is one-sided , which may be demonstrated by drawing a curve along the equator of M with a pencil. EXTENSION . Frequently we shall need to analyze a surface M R 3 which is not actually orientable in the above sense, but is close enough. The surface may consist of finitely many surfaces with the proper orientability. A few examples should suffice for a good explanation....
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