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Unformatted text preview: Integration on manifolds 1 Chapter 11 Integration on manifolds We are now almost ready for our concluding chapter on the great theorems of classical vector calculus, the theorems of Green and Gauss and Stokes. The final thing we need to understand is the correct procedure for integrating over a manifold. Of course, manifolds are typically curved objects, so there are significant issues here that we did not have to face in dealing with integration over (flat) Euclidean space. Even though we have been able to use integration to compute such things as the volume of a ball in R n , yet the integration involved is essentially flat. This is reflected in a formula such as d vol n = dx 1 dx 2 ...dx n . As a specific example, we have for the ball B (0 ,r ) in R 3 , vol 3 ( B (0 ,r )) = 4 3 r 3 , but we have not yet even defined what we should mean by the area of the sphere, vol 2 ( S (0 ,r )) . It will in fact turn out that there is a very reasonable way to accomplish this task, and it will be based on our knowledge of flat integration. We choose to employ the parametric presentation of manifolds as discussed thoroughly in Section 6E. It is in this context that we shall see how to define volume and integration. Though our definition is to be based on a particular choice of parameters, we shall be able to prove that the integration we define is actually invariant under a change of parameters and is thus intrinsic to the manifold. We denote by M an m-dimensional manifold contained in R n . In what follows we shall first investigate how to define the m-dimensional volume of subsets of M , and after that we shall easily define integrals of real-valued functions defined on M . We begin with sort of a warm up case, that of one-dimensional manifolds in R n . A. The one-dimensional case We assume in this section that m = 1, so we are dealing essentially with a curve M R n . We effectively already know exactly how to deal with this case, thanks to Section 2B. If M is represented parametrically as the image of a one-to-one function of class C 1 , ( a,b ) F- M, 2 Chapter 11 then the length of M is given as vol 1 ( M ) = Z b a k F ( t ) k dt. Of course, M R n and F ( t ) R n and k F ( t ) k is the norm of the vector F ( t ). We really need to say no more about this definition except to remark that if we imagine the interval ( a,b ) to be partitioned into small pieces ( , ), then F sends ( , ) to an approximate interval in R n whose length should be approximately k F ( ) k ( - ). Thus k F ( ) k represents a scale factor relating the parameter length - to a length in R n . A useful way to represent this definition is d vol 1 = k F ( t ) k dt....
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