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Unformatted text preview: Section 17.8 Stokes Theorem Making Flux Integrals Easy Our first approach to line integrals was the brute force method - pa- rameterize the curve, take the dot product and then integrate. Though this method is always guaranteed to work, we saw very quickly that it can take a very long time to do some fairly simple integrals. This led us to two other ways to calculate line integrals - Greens Theorem and the Fundamental Theorem of Calculus for line integrals, both meth- ods which were much easier than brute force. In the next two sections we shall consider two different methods to evaluate flux integrals, one generalizing FTC and the other generalizing Greens Theorem. 1. Stokes Theorem Before we state Stokes Theorem, we need a definition. Definition 1.1. Suppose S is an oriented surface with orientation vectorn and with boundary B (which is a space curve). Then we define an ori- entation on B called the induced orientation on B as follows: if we start walking around B standing in the same direction as the orientation vectorn , then the surface is always on our left. We look at a couple of examples. Example 1.2. ( i ) Find the boundary and the orientation of the boundary for the unit sphere if it has outward orientation. The unit sphere has no boundary (it is called a closed sur- face), so obviously there is no orientation on it. ( ii ) Find the boundary and the orientation of the boundary for the surface z = x 2 + y 2 over the rectangle [ 5 , 5] [ 5 , 5] sketched below with downward pointing orientation.-5.0 x-5.0-2.5 y-2.5 0.0 0.0 10 2.5 20 2.5 5.0 30 40 5.0 50 The boundary of the surface is the edge which consists of four connected parabolas. Since it is downard orientation, to 1 2 guarantee that the surface is always on the left as we walk along the boundary, we need to walk in a clcokwise direction. We are now ready to state Stokes Theorem. Result 1.3. Let S be an oriented piecewise-smooth surface that is bounded by a simple closed piecewise smooth boundary B with induced orientation. Let vector F be a vector field whose components have continuous partial derivatives on an open region in R 3 which contains...
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