This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Gauss theorem 1 Chapter 14 Gauss theorem We now present the third great theorem of integral vector calculus. It is interesting that Greens theorem is again the basic starting point. In Chapter 13 we saw how Greens theorem directly translates to the case of surfaces in R 3 and produces Stokes theorem. Now we are going to see how a reinterpretation of Greens theorem leads to Gauss theorem for R 2 , and then we shall learn from that how to use the proof of Greens theorem to extend it to R n ; the result is called Gauss theorem for R n . A. Greens theorem reinterpreted We begin with the situation obtained in Section 12C for a region R in R 2 . With the positive orientation for bd R , we have ZZ R f x dxdy = Z bd R fdy, ZZ R g y dxdy = Z bd R gdx. We now consider a vector field F = ( F 1 ,F 2 ) on R , and we obtain from Greens theorem ZZ R F 1 x + F 2 y dxdy = Z bd R F 1 dy F 2 dx. (Notice if we were thinking in terms of Stokes theorem, we would put the minus sign on the left side instead of the right.) We now work on the line integral, first writing it symbolically in the form Z bd R F ( dy, dx ) . We now express what this actually means. Of course, bd R may come in several disjoint pieces, so we focus on just one such piece, say the closed curve = ( t ) = ( 1 ( t ) , 2 ( t )), a t b . Then the part of the line integral corresponding to this piece is actually Z b a F ( ( t )) ( 2 ( t ) , 1 ( t )) dt . R 2 Chapter 14 This in turn equals Z b a F ( ( t )) ( 2 ( t ) , 1 ( t )) k ( t ) k k ( t ) k dt. The unit vector in the dot product has a beautiful geometric interpretation. Namely, it is orthogonal to the unit tangent vector ( 1 ( t ) , 2 ( t )) k ( t ) k , and it is oriented 90 clockwise from the tangent vector. (You can use the matrix J of Section 8D to see this, as J 1 2 = 2 1 . You can also think in complex arithmetic: i ( 1 + i 2 ) = 2 i 1 , and multiplication by i rotates 90 clockwise.) Because the tangent direction ( t ) has the region R placed 90 counterclockwise from it, we conclude that the normal vector ( 2 ( t ) , 1 ( t )) k ( t ) k points away from R : N Let us name this unit normal vector b N . Thus at each point ( x,y ) bd R , the vector b N ( x,y ) at that point is determined by these requirements: b N has unit norm, b N is orthogonal to bd R , b N points outward from R . Greens theorem has now been rewritten in the form ZZ R F 1 x + F 2 y dxdy = Z bd R F b Nd vol 1 . Gauss theorem 3 This result is precisely what is called Gauss theorem in R 2 . The integrand in the integral over R is a special function associated with a vector field in R 2 , and goes by the name the divergence of F : div F = F 1 x + F 2 y ....
View
Full
Document
This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 Hatcher
 Calculus, Vector Calculus

Click to edit the document details