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chap14 - gauss theorem

# chap14 - gauss theorem - Gauss theorem 1 Chapter 14 Gauss...

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Gauss’ theorem 1 Chapter 14 Gauss’ theorem We now present the third great theorem of integral vector calculus. It is interesting that Green’s theorem is again the basic starting point. In Chapter 13 we saw how Green’s 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 Green’s theorem leads to Gauss’ theorem for R 2 , and then we shall learn from that how to use the proof of Green’s theorem to extend it to R n ; the result is called Gauss’ theorem for R n . A. Green’s 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 Green’s 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 γ = γ ( t ) = ( γ 1 ( t ) , γ 2 ( t )), a t b . Then the part of the line integral corresponding to this piece is actually Z b a F ( γ ( t )) ( γ 0 2 ( t ) , - γ 0 1 ( t )) dt . R

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2 Chapter 14 This in turn equals Z b a F ( γ ( t )) ( γ 0 2 ( t ) , - γ 0 1 ( t )) k γ 0 ( t ) k k γ 0 ( t ) k dt. The unit vector in the dot product has a beautiful geometric interpretation. Namely, it is orthogonal to the unit tangent vector ( γ 0 1 ( t ) , γ 0 2 ( t )) k γ 0 ( 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 γ 0 1 γ 0 2 = - γ 0 2 γ 0 1 . You can also think in complex arithmetic: - i ( γ 0 1 + 0 2 ) = γ 0 2 - 0 1 , and multiplication by - i rotates 90 clockwise.) Because the tangent direction γ 0 ( t ) has the region R placed 90 counterclockwise from it, we conclude that the normal vector ( γ 0 2 ( t ) , - γ 0 1 ( t )) k γ 0 ( 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 . Green’s 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 . Again we can use the symbolic “del” vector = ∂x , ∂y to write div F = ∇ • F.

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chap14 - gauss theorem - Gauss theorem 1 Chapter 14 Gauss...

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