Multivariable Advanced Calculus

11.8. THE DIVERGENCE THEOREM 315The vertical arrow at the end indicates the direction of increasing u 2 . The vertical sideof R (W ) shown there corresponds to the curved side in R (W ) which corresponds tothe part of ∂Ω which is selected by Q as shown in the picture. Here R is an orthogonaltransformation which has determinant equal to 1. Now the shear which goes from thediagram on the right to the one on its left preserves the direction of motion relative tothe surface the curve is bounding. This is geometrically clear. Similarly, the orthogonaltransformation R ∗ which goes from the curved part of the boundary of R (W ) to thecorresponding part of ∂Ω preserves the direction of motion relative to the surface. Thisis because orthogonal transformations in R 2 whose determinants are 1 correspond torotations. Thus increasing u 2 corresponds to counter clockwise motion around R(W )along the vertical side of R(W ) which corresponds to counter clockwise motion aroundR(W ) along the curved side of R(W ) which corresponds to counter clockwise motionaround Ω in the sense that the direction of motion along the curve is always such thatif you were walking in this direction, your left hand would be over the surface. In otherwords this agrees with the usual calculus conventions.11.8 The Divergence TheoremFrom Green’s theorem, one can quickly obtain a general Divergence theorem for Ω asdescribed above in Section 11.7.1. First note that from the above description of the R j ,∂ ( x k , x i1 , · · · x in−1)∂ (u 1 , · · · , u n )= sgn (k, i 1 · · · , i n−1 ) .Let F (x) be a C ( 1 Ω; R n) vector field. Say F = (F 1 , · · · , F n ) . Consider the differentialformn∑ω (x) ≡ F k (x) (−1) k−1 dx 1 ∧ · · · ∧ ̂dx k ∧ · · · ∧ dx nk=1where the hat means dx k is being left out. Thendω (x) ==≡n∑n∑k=1 j=1n∑k=1∂F k∂x j(−1) k−1 dx j ∧ dx 1 ∧ · · · ∧ ̂dx k ∧ · · · ∧ dx n∂F k∂x kdx 1 ∧ · · · ∧ dx k ∧ · · · ∧ dx ndiv (F) dx 1 ∧ · · · ∧ dx k ∧ · · · ∧ dx nThe assertion between the first and second lines follows right away from properties ofdeterminants and the definition of the integral of the above wedge products in termsof determinants. From Green’s theorem and the change of variables formula applied tothe individual terms in the description of ∫ Ω dωp∑∫j=1n ∑B j k=1∫Ωdiv (F) dx =(−1) k−1 ∂ (x 1 , · · · ̂x k · · · , x n )∂ (u 2 , · · · , u n )(ψj F k)◦ R−1j (0, u 2 , · · · , u n ) du 1 ,du 1 short for du 2 du 3 · · · du n .I want to write this in a more attractive manner which will give more insight. Theabove involves a particular partition of unity, the functions being the ψ i . Replace F in