M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM


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1 68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof of the Divergence Theorem Let F be a smooth vector field defined on a solid region with boundar surface oriented outward. e wish to show that F d div F d. For the Divergence Theorem, we use the same approach as we used for Green s Theorem; first prove the theorem for rectangular regions, then use the change of variables formula to prove it for regions parameteried b rectangular regions, finall paste such regions together to form general regions. Proof for ectangular Solids with Sides Parallel to the es onsider a smooth vector field F defined on the rectangular solid : a b, c d, e f. (See Figure M.50). e start b computing the flu of F through the two faces of perpendicular to the ais, 1 2, both oriented outward: F d F d f d e c f d e c F 1 (a,, ) d d + f d e c (F 1 (b,, ) F 1 (a,, )) d d. F 1 (b,, ) d d the Fundamental Theorem of alculus, F 1 (b,, ) F 1 (a,, ) b a F 1 d, so F d + F d 1 2 f d b e c a F 1 d d d F 1 d. a similar argument, we can show 4 F d + F d F 2 d 5 6 F d + F d F d. dding these, we get F d This is the Divergence Theorem for the region. ( F1 + F 2 + F ) d div F d.
2 Theor Supplement Section M 69 1 ) left) (bottom) 4 1 right) Figure M.50: ectangular solid in space S left) s u S 2 S 6 S 5 (bottom) S 4 t S1 right) (left) 2 5 (bottom) 6 4 Figure M.51: rectangular solid in stuspace the corresponding curved solid in space Proof for egions Parameteried b ectangular Solids Now suppose we have a smooth change of coordinates (s, t, u), (s, t, u), (s, t, u). onsider a curved solid in space corresponding to a rectangular solid in stuspace. See Figure M.51. e suppose that the change of coordinates is onetoone on the interior of, that its Jacobian determinant is positive on. e prove the Divergence Theorem for using the Divergence Theorem for. Let be the boundar of. To prove the Divergence Theorem for, we must show that F d div F d. First we epress the flu through as a flu integral in stuspace over S, the boundar of the rectangular region. In vector notation the change of coordinates is The face 1 of is parameteried b so on this face r r (s, t, u) (s, t, u) i + (s, t, u) j + (s, t, u) k. r r (a, t, u), c t d, e u f, d ± r r dt du. In fact, in order to make d point outward, we must choose the negative sign. (Problem on page 7 shows how this follows from the fact that the Jacobian determinant is positive.) Thus, if S 1 is the face s a of, F d r F S 1 r dt du, 1 The outward pointing area element on S 1 is d S i dt du. Therefore, if we choose a vector field G on stuspace whose component in the sdirection is we have G 1 F r r, F d G ds. 1 S 1
3 70 Theor Supplement Section M Similarl, if we define the t u components of G b then G 2 F r r G F r r, i S i F d G ds, i 2,..., 6. (See Problem 4.) dding the integrals for all the faces, we find that F d G ds. where Since we have alread proved the Divergence Theorem for the rectangular region, we have G ds div G d, div G + G 2 Problems 5 6 on page 7 show that + G 2 + G (,, ) (s, t, u) S S + G. ( F1 + F 2 + F ). So, b the threevariable change of variables formula on page 61, ( div F F1 d + F 2 + F ) d d d ( F1 + F 2 + F ) (,, ) (s, t, u) ds dt du ( G1 + G 2 + G ) ds dt du div G d. In summar, we have shown that F d div F d S G d S div G d. the Divergence Theorem for rectangular solids, the righth sides of these equations are equal, so the lefth sides are equal also. This proves the Divergence Theorem for the curved region. Pasting egions Together s in the proof of Green s Theorem, we prove the Divergence Theorem for more general regions b pasting smaller regions together along common faces. Suppose the solid region is formed b pasting together solids 1 2 along a common face, as in Figure M.52. The surface which bounds is formed b joining the surfaces 1 2 which bound 1 2, then deleting the common face. The outward flu integral of a vector field F through 1 includes the integral across the common face, the outward flu integral of F through 2
4 Theor Supplement Section M 71 ommon face 2 1 Figure M.52: egion formed b pasting together 1 2 includes the integral over the same face, but oriented in the opposite direction. Thus, when we add the integrals together, the contributions from the common face cancel, we get the flu integral through. Thus we have F d F d + F d. 1 2 ut we also have div F d div F d + div F d. 1 2 So the Divergence Theorem for follows from the Divergence Theorem for 1 2. Hence we have proved the Divergence Theorem for an region formed b pasting together regions that can be smoothl parameteried b rectangular solids. Eample 1 Let be a spherical ball of radius 2, centered at the origin, with a concentric ball of radius 1 removed. Using spherical coordinates, show that the proof of the Divergence Theorem we have given applies to. Solution e cut into two hollowed hemispheres like the one shown in Figure M.5,. In spherical coordinates, is the rectangle 1 ρ 2, 0 φ π, 0 θ π. Each face of this rectangle becomes part of the boundar of. The faces ρ 1 ρ 2 become the inner outer hemispherical surfaces that form part of the boundar of. The faces θ 0 θ π become the two halves of the flat part of the boundar of. The faces φ 0 φ π become line segments along the ais. e can form b pasting together two solid regions like along the flat surfaces where θ constant. θ π φ π θ 0 φ π ρ 1 ρ 2 φ π 2 ρ Figure M.5: The hollow hemisphere the corresponding rectangular region in ρθφspace 1 π θ
5 72 Theor Supplement Section M Proof of Stokes Theorem onsider an oriented surface, bounded b the curve. e want to prove Stokes Theorem: curl F d F d r. e suppose that has a smooth parameteriation r r (s, t), so that corresponds to a region in the stplane, corresponds to the boundar of. See Figure M.54. e prove Stokes Theorem for the surface a continuousl differentiable vector field F b epressing the integrals on both sides of the theorem in terms of s t, using Green s Theorem in the stplane. First, we convert the line integral F d r into a line integral around : F d r F r ds + F r dt. So if we define a 2dimensional vector field G (G 1, G 2 ) on the stplane b then G 1 F r F d r G 2 F r, G d s, using s to denote the position vector of a point in the stplane. hat about the flu integral curl F d that occurs on the other side of Stokes Theorem? In terms of the parameteriation, curl F d In Problem 7 on page 74 we show that Hence e have alread seen that curl F r r ds dt. curl F r r G 2. curl F d F d r ( G2 G ) 1 ds dt. G d s. Green s Theorem, the righth sides of the last two equations are equal. Hence the lefth sides are equal as well, which is what we had to prove for Stokes Theorem. t s Figure M.54: region in the stplane the corresponding surface in space; the curve corresponds to the boundar of
6 Theor Supplement Section M 7 Problems for Section M 1. Let be a solid circular clinder along the ais, with a smaller concentric clinder removed. Parameterie b a rectangular solid in rθspace, where r, θ, are clindrical coordinates. 2. In this section we proved the Divergence Theorem using the coordinate definition of divergence. Now we use the Divergence Theorem to show that the coordinate definition is the same as the geometric definition. Suppose F is smooth in a neighborhood of ( 0, 0, 0), let U be the ball of radius with center ( 0, 0, 0). Let m be the minimum value of div F on U let M be the maimum value. (a) Let S be the sphere bounding U. Show that S F d m M. olume of U (b) Eplain wh we can conclude that lim 0 S F d div F olume of U ( 0, 0, 0). (c) Eplain wh the statement in part (b) remains true if we replace U with a cube of side, centered at ( 0, 0, 0). Problems 6 fill in the details of the proof of the Divergence Theorem.. Figure M.51 on page 69 shows the solid region in space parameteried b a rectangular solid in stuspace using the continuousl differentiable change of coordinates r r (s, t, u), a s b, c t d, e u f. ( ) Suppose that r r r is positive. (a) Let 1 be the face of corresponding to the face s a of. Show that r, if it is not ero, points into. (b) Show that r r is an outward pointing normal on 1. (c) Find an outward pointing normal on 2, the face of where s b. 4. Show that for the other five faces of the solid in the proof of the Divergence Theorem (see page 70): i F d S i G d S, i 2,, 4, 5, Suppose that F is a continuousl differentiable vector field that a, b, c are vectors. In this problem we prove the formula grad( F b c ) a + grad( F c a ) b + grad( F a b ) c ( a b c ) div F. (a) Interpreting the divergence as flu densit, eplain wh the formula makes sense. [Hint: onsider the flu out of a small parallelepiped with edges parallel to a, b, c.] (b) Sa how man terms there are in the epansion of the lefth side of the formula in artesian coordinates, without actuall doing the epansion. (c) rite down all the terms on the lefth side that contain F 1/. Show that these terms add up to a b c F1. (d) rite down all the terms that contain F 1/. Show that these add to ero. (e) Eplain how the epressions involving the other seven partial derivatives will work out, how this verifies that the formula holds. 6. Let F be a smooth vector field in space, let (s, t, u), (s, t, u), (s, t, u) be a smooth change of variables, which we will write in vector form as r r (s, t, u) (s, t, u) i +(s, t, u) j +(s, t, u) k. Define a vector field G (G 1, G 2, G ) on stuspace b G 1 F r r G F r r. (a) Show that + G2 + G G 2 F r r F r r + F r r + F r r. (b) Let r 0 r (s 0, t 0, u 0), let a r ( r 0), r b ( r 0), Use the chain rule to show that ( ) G1 + G2 + G r r 0 r c ( r 0). grad( F b c ) a + grad( F c a ) b + grad( F a b ) c.
7 74 INDEX (c) Use Problem 5 to show that + G2 + G ( ) (,, ) F1 (s, t, u) + F2 + F. 7. This problem completes the proof of Stokes Theorem. Let F be a smooth vector field in space, let S be a surface parameteried b r r (s, t). Let r 0 r (s 0, t 0) be a fied point on S. e define a vector field in stspace as on page 72: G 1 F r G 2 F r. (a) Let a r ( r 0), r b ( r 0). Show that G2 ( r 0) ( r 0) grad( F a ) b grad( F b ) a. (b) Use Problem 0 on page 961 of the tetbook to show curl F r r G2 G1.
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