Green s, Divergence, Stokes: Statements and First Applications
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1 Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane and let be its boundary. Let n be the outward normal to, let ds be an element of arclength, and let da be an element of area. Then < F, n > ds = div(f )da. We use the notation from the theorem above. Theorem 2 (Green s). Let P, Q be functions on the plane. Assume that is parametrized counterclockwise. Then ( Q P dx + Qdy = x P ) dxdy. Let F = (F 1, F 2 ) be a vector field in the plane so F 1, F 2 are functions of two variables. We view this as a vector field F = (F 1, F 2, 0) with three components and take its curl. We get curlf = (0, 0, F 2 x F 1 ). In the theorem below we write curl F = F 2 x F 1. Theorem 3 (tokes(planar version)). Let u be a unit tangent vector to the curve parametrized counterclockwise. Then curl F da = < F, u > ds. We can also write this as curl F da = F 1 dx + F 2 dy. Theorem 4 (Divergence). Let be a region in 3 with nice boundary. Let F be a vector field on an open set containing and. Let n be an outward unit normal to. Let DA denote an element of surface area of. Then < F, n > da = divf dv. Theorem 5 (tokes). Let be a nice surface in 3 bounded by a closed curve. Assume that the surface is orientable and that the boundary is chosen so that the surface lies to the left of the boundary. Let F be a vector field defined on an open set containing the surface and its boundary. Then < (curlf ), n > da = < F, ds >. 1
2 emarks: We say that a surface is orientable if we can choose, in a continuous fashion, a unit normal to the surface. An orientation of a curve is a choice of direction on the curve. Physical Interpretation of Curl and Divergence We gave heuristic arguments interpreting the divergence and curl of a vector field. particuclar divergence measures: If the vector field represents heat flow, then the divergence of a vector field at a point measures the amount of heat created at that point. If the vector field is an electrical field generated by static charges, then the divergence at a point measures charge density at that point. On the other hand, the curl of a vector field measures: If the vector field represents a fluid flow, then the curl of that vector field at a point measures the circulation of the flow about that point. Using the divergence theorem, stokes theorem we can give a more rigorous argument justifying our interpretations of divergence and curl. Let p 2, a connected region of 2 that contains p, C the boundary of which is a simple closed curve containing p, and F a vector field in 2. Let n be an outward normal to C. By the divergence theorem we have that the net flux out of C is F n ds. By the Divergence Theorem this is equal to div(f )da where da denotes an element of area. Let get smaller and smaller. In the limit we have that div(f )da div(f )(p) (area of ) In particular, the divergence measures the net outflow from p per unit area. Let p be a point on a surface and let be a small connected region of that contains p. Let C denote the boundary of. Let F be a vector field on and n a normal to. By toke s Theorem we have (curlf ) n da = F d s C Let shrink to p and divide the above by the area of. Upon taking the limit we get 1 curlf (p) = lim F d s p (Area of ) C In 2
3 Green s Theorem: Implications, Examples Green s theorem implies the planar tokes theorem. Let be a bounded, simply connected region with a nice boundary. We traverse F in a counter clockwise route in the line integrals. et F = (P, Q). Then P dx + Qdy = < (P, Q), ( dx dt, dy )dt >= dt < F, ds >. On the other hand ( Q x P ) dxdy = curl F dxdy. Green s theorem also implies the planar divergence theorem. Let F = (F 1, F 2 ) be a vector field. Again let be a bounded, simply connected region with a nice boundary. et P = F 2, Q = F 1 in Green s theorem. We obtain P dx + Qdy = F 2 dx + F 1 dy = < (F 1, F 2 ), (dy, dx) >= < F, n > ds. On the other hand ( Q x P ) ( F1 dxdy = x F ) 2 dxdy = (div F )dxdx. Example: We calculate the integral of the vector field F (x, y) = (y + 3x, 2y x) around the ellipse 4x 2 +y 2 = 4. Denote the region inside the ellipse by E. etting P = y +3x, Q = 2y x we have by Green s theorem E < F, ds >= E Q x P dxdy = ( 1 1)dxdy = 2 area of E = 2π4. The last calculation is an exercise. Example: We verify Green s theorem for P (x, y) = x, Q(x, y) = xy and is the unit disk. To verify means that we independently calculate both sides of the equality in Green s theorem and see that the two sides are indeed equal. We parametrize the unit circle by t (cos(t), sin(t), 0 t 2π. We get t=2π P dx + Qdy = (cos(t)( sin(t)) + cos(t) sin(t) cos(t)) dt = On the other hand by symmetry. cos(t) 2 2 ( Q x P ) dxdy = ydxdy = 0 2π cos(t)3 3 t=2π = 0 3
4 Divergence Theorem Examples: Example 1: Consider the vector field F (x, y, z) = ((x, y, z). We compute the flow of the vector field out of the unit ball in two ways using the divergence theorem. Denote the sphere by B. ince F is orthogonal to the sphere at each point and of length 1 we have < F (x, y, z), n(x, y, z) >= 1 at every point. Hence the flow out is < F, n > da = 4π B since the surface area of a sphere of radius 1 is 4π. On the other hand the divergence of F is 3 at every point. Hence Bdiv F dv = 3 volume of the unit sphere = 4π. Example 2: We need spherical coordinates to do this example. o we give a brief introduction. The coordinates are φ which runs from 0 to π, θ which runs from 0 to 2π, and which runs from 0 to. They are explained in the diagram called spherical coordinates. We have a change of coordinates P : (, φ, θ) x = sin(φ) cos(θ) P : (, φ, θ) y = sin(φ) sin(θ) P : (, φ, θ) z = cos(φ). If we parametrize a region 3 by spherical coordinates we need to know the formula for an element of area. This comes from the change of variables theorem. Given a function f : 3 we obtain the diagram D 2 P 3 f The change of variables theorem says that fdxdydz = D A lengthy calculation gives det(p ) = 2 sin(φ). (f P ) det(p ) ddθdφ. 4
5 Figure 1: pherical Coordinates 5
6 Problem: Let B be the unit ball and let F be the vector field (x 3, y 3, z 3 ). We wish to find < f, n > da. partialb By the Divergence Theorem this is equal to div F dv = B (3x 2 + 3y 2 + 3z 2 )dxdydz. Here dv is an element of volume. To calculate this integral we use spherical coordinates. It is equal to θ=2π φ=π =1 θ=0 tokes Theorem Examples φ=0 =0 2 2 sin(φ)ddφdθ. Example 3: Let F (x, y, z) = (z, x, y). Let be the surface z = x 2 + y 2, z 1. We verify tokes Theorem for F and. In particular we show that < curl F, n > da = < F, ds >. The first step is to sketch the surface. We first compute < curl F, n > da. ince this is a graph we have the parametrization ( ) u u P : v. v u 2 + v 2 The variables must lie in the domain u 2 + v 2 1. We calculate n. We have P u P 2u v = 2v. 1. This is not n. We have i j k 1 curl F = det x y z = 1. z x y 1 We can now calculate the first integral. We have 1 < curl F, n > da = 2u 2v, 1 dudv = ( 2u 2v+1)dudv. u 2 +v u 2 +v 2 1 By symmetry the integrals over u and v are zero. The integral over 1 is just the area of u 2 + v 2 1, that is. It is π. 6
7 We calculate the integral < F, ds >. We parametrize the boundary of, the unit circle in the plane z = 1, by cos(t) t sin(t). 1 Hence sin(t) ds = cos(t). 0 We calculate the integral t=2π < F, ds 1 sin(t) >= < cos(t), cos(t) > dt = sin(t) 0 2π 0 ( sin(t) + cos 2 (t) ) dt = π 7
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