ES.182A Topic 45 Notes Jeremy Orloff

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1 E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed surface is one the encloses a solid region. Here are some examples: Each of the examples is drawn with outward normals, i.e. they are oriented outwards. 45. ivergence efinition. The divergence of a vector field is essentially the same as divergence in. If F = M, N, P = Mi + Nj + P k then divf = M x + N y + P z = M x + N y + P z. As before, note that the divergence of a vector field is a scalar function. After we state the divergence theorem we will show that the divergence can be interpreted as the infinitesimal source rate of the field. The divergence theorem is exactly like the flux form of Green s theorem. Theorem. ivergence theorem. (Also called Gauss theorem.) uppose is a closed surface with interior and outward pointing normals. uppose F is a vector field which is continuously differentiable on all of. Then F n d = divf dv. () ee below for a physical interpretation of the divergence theorem. 45. Notation ometimes we write d for n d.

2 45 MORE URFACE INTEGRAL; IVERGENCE THEOREM 45.4 Examples Example 45.. Verify the divergence theorem for F = xi + yj + zk and = sphere of radius a. answer: ince F is radial, F n = F = a (on the sphere). Thus, the right-hand side of Equation becomes F n d = a area of = a 4πa = 4πa. We have divf =, so the left-hand side of Equation becomes divf dv = volume of sphere = 4πa. QE Example 45.. Use the divergence theorem to evaluate the flux of F = x i + y j + z k across the sphere ρ = a. answer: Here, divf = (x +y +z ) = ρ. Therefore by the divergence theorem Equation π π a F n d = ρ dv = ρ ρ sin(φ) dρ dφ dθ = πa Example 45.. Let = part of the paraboloid z = x y above the xy-plane and let = the unit disk in the xy-plane. Take F = yz, xz, xy and find the flux of F upward through using the divergence theorem. answer: Note that is not a closed surface, but + is closed. To use the divergence theorem we ll need to work with the closed surface. Write F = Mi+Nj+P k, where M = yz, N = xz, P = xy. o, divf = M x +N y +P z = 0. The divergence theorem says: flux = F n d = + divf dv = 0 dv = 0

3 45 MORE URFACE INTEGRAL; IVERGENCE THEOREM Thus, F n d + F n d = 0 F n d = F n d. Therefore to find what we want we only need to compute the flux through. But is in the xy-plane, so d = dx dy and the outward normal n = k. Thus, F n d = xy dx dy on. ince is the unit disk, symmetry gives xy dx dy = 0. o, we have our answer: F n d = F n d = 0. Example Let be the part of the plane shown. Compute the upward flux of F = y j two ways. (i) directly; (ii) using the divergence theorem. z y x y R x answer: (i) The plane has equation x + y/ + z/ = A normal to is N =, /, /. Making the z-component we get n d =, /, dx dy. Thus F n d = y dx dy. o, flux = y dx dy = R x (ii) To use the divergence theorem we need a closed surface. 0 0 y dy dx =... =. We use the tetrahedron with faces,,,. Computing flux: : n = j F n = y = 0 on flux = 0. : n = i F n = 0 on flux = 0. : n = k F n = 0 on flux = 0. Now using all these zeros: F n d = F n d Using the divergence theorem F n d = dv = volume = base height = Combining these two equations we have F n d =.

4 45 MORE URFACE INTEGRAL; IVERGENCE THEOREM Proof of the divergence theorem First we look at a small box. Then just like for Green s theorem we handle the general volume by adding together a lot of boxes. Let F = M, N, P, P 0 = (,, z 0 ), the box as shown. The box has limits: x: to + x, y: to + y, z: z 0 to z 0 + z. Look at the flux through the top and bottom. Top: n = k F n d = P d = P dx dy flux = z) dy dx. Bottom: n = k F n d = P d = P dx dy flux = net flux through top and bottom = P (x, y, z 0 + z) P (x, y, z 0 ) dy dx = P (x, y, z 0 + z0 + z z 0 P (x, y, z 0 ) dy dx. P (x, y, z) dz dy dx. z In brief, net flux through top and bottom = P z dv. Likewise, net flux through front and back = M x dv. Likewise, net flux through left and right = N y dv. net flux out of the box = M x + N y + P z dv. QE 45.6 Interpretation of the divergence theorem Physically, the divergence theorem is interpreted just like the normal form for Green s theorem. Think of F as a three-dimensional velocity field. Look first at the left side of Equation. The surface integral represents the rate fluid flows across the closed surface (with flow out of considered as positive, flow into as negative). Look now at the right side of Equation. We will focus on a small volume around a point P 0 and see that we can interpret div F P0 as the infinitesimal source rate of F at P 0. o, consider a point P 0 inside a small box (really any shaped volume) with surface. If is small enough then divf dv div F P0 volume().

5 45 MORE URFACE INTEGRAL; IVERGENCE THEOREM 5 o, the divergence theorem says that flux across div F P0 volume() div F P0 flux across volume(). Letting shrink to the point P 0, we can interpret the right-hand side of this as the (infinitesimal) source rate at P 0, i.e. div F P0 = source rate of F at P 0. Note. If F is a velocity field then divf has units of /time. o, integrating divf gives units of volume/time, i.e. flux. o, the divergence theorem says total flux across = integral of the infinitesimal source rate of F over. i.e., the net flow outward across is the same as the rate at which fluid is being produced (or added to the flow) inside.