Divergence Theorem. Catalin Zara. May 5, UMass Boston. Catalin Zara (UMB) Divergence Theorem May 5, / 14
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1 Divergence Theorem Catalin Zara UMass Boston May 5, 2010 Catalin Zara (UMB) Divergence Theorem May 5, / 14
2 Vector Fields in pace X: smooth vector field defined on an open region D in space; p: point in D. Questions: What is the effect of X around p? What is the infinitesimal effect of X at p? Infinitesimal: limit of effect on regions shrinking to p. Catalin Zara (UMB) Divergence Theorem May 5, / 14
3 Vector Fields in pace X: smooth vector field defined on an open region D in space; p: point in D. Questions: What is the effect of X around p? What is the infinitesimal effect of X at p? Infinitesimal: limit of effect on regions shrinking to p. Two types of effects: radial effect rotational effect Catalin Zara (UMB) Divergence Theorem May 5, / 14
4 Vector Fields in pace X: smooth vector field defined on an open region D in space; p: point in D. Questions: What is the effect of X around p? What is the infinitesimal effect of X at p? Infinitesimal: limit of effect on regions shrinking to p. Two types of effects: Radial effect: radial effect rotational effect : a piecewise smooth surface, bounding a region B around p; Questions: How much matter does X carry across? What happens when B {p}? Catalin Zara (UMB) Divergence Theorem May 5, / 14
5 Orientations of urfaces We first need to decide which way do we measure: inward or outward? Catalin Zara (UMB) Divergence Theorem May 5, / 14
6 Orientations of urfaces We first need to decide which way do we measure: inward or outward? : smooth surface, not necessarily boundary of a domain; We need a consistent choice of normal direction at each point of N: a continuous unit vector field normal to. uch a normal field doesn t always exist! (Möebius band) Catalin Zara (UMB) Divergence Theorem May 5, / 14
7 Orientations of urfaces We first need to decide which way do we measure: inward or outward? : smooth surface, not necessarily boundary of a domain; We need a consistent choice of normal direction at each point of N: a continuous unit vector field normal to. uch a normal field doesn t always exist! (Möebius band) Definitions: is orientable if it has a continuous normal unit vector field. Each choice of such a normal field endows with an orientation; An oriented surface is a surface with a predetermined orientation. Catalin Zara (UMB) Divergence Theorem May 5, / 14
8 Orientations of urfaces We first need to decide which way do we measure: inward or outward? : smooth surface, not necessarily boundary of a domain; We need a consistent choice of normal direction at each point of N: a continuous unit vector field normal to. uch a normal field doesn t always exist! (Möebius band) Definitions: is orientable if it has a continuous normal unit vector field. Each choice of such a normal field endows with an orientation; An oriented surface is a surface with a predetermined orientation. If the surface bounds a domain in space: outward normal gives the positive orientation; inward normal gives the negative orientation. Catalin Zara (UMB) Divergence Theorem May 5, / 14
9 Flux and Divergence : an oriented surface, orientation given by the unit normal field N; X: smooth vector field on Definition: The flux of X across is X Nd = X d. Catalin Zara (UMB) Divergence Theorem May 5, / 14
10 Flux and Divergence : an oriented surface, orientation given by the unit normal field N; X: smooth vector field on Definition: The flux of X across is X Nd = Radial effect of X at p: D: region around p, X d. Boundary of D: = D, piecewise smooth parametrized surface. Definition: The divergence of X at p is the density of flux at p: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. Catalin Zara (UMB) Divergence Theorem May 5, / 14
11 Computations Using Parametrizations ϕ: D, P = ϕ(u, v): smooth parametrization of ; ϕ u and ϕ v are tangent vectors; ϕ u ϕ v is a normal vector; may point in the direction of N or not; parametrization ϕ is compatible with the orientation given by N if ϕ u ϕ v and N point in the same direction Equivalently: the frame {N, ϕ u, ϕ v } is right hand oriented. If ϕ is a parametrization compatible with the orientation, then N = ϕ u ϕ v ϕ u ϕ v d = N d = ϕ u ϕ v ϕ u ϕ v ϕ u ϕ v du dv = ϕ u ϕ v du dv and therefore X Nd = X d = D X(ϕ(u, v)) (ϕ u ϕ v )du dv. Catalin Zara (UMB) Divergence Theorem May 5, / 14
12 Example Compute the flux of X = axi across : sphere of radius R centered at the origin, positively oriented Catalin Zara (UMB) Divergence Theorem May 5, / 14
13 Example Compute the flux of X = axi across : sphere of radius R centered at the origin, positively oriented A parametrization of is ϕ: [0, π] [0, 2π] R 3, ϕ(u, v) = (R sinu cosv, R sin u sinv, R cosu). Catalin Zara (UMB) Divergence Theorem May 5, / 14
14 Example Compute the flux of X = axi across : sphere of radius R centered at the origin, positively oriented A parametrization of is ϕ: [0, π] [0, 2π] R 3, Then ϕ(u, v) = (R sinu cosv, R sin u sinv, R cosu). ϕ u = R cosucosv, R cosusinv, R sin u ϕ v = R sin u sinv, R sin u cosv, 0 ϕ u ϕ v = R 2 sin 2 u cosv, R 2 sin 2 u sin v, R 2 sin u cosu = R sinur = R 2 sin un X d = X(ϕ(u, v)) (ϕ u ϕ v )du dv = = D u=π v=2π u=0 v=0 =ar 3 ( u=π u=0 a R sinu cosv R 2 sin 2 u cosv du dv = )( v=2π ) sin 3 u du cos 2 v dv = cr 3 4 v=0 3 π = 4πaR3. 3 Catalin Zara (UMB) Divergence Theorem May 5, / 14
15 Example: Continued axi d = 4πaR3, 3 byj d = 4πbR3, 3 czk d = 4πcR3 3 Catalin Zara (UMB) Divergence Theorem May 5, / 14
16 Example: Continued axi d = 4πaR3, 3 byj d = 4πbR3, 3 czk d = 4πcR3 3 If X = axi + by j + cz k, then X d = 4πR3 (a + b + c) = (a + b + c)vol(b), 3 where B is the volume of the ball enclosed by the sphere. Catalin Zara (UMB) Divergence Theorem May 5, / 14
17 Example: Continued axi d = 4πaR3, 3 byj d = 4πbR3, 3 If X = axi + by j + cz k, then X d = 4πR3 (a + b + c) = (a + b + c)vol(b), 3 where B is the volume of the ball enclosed by the sphere. czk d = 4πcR3 3 If the divergence of X at 0 exists, then: 3 divx(0) = lim R 0 4πR 3 X d = lim (a + b + c) = a + b + c. R 0 R(0) Catalin Zara (UMB) Divergence Theorem May 5, / 14
18 Another Example Let be the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward, and X = ai + bj + ck. Compute X d. Catalin Zara (UMB) Divergence Theorem May 5, / 14
19 Another Example Let be the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward, and X = ai + bj + ck. Compute X d. Parametrization: ϕ: B R 3, ϕ(u, v) = (u, v, 4 u 2 v 2 ) i j k ϕ u ϕ v = 1, 0, 2u 0, 1, 2v = 1 0 2u = 2ui + 2vj k v N = ϕ u ϕ v ϕ u ϕ v. ( X d = X n d = X ϕ ) u ϕ v ϕ u ϕ v du dv = ϕ u ϕ v hence = (ai + bj + ck) ( 2ui 2vj + k)du dv = ( 2au 2bv + c)du dv, X d = B ( 2au 2bv + c)du dv = c B du dv = c 4π = 4π c. Catalin Zara (UMB) Divergence Theorem May 5, / 14
20 Back to Divergence Recall: The divergence of X at p is the density of flux: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. Catalin Zara (UMB) Divergence Theorem May 5, / 14
21 Back to Divergence Recall: The divergence of X at p is the density of flux: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. The limit does exist when X is reasonably smooth. Catalin Zara (UMB) Divergence Theorem May 5, / 14
22 Back to Divergence Recall: The divergence of X at p is the density of flux: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. The limit does exist when X is reasonably smooth. We have seen similar limits when we talked about average values: 1 f(p) = lim f(q)dv. D {p} vol(d) D Catalin Zara (UMB) Divergence Theorem May 5, / 14
23 Back to Divergence Recall: The divergence of X at p is the density of flux: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. The limit does exist when X is reasonably smooth. We have seen similar limits when we talked about average values: 1 f(p) = lim f(q)dv. D {p} vol(d) However: the definition of divx involves a surface integral the definition of average involves a triple integral D Catalin Zara (UMB) Divergence Theorem May 5, / 14
24 Back to Divergence Recall: The divergence of X at p is the density of flux: 1 (div X)(p) = lim X Nd, D {p} vol(d) if the limit exists. The limit does exist when X is reasonably smooth. We have seen similar limits when we talked about average values: 1 f(p) = lim f(q)dv. D {p} vol(d) However: the definition of divx involves a surface integral the definition of average involves a triple integral We should somehow transform the surface integral into a triple integral. D Catalin Zara (UMB) Divergence Theorem May 5, / 14
25 Divergence Theorem Theorem Let D be a compact set in space with boundary a piecewise smooth parametrized surface, oriented by the outward normal, and let X(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k be a smooth vector field defined on D. Then ( P X d = x + Q y + R ) dv z D Catalin Zara (UMB) Divergence Theorem May 5, / 14
26 Divergence Theorem Theorem Let D be a compact set in space with boundary a piecewise smooth parametrized surface, oriented by the outward normal, and let X(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k be a smooth vector field defined on D. Then ( P X d = x + Q y + R ) dv z Consequence: 1 (div X)(p) = lim D {P } vol(d) = lim D {P } 1 vol(d) D X d = ( P x + Q y + R ) dv = P z x + Q y + R z. D Catalin Zara (UMB) Divergence Theorem May 5, / 14
27 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Catalin Zara (UMB) Divergence Theorem May 5, / 14
28 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Catalin Zara (UMB) Divergence Theorem May 5, / 14
29 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv Catalin Zara (UMB) Divergence Theorem May 5, / 14
30 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (div X)(p) > 0, then Catalin Zara (UMB) Divergence Theorem May 5, / 14
31 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (divx)(p) > 0, then p acts as a source; Catalin Zara (UMB) Divergence Theorem May 5, / 14
32 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (divx)(p) > 0, then p acts as a source; If (div X)(p) < 0, then Catalin Zara (UMB) Divergence Theorem May 5, / 14
33 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (divx)(p) > 0, then p acts as a source; If (divx)(p) < 0, then p acts as a sink; Catalin Zara (UMB) Divergence Theorem May 5, / 14
34 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (divx)(p) > 0, then p acts as a source; If (divx)(p) < 0, then p acts as a sink; If divx 0 on some domain D, then Catalin Zara (UMB) Divergence Theorem May 5, / 14
35 Divergence Theorem If X = P i + Qj + Rk, then divx = P x + Q y + R z = x, y, z P, Q, R = X. Intuitive notation: divx = X. Confirmed by the example of X = axi + by j + cz k = divx = a + b + c. Theorem X d = D divxdv If (divx)(p) > 0, then p acts as a source; If (divx)(p) < 0, then p acts as a sink; If divx 0 on some domain D, then X is incompressible on D. Catalin Zara (UMB) Divergence Theorem May 5, / 14
36 Example is the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward; X = ai + bj + ck. Use the Divergence Theorem to compute X d. Catalin Zara (UMB) Divergence Theorem May 5, / 14
37 Example is the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward; X = ai + bj + ck. Use the Divergence Theorem to compute X d. Problem: The surface does not enclose a region in space; Catalin Zara (UMB) Divergence Theorem May 5, / 14
38 Example is the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward; X = ai + bj + ck. Use the Divergence Theorem to compute X d. Problem: The surface does not enclose a region in space; We add the disk D of radius 2 centered at the origin in the plane z = 0; X d = divx dv = 0, D R Catalin Zara (UMB) Divergence Theorem May 5, / 14
39 Example is the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward; X = ai + bj + ck. Use the Divergence Theorem to compute X d. Problem: The surface does not enclose a region in space; We add the disk D of radius 2 centered at the origin in the plane z = 0; X d = divx dv = 0, D R orients D with the downward normal, hence X d = X d R D Catalin Zara (UMB) Divergence Theorem May 5, / 14
40 Example is the part of the paraboloid z = 4 x 2 y 2 above the xy plane, oriented upward; X = ai + bj + ck. Use the Divergence Theorem to compute X d. Problem: The surface does not enclose a region in space; We add the disk D of radius 2 centered at the origin in the plane z = 0; X d = divx dv = 0, D R orients D with the downward normal, hence X d = X d The upward normal to D is k, hence X d = X kd = c d. Therefore X d = X d = c d = c area(d) = 4πc. D D R D Catalin Zara (UMB) Divergence Theorem May 5, / 14
41 Application Catalin Zara (UMB) Divergence Theorem May 5, / 14
42 Application F: total force due to the difference in pressure between the interior of an inflated balloon and the exterior; F = df = pnd = p d. Catalin Zara (UMB) Divergence Theorem May 5, / 14
43 Application F: total force due to the difference in pressure between the interior of an inflated balloon and the exterior; F = df = pnd = p d. For every unit vector u we have ( ) F u = pn d u = pu Nd = D div(pu)dv = 0 because div(pu) = 0 since the vector field X = pu is constant on D. Therefore F u = 0 for every unit vector u; Catalin Zara (UMB) Divergence Theorem May 5, / 14
44 Application F: total force due to the difference in pressure between the interior of an inflated balloon and the exterior; F = df = pnd = p d. For every unit vector u we have ( ) F u = pn d u = pu Nd = D div(pu)dv = 0 because div(pu) = 0 since the vector field X = pu is constant on D. Therefore F u = 0 for every unit vector u; Which implies F = 0. Catalin Zara (UMB) Divergence Theorem May 5, / 14
45 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Catalin Zara (UMB) Divergence Theorem May 5, / 14
46 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Body occupies a region D, exterior boundary ; Unit outward normal field N; Catalin Zara (UMB) Divergence Theorem May 5, / 14
47 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Body occupies a region D, exterior boundary ; Unit outward normal field N; Magnitude of pressure at depth a below the surface is p 0 + ρag, where g is the magnitude of the gravitational acceleration. p0 is the pressure at surface of liquid Catalin Zara (UMB) Divergence Theorem May 5, / 14
48 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Body occupies a region D, exterior boundary ; Unit outward normal field N; Magnitude of pressure at depth a below the surface is p 0 + ρag, where g is the magnitude of the gravitational acceleration. p0 is the pressure at surface of liquid Infinitesimal force acting on is df = (p 0 + ρag)nd, Catalin Zara (UMB) Divergence Theorem May 5, / 14
49 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Body occupies a region D, exterior boundary ; Unit outward normal field N; Magnitude of pressure at depth a below the surface is p 0 + ρag, where g is the magnitude of the gravitational acceleration. p0 is the pressure at surface of liquid Infinitesimal force acting on is df = (p 0 + ρag)nd, The total force is F = df = (p 0 +ρag)nd = ρag N d = ρgz N d Catalin Zara (UMB) Divergence Theorem May 5, / 14
50 Another Application A solid body is submerged into a tank containing a liquid of constant density ρ. What is the buoyant force? Body occupies a region D, exterior boundary ; Unit outward normal field N; Magnitude of pressure at depth a below the surface is p 0 + ρag, where g is the magnitude of the gravitational acceleration. p0 is the pressure at surface of liquid Infinitesimal force acting on is df = (p 0 + ρag)nd, The total force is F = df = (p 0 +ρag)nd = ρag N d = EC: Use the Divergence Theorem to show that F = ρv g k (V : volume of the region enclosed by.) ρgz N d Catalin Zara (UMB) Divergence Theorem May 5, / 14
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