18.02A Topic 47: Line integrals in 3D. Conservative vector fields.Author: Jeremy OrloffRead: SN: V11, V12Conservative vector fields in 3D is just like the 2D case.Curves in space: x = x(t), y = y(t), z = z(t).Or r(t) = x(t)i + y(t)j + z(t)k = hx, y, zi.As before, velocity = v =drdt=dsdtT, wheredsdt= |v| = speed and T =v|v|.Line integrals: Take F = Mi + Nj + P k and C a curve.Then the work integral =ZCF · dr =ZCM dx + N dy + P dz.Example: Find the work done by the force F = yi + zj + xk in moving a particle from(1,1,1) to (2,4,8) a) along a line b) along the twisted cubic r = ti + t2j + t3k.answer: a) x = 1 + t, y = 1 + 3t, z = 1 + 7t; t goes from 0 to 1.⇒ W =ZCM dx + N dy + P dz =ZCy dx + z dy + x dz=Z10(1 + 3t) dt + (1 + 7t) 3dt + (1 + t) 7dt =Z1011 + 31t dt = 11 +312=532.answer: b) x = t, y = t2, z = t3, t goes from 1 to 2.⇒ W =ZCy dx + z dy + x dz =Z21t2dt + t2· 2t dt + t · 3t2dt =t33+3t44+2t5521=155960.Curl: curlF = ∇ × F. If F = hM, N, P i thencurlF =D∂∂x,∂∂y,∂∂zE×hM, N, Mi =i j k∂∂x∂∂y∂∂zM N P= (Py−Nz)i+(Mz−Px)j+(Nx−My)k.We will compute this in some examples below.Notes: 1. The x-component of curlF involves only the j and k components of F andderivatives in y and z, etc.2. For a vector field in the plane we had P = 0 and M and N are functions of x and y⇒ curlF = (Nx− My)k. This is consistent with our previous definition of curl.3. curl gradf = ∇ × ∇f = 0. (This is easy to compute. It’s easier to note ∇ × ∇ = 0.)Gradient fields and path independenceTheorem (fundamental theorem for line integrals)If F = ∇f thenZP2P1F · dr = f(P2) − f(P2). That is, the integral is path independent.Equivalently,ICF · dr = 0 for any closed path C. In this case we call F conservative justas in two dimensions.118.02A topic 47 2Theorem: (A) If F = ∇f then curlF = 0.(B) If F is continuously differentiable on all x, y, z and curlF = 0 then F = ∇f for somepotential function f.(B’) If D is a simply connected volume (defined below), F is continuously differentiable inD and curlF = 0 then in D, F = ∇f for some potential function f.Simply connected volumes in spaceThis is a little harder to define than for the plane because there is a lot more room in space.We say a volume D is simply connected if every closed loop in D can be continuosly shrunkto a point with out leaving D.Examples:Simply connected: R3, unit ball, R3− {0}, R3minus unit ball, R2minus a linesegment.Not simply connected: R3minus the z-axis, a solid torus R3minus a circle.Example: For what value(s) of a is F = yi + (x + ayz)j + (y2+ 1)k conservative?For this value, find the potential function.answer: Write F = hM, N, P i, where M = y, N = x + ayz, P = y2+ 1.curlF = ∇ × F =i j k∂∂x∂∂y∂∂zM N P=i j k∂∂x∂∂y∂∂zy x + ayz y2+ 1= i(Py− Nz) + j(Mz− Px) + k(Nx− My) = (2y − ay)i + 0 j + 0 k.Thus, curlF = 0 ⇔ a = 2 ⇒ F is conservative ⇔ a = 2.Now, take a = 2 and find a potential function f.Method 1: Compute the path integral. f(x1, y1, z1) =RC1+C2+C3F · dr + CParametrizing: C1: x = x, y = 0, z = 0 x from 0 to x1C2: x = x1, y = y, z = 0 y from 0 to y1C1: x = x1, y = y1, z = z z from 0 to z1.f(x1, y1, z1) =ZC1M dx +ZC2N dy +ZC3P dz + C=Zx100 dx +Zy10x1dy +Zz10y21+ 1 dz + C= x1y1+ (y21+ 1)z1+ C⇒ f(x, y, z) = xy + y2z + z + C.18.02A topic 47 3Method 2: fx= M = y ⇒ f = xy + g(y, z).fy= N ⇒ x + gy= x + 2yz ⇒ g = y2z + h(z) ⇒ f = xy + y2z + h(z).fz= P ⇒ y2+ h0(z) = y2+ 1 ⇒ h = z + C. ⇒ f = xy + y2z + z + C.Note, method 1 is well-suited for numerical methods while method 2 relies on symbolicmanipulation.Example: F = h−y/r2, x/r2, 0i has curlF = 0, but it is not defined when r = 0, i.e, onthe z-axis. So, the theorem doesn’t apply and, in fact, we know F is not conservative.Exact differentials: As in 2D we can rephrase our theorems in terms of differentials.If the differential M dx + N dy + P dz = df for some function f we say it is exact.Theorem: (A) M dx + N dy + P dz = df ⇒ Nz= Py, Mz= Px, Nx= My.(B) If M, N, P are continuously differentiable on all x, y, z and Nz= Py, Mz= Px, Nx= Pythen M dx + N dy + P dz is exact.(B’) (B) holds if we restrict our attention to a simply connected volume D.Example: Gravitation. The gravitational field F = −hx,y,ziρ3the F is conservative andF = ∇(1/ρ).Note in this case, curlF = 0 and F is defined everywhere but the origin. Since this regionis simply connected Theorem B’ guarantees that there is a potential function.End of topic 47
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