18 02A Topic 47 Line integrals in 3D Conservative vector fields Author Jeremy Orloff Read SN V11 V12 Conservative 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 dr ds ds v As before velocity v T where v speed and T dt dt dt v Line integrals Take F M i N j P k and C a curve Z Z M dx N dy P dz F dr Then the work integral C C 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 t2 j t3 k answer a x 1 t y 1 3t z 1 7t t goes from 0 to 1 Z Z W M dx N dy P dz y dx z dy x dz C C Z 1 Z 1 31 53 1 3t dt 1 7t 3dt 1 t 7dt 11 31t dt 11 2 2 0 0 answer b x t y t2 z t3 t goes from 1 to 2 Z 2 Z t3 3t4 2t5 y dx z dy x dz t2 dt t2 2t dt t 3t2 dt W 3 4 5 1 C If F hM N P i then i j k D E curlF x y z hM N M i x y z M N P Curl 2 1 1559 60 curlF F 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 and derivatives 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 independence Theorem fundamental theorem for line integrals Z P2 If F f then F dr f P2 f P2 That is the integral is path independent P1 I Equivalently F dr 0 for any closed path C In this case we call F conservative just C as in two dimensions 1 18 02A topic 47 2 Theorem 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 some potential function f B If D is a simply connected volume defined below F is continuously differentiable in D and curlF 0 then in D F f for some potential function f Simply connected volumes in space This 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 shrunk to a point with out leaving D Examples Simply connected R3 unit ball R3 0 R3 minus unit ball R2 minus a line segment Not simply connected R3 minus the z axis a solid torus R3 minus a circle Example For what value s of a is F yi x ayz j y 2 1 k conservative For this value find the potential function answer Write F hM N P i where M y N x ayz P y 2 1 i j k i j k curlF F x y z x y z M N P y x ayz y 2 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 R Method 1 Compute the path integral f x1 y1 z1 C1 C2 C3 F dr C Parametrizing C1 x x y 0 z 0 x from 0 to x1 C2 x x1 y y z 0 y from 0 to y1 C1 x x1 y y1 z z z from 0 to z1 Z Z Z f x1 y1 z1 M dx N dy P dz C Z Cx11 Z yC12 Z Cz31 0 dx x1 dy y12 1 dz C 0 0 2 x1 y1 y1 1 z1 f x y z xy y 2 z z 0 C C 18 02A topic 47 3 Method 2 fx M y f xy g y z fy N x gy x 2yz g y 2 z h z f xy y 2 z h z fz P y 2 h0 z y 2 1 h z C f xy y 2 z z C Note method 1 is well suited for numerical methods while method 2 relies on symbolic manipulation Example F h y r2 x r2 0i has curlF 0 but it is not defined when r 0 i e on the 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 Py then M dx N dy P dz is exact B B holds if we restrict our attention to a simply connected volume D the F is conservative and Example Gravitation The gravitational field F hx y zi 3 F 1 Note in this case curlF 0 and F is defined everywhere but the origin Since this region is simply connected Theorem B guarantees that there is a potential function End of topic 47 notes
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