Laplace s Equation in Spherical Polar Coördinates

Transcript

1 Laplace s Equation in Spheical Pola Coödinates C. W. David Dated: Januay 3, 001 We stat with the pimitive definitions I. x = sin θ cos φ y = sin θ sin φ z = cos θ thei inveses = x y z θ = cos 1 z = z cos1 x y z attempt to wite using the chain ule x = x y = z = θ,φ y θ,φ z θ,φ The needed above patial deivatives ae: we have as a stating point fo doing the θ tems, φ = tan 1 y x θ x θ,φ θ y θ,φ θ z θ,φ φ x φ,θ φ y φ,θ φ z φ,θ = sin θ cos φ 1 x = sin θ sin φ y = cos θ 3 z d cos θ = sin θdθ = dz z 1 xdx ydy zdz

2 so that, fo example which is so that but, fo the z-equation, we have which is so one has Next, we have as an example so o which leads to sin θdθ = z x dx sin θdθ = cos θ sin θdθ = sin θdθ = sin θ cos φdx θ cos θ cos φ = x θ cos θ sin φ = y sin θdθ = dz z 1 zdz 1 z 3 dz = z 3 dz 1 z 3 dz = sin θ 3 dz θ = sin θ z tan φ = sin φ cos φ = tan1 y x 1 sin φ cos dφ = dy φ x y x dx 1 cos dφ = dy φ x y x dx φ = cos φ y sin θ φ = sin φ x sin θ

3 φ = 0 9 z 3 Given these esults above we wite z = cos θ sin θ θ 10 y = sin θ sin φ x = sin θ cos φ cos θ sin φ cos θ cos φ cos φ θ sin θ φ θ sin φ sin θ φ 11 1 Fom Equation 10 we fom z = cos θ while fom Equation 11 we obtain y = sin θ sin φ cos θ sin φ [ cos θ sin θ θ [ sin θ sin θ sin φ cos θ sin φ [sin θ sin φ cos θ sin φ θ cos φ [sin θ sin φ cos θ sin φ sin θ φ fom Equation 1 we obtain [ sin θ cos φ x = sin θ cos φ cos θ cos φ cos θ cos φ [sin θ cos φ cos θ cos φ θ sin φ [sin θ cos φ cos θ cos φ sin θ φ cos θ sin θ θ θ θ cos φ sin θ φ θ cos φ sin θ θ cos φ sin θ θ sin φ sin θ θ sin φ sin θ θ sin φ sin θ φ φ φ φ φ Exping, we have while fo the y-equation we have z = cos θ cos θ sin θ sin θ cos θ θ θ sin θ sin θ cos θ sin θ cos θ sin θ θ θ θ 16 y = sin θ sin φ 17 [ cos θ sin φ cos θ sin φ sin θ sin φ θ 18 θ

4 finally [ sin θ sin φ cos φ cos φ sin θ φ sin θ φ cos θ sin φ [cos θ sin φ sin θ sin φ θ [ cos θ sin φ sin θ sin φ cos θ sin φ θ θ [ cos θ sin φ cos φ cos θ cos φ sin θ φ sin θ φθ cos φ [sin θ cos φ sin θ sin φ sin θ φ [ cos φ cos θ cos φ cos θ sin φ sin θ θ θφ [ cos φ sin φ cos φ cos φ sin θ sin θ φ sin θ φ = sin θ cos φ sin θ cos φ x [ cos θ cos φ cos θ cos φ sin θ cos φ θ θ [ sin φ sin φ sin θ cos φ sin θ φ sin θ φ cos θ cos φ [cos θ cos φ sin θ cos φ θ [ cos θ cos φ sin θ cos φ cos θ cos φ θ θ [ cos θ cos φ sin φ sin φ sin θ φ sin θ φθ sin φ [sin θ sin φ sin θ cos φ sin θ φ [ sin φ cos θ sin φ cos θ cos φ sin θ θ θφ [ sin φ cos φ sin φ sin θ sin θ φ sin θ φ Now, one by one, we exp completely each of these thee tems. We have z = cos θ 34 cos θ sin θ 35 θ sin θ cos θ 36 θ sin θ 37 sin θ cos θ 38 θ sin θ cos θ 39 θ sin θ θ 40

5 5, fo the y-equation: y = sin θ sin φ 41 sin θ cos θ sin φ 18 4 θ cos θ sin θ sin φ 43 θ sin φ cos φ φ cos φ sin φ 45 φ cos θ sin φ 0 46 cos θ sin θ sin φ 47 θ sin θ cos θ sin φ 48 θ cos θ sin φ 1 θ 49 cos θ cos φ sin φ sin 50 θ φ cos θ cos φ sin φ 51 sin θ φθ cos φ 5 cos φ sin φ 53 φ cos φ cos θ 54 sin θ θ cos θ cos φ sin φ 4 55 sin θ θφ cos φ sin φ 5 sin 56 θ φ cos φ sin θ φ 57 finally, fo the x-equation, we have x = sin θ cos φ 58 sin θ cos θ cos φ 6 59 θ sin θ cos θ cos φ 6 60 θ cos φ sin φ 61 φ sin φ cos φ 6 φ

6 cos θ cos φ 7 63 sin θ cos θ cos φ 64 θ sin θ cos θ cos φ 7 65 θ cos θ cos φ θ 66 cos θ cos φ cos φ sin φ 8 67 sin θ φ sin φ cos φ cos θ 68 sin θ φθ sin φ 9 69 sin φ cos φ 70 φ cos θ sin φ sin θ θ cos θ sin φ cos φ 7 sin θ θφ sin φ cos φ 3 sin 73 θ φ sin φ sin θ φ 74 Gatheing tems as coefficients of patial deivatives, we obtain fom Equations 34, fom Equations 35, 38, 4, 48, 54, 59, 65, 71 θ cos θ sin θ sin φ sin θ cos φ cos θ sin θ sin θ cos θ sin θ cos θ sin φ sin θ cos θ sin φ cos φ cos θ sin θ while we obtain fom Equations 40, 49, 66: sin θ cos θ cos φ sin θ cos θ cos φ cos θ sin φ sin θ cos θ sin θ θ 75 θ Fom Equations 37, 46, 5, 63, 69, sin θ cos θ sin φ sin θ cos θ sin φ cos φ cos θ cos φ sin φ Fom Equations 44, 50, 56, 61, we obtain φ sin φ cos φ cos θ cos φ sin φ sin cos θ cos φ sin φ θ sin θ cos θ cos φ 1 θ cos φ sin φ cos θ cos φ cos θ cos φ sin φ sin φ cos φ sin θ sin 0 78 θ

7 Fom Equations we obtain φ cos φ sin θ sin φ sin θ 1 sin θ φ 79 The mixed deivatives yield, fist, fom Equations 45, 53, 6, 70 leading to cos φ sin φ cos φ sin φ sin φ cos φ sin φ cos φ 0 80 φ Fom Equations 36, 39, 47, 43 64, 60 Fom Equations θ sin θ cos θ cos φ sin θ cos θ sin θ cos θ cos θ sin θ sin φ cos θ sin θ sin φ Gatheing togethe the non-vanishing tems, we obtain sin θ cos θ cos φ 0 81 cos θ cos φ sin φ cos φ sin φ φθ sin θ sin θ sin φ cos φ cos θ cos θ sin φ cos φ sin θ 0 8 sin θ 1 θ cos θ sin θ θ 1 sin θ which is one of the two classic foms fo. The othe is 1 1 sin sin θ sin θ θ θ θ φ φ 7 II. MAPLE EQUIVALENT Hee is a set of Maple instuctions which will get you the same esult: estat; f:=g,theta,phi; tx := sintheta*cosphi*difff,costheta*cosphi/*difff,theta -sinphi/*sintheta*difff,phi; tx:=exp sintheta*cosphi*difftx,costheta*cosphi/*difftx,theta -sinphi/*sintheta*difftx,phi; ty := sintheta*sinphi*difff,costheta*sinphi/*difff,theta cosphi/*sintheta*difff,phi; ty:=expsintheta*sinphi*diffty,costheta*sinphi/ *diffty,thetacosphi/*sintheta*diffty,phi; tz := costheta*difff, -sintheta/*difff,theta; tz := expcostheta*difftz,-sintheta/*difftz,theta; del := txtytz: del := algsubs costheta^=1-sintheta^, del : del := expalgsubs cosphi^=1-sinphi^, del ;