A Review: Operators involving in Curvilinear Orthogonal Coordinates r and Vector Identities and Product Rules r Markus Selmke Student at the Universität Leipzig, IPSP,[email protected] 22.7.2006, updated 15.02.2008 Contents Abstract I 1 The Laplacian in Spherical Coordinates - The Tedious Way 1 2 The Angular Momentum Operators in Spherical Coordinates 4 3 Operators in Arbitrary Orthogonal Cuvilinear Coordinate Systems 5 r Summary of Important Formulas 11 4 Examples using the General Expressions 12 5 Proofs for the Product Rules 16 r 6 Gauss Divergence Theorem variants 18 References 19 Abstract This document provides easily understandable derivations for all important operators applicable to arbitrary orthog- 3 r onal coordinate systems in R (like spherical, cylindrical, elliptic, parabolic, hyperbolic,..). These include the laplacian (of a scalar function and a vector field), the gradient and its absolute value (of a scalar function), the divergence (of a vector field), the curl (of a vector field) and the convective operator (acting on a scalar function and a vector field) (Section 3). For the first two mentioned special coordinate systems the operators are calculated using these expressions along with some remarks on their simplifications in cartesian coordinates (Section 4). Alongside these calculations, some useful derivations are presented, including the volume element and the velocity components. The Laplacian as well as the absolute value of the gradient are also calculated the clumsy way (ab)using the chain-rule (Section 1). Using the same tedious way as in section 1, the angular momentum operators arising in quantum mechanics are derived for spherical coodinates (Section 2) only. Furthermore, all important product rules for the operator, which are utilized throughout the derivations in section 3, are proven using the Levi-Civita-Symbol and the Kronecker-Deltar (Section 5). I 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 1 1 The Laplacian in Spherical Coordinates - The Tedious Way In this section the Laplacian operator acting on a scalar function is derived for the special case of spherical coordinates only. The way presented here and often described in physic books is very tedious and involved, though only the very basic concepts of partial differential calculus are used here. However, the operator may be calculated way easier using the method decribed in (Section 3) and carried out in (Section 4)! Recall the definition of the Laplacian in cartesian coordonates: @ 2 f @ 2 f @ 2 f ∆f := + + (1) @ x 2 @ y 2 @ z 2 Some things we need and the transforms: x = r sinθ cosφ y = r sinθ sinφ z = r cosθ (2) ! z y p 2 2 2 θ = arccos φ = arctan r = x + y + z (3) p 2 2 2 x x + y + z f = f x r,θ ,φ ,y r,θ ,φ ,z r,θ ,φ (4) @ f @ f @ r @ f @ θ @ f @ φ = + + (5) @ x @ r @ x @ θ @ x @ φ @ x @ f @ f @ r @ f @ θ @ f @ φ = + + (6) @ y @ r @ y @ θ @ y @ φ @ y @ f @ f @ r @ f @ θ @ f @ φ = + + (7) @ z @ r @ z @ θ @ z @ φ @ z 1 1 1 1 0 1 arccosx x arctanx arctan (8) ( )0 = p p 0 = ( )0 = 2 = 2 − 1 x 2 2px 1 + x x − 1 + x − p y y y x @ x 2 y 2 z 2 1 2x 0 0 + + arctan = 2 2 arctan = 2 2 = (9) x y y x x x x y @ x 2 p 2 2 2 − + + x + y + z Now we begin: 2 3 2 3 @ f @ f 1 2r sinθ cosφ @ f 1 1 2x @ f r sinθ sinφ 6 z 7 4 5 = + 4 Æ 2 2 − 23=2 5 + 2 2 2 2 @ x @ r 2 r @ θ − r cos θ 2 r @ φ − r sin θ cos φ + sin φ 1 r 2 − @ f @ f 1 @ f sinφ = sinθ cosφ + cosθ cosφ (10) @ r @ θ r − @ φ r sinθ 2 3 2 3 @ f @ f 1 2r sinθ sinφ @ f 1 1 2y @ f r sinθ cosφ 6 z 7 4 5 = + 4 Æ 2 2 − 23=2 5 + 2 2 2 2 @ y @ r 2 r @ θ − r cos θ 2 r @ φ − r sin θ cos φ + sin φ 1 r 2 − @ f @ f 1 @ f cosφ = sinθ sinφ + cosθ sinφ + (11) @ r @ θ r @ φ r sinθ 2 1 3 @ f @ f 1 2r cosθ @ f 1 r r cosθ 2r 2r cosθ @ f = + 4 − 2 5 + [0] @ z @ r 2 r @ θ − sinθ r @ φ @ f @ f sinθ = cosθ (12) @ r − @ θ r 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 2 @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (13) @ x 2 @ x @ x @ r @ x @ x @ θ @ x @ x @ φ @ x @ x @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (14) @ y 2 @ y @ y @ r @ y @ x @ θ @ x @ y @ φ @ y @ y @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (15) @ z 2 @ z @ z @ r @ z @ z @ θ @ z @ z @ φ @ z @ z Plugging equations (10),(11),(12) into (13),(14),(15) we get: @ 2 f @ @ f @ f 1 @ f sinφ 2 = sinθ cosφ + cosθ cosφ sinθ cosφ + ... @ x @ r @ r @ θ r − @ φ r sinθ @ @ f @ f 1 @ f sinφ cosθ cosφ sinθ cosφ + cosθ cosφ + ... @ θ @ r @ θ r − @ φ r sinθ r @ @ f @ f 1 @ f sinφ sinφ sinθ cosφ + cosθ cosφ − @ φ @ r @ θ r − @ φ r sinθ r sinθ @ 2 f 1 @ f sinφ @ f = sinθ cosφ 2 cosθ cosφ 2 + 2 sinθ cosφ + ... @ r − r @ θ r sinθ @ φ @ f 1 @ f 1 @ 2 f cosθ sinφ @ f cosθ cosφ cosθ cosφ sinθ cosφ + cosθ cosφ 2 + 2 + ... @ r − r @ θ r @ θ r sin θ @ φ r @ f 1 @ f cosφ @ f sinφ @ 2 f sinφ sinθ sinφ cosθ sinφ 2 − − @ r − r @ θ − r sinθ @ φ − r sinθ @ φ sinθ 2 2 2 2 @ f sinθ cosθ cos φ @ f sinφ cosφ @ f = sin θ cos φ 2 2 + 2 + ... @ r − r @ θ r @ φ cos2 θ cos2 φ @ f sinθ cosθ cos2 φ @ f cos2 θ cos2 φ @ 2 f cos2 θ cosφ sinφ @ f ... 2 + 2 2 + 2 2 + r @ r − r @ θ r @ θ r sin θ @ φ sin2 φ @ f cosθ sin2 φ @ f cosφ sinφ @ f sin2 φ @ 2 f + + + (16) r @ r r 2 sinθ @ θ r 2 sin2 θ @ φ r 2 sin2 θ @ φ2 @ 2 f @ @ f @ f 1 @ f cosφ = sinθ sinφ + cosθ sinφ + sinθ sinφ + ... @ y 2 @ r @ r @ θ r @ φ r sinθ @ @ f @ f 1 @ f cosφ cosθ sinφ sinθ sinφ + cosθ sinφ + + ... @ θ @ r @ θ r @ φ r sinθ r @ @ f @ f 1 @ f cosφ cosφ sinθ sinφ + cosθ sinφ + @ φ @ r @ θ r @ φ r sinθ r sinθ 2 2 2 2 @ f cosθ sinθ sin φ @ f cosφ sinφ @ f = sin θ sin φ 2 2 2 + ... @ r − r @ θ − r @ φ cos2 θ sin2 φ @ f sinθ cosθ sin2 φ @ f cos2 θ sin2 φ @ 2 f cosφ cos2 θ sinφ @ f ... 2 + 2 2 2 2 + r @ r − r @ θ r @ θ − r sin θ @ φ cos2 φ @ f cos2 φ cosθ @ f sinφ cosφ @ f cos2 φ @ 2 f (17) + 2 2 + 2 2 2 r @ r r sinθ @ θ − r sin θ @ φ r sin θ @ φ @ 2 f @ @ f @ f sinθ 2 = cosθ cosθ + ... @ z @ r @ r − @ θ r @ @ f @ f sinθ sinθ cosθ − @ θ @ r − @ θ r r 2 2 @ f sinθ cosθ @ f = cos θ + + ... @ r 2 r 2 @ θ sin2 θ @ f cosθ sinθ @ f sin2 θ @ 2 f + + (18) r @ r r 2 @ θ r 2 @ θ 2 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 3 Using equation (16),(17),(18) we obtain the Laplacian: @ 2 f @ 2 f @ 2 f ∆f = + + @ x 2 @ y 2 @ z 2 @ 2 f sinθ cosθ @ f cos2 θ @ f sinθ cosθ @ f cos2 θ @ 2 f 1 @ f cosθ @ f = 2 2 + 2 + 2 2 + + 2 + ... @ r − r @ θ r @ r − r @ θ r @ θ r @ r r sinθ @ θ 1 @ 2 f sinθ cosθ @ f sin2 θ @ f cosθ sinθ @ f sin2 θ @ 2 f + + + + r 2 sin2 θ @ φ2 r 2 @ θ r @ r r 2 @ θ r 2 @ θ 2 @ 2 f 2 @ f 1 @ 2 f 1 @ 2 f cosθ @ f = + + + + @ r 2 r @ r r 2 @ θ 2 r 2 sin2 θ @ φ2 r 2 sinθ @ θ @ 2 f 2 @ f 1 @ 2 f 1 @ 2 f @ f = + + + sinθ + cosθ @ r 2 r @ r r 2 sin2 θ @ φ2 r 2 sinθ @ θ 2 @ θ @ 2 f 2 @ f 1 @ 2 f 1 @ @ f = + + + sinθ @ r 2 r @ r r 2 sin2 θ @ φ2 r 2 sinθ @ θ @ θ @ 2 2 @ 1 @ @ 1 @ 2 ∆ = + + sinθ + (19) @ r 2 r @ r r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 We also note the following relations: 2 2 1 @ 2 @ f 1 @ f 2 @ f 2 @ f @ f r = 2r + r = + (20) r 2 @ r @ r r 2 @ r @ r 2 r @ r @ r 2 2 2 2 1 @ 1 @ @ f 1 @ f @ f @ f 2 @ f @ f r f (r ) = f (r ) + r = + + r = + (21) r @ r 2 r @ r @ r r @ r @ r @ r 2 r @ r @ r 2 And hence obtain the Laplacian operator in spherical coordinates: 2 1 @ 2 @ 1 @ @ 1 @ ∆ = r + sinθ + (22) r 2 @ r @ r r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 Or equivalently: 1 @ 2 1 @ @ 1 @ 2 ∆ = r + sinθ + (23) r @ r 2 r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 The absolute value of the gradient in cartesian coordinates is: 2 2 2 2 @ f @ f @ f f = + + (24) jr j @ x @ y @ z 2 2 2 2 Using equations (10),(11),(12) and x + y + z = x + y + z + 2x y + 2xz + 2y z we get: 2 2 2 @ f @ f 1 @ f sinφ @ f @ f 1 @ f cosφ f = sinθ cosφ + cosθ cosφ + sinθ sinφ + cosθ sinφ + + ..