Curl, Divergence, and Gradient in Cylindrical and Spherical Coordinate Systems

Transcription

1 APPENDIX B Cul, Divegence, and Gadient in Cylindical and Spheical Coodinate Systems In Sections 3., 3.4, and 6., we intoduced the cul, divegence, and gadient, espectively, and deived the expessions o them in the Catesian coodinate system. In this appendix, we shall deive the coesponding expessions in the cylindical and spheical coodinate systems. Consideing ist the cylindical coodinate system, we ecall om Appendix A that the ininitesimal box deined by the thee othogonal suaces intesecting at point P(, u, ) and the thee othogonal suaces intesecting at point Q( + d, + d, z + dz) is as shown in Figue B.. c d h Q( d, d, z dz) dz b e a P(,, z) d d g ( d) d FIGURE B. Ininitesimal box omed by incementing the coodinates in the cylindical coodinate system. 42

2 Appendix B 42 Fom the basic deinition o the cul o a vecto intoduced in Section 3.3 and given by : A c A C A dl d a S: S n max (B.) we ind the components o : A as ollows with the aid o Figue B.: Aabcda ( : A) A dl d: aea abcd dz: d: dz: e [A ] (, z) d + [A z ] (, + d) dz - [A ] (, z + dz) d - [A z ] (, ) dz d dz [A z ] (, + d) - [A z ] (, ) d: d [A ] (, z) - [A ] (, z + dz) dz: dz = A z - A z (B.2a) Aadea A dl ( : A) dz: aea ade d: dz: d: e [A z] (, ) dz + [A ] (, z + dz) d - [A z ] ( + d, ) dz - [A ] (, z) d d dz [A ] (, z + dz) - [A ] (, z) dz: dz [A z ] (, ) - [A z ] ( + d, ) d: d = A z - A z (B.2b) Aagba A dl ( : A) z d: aea agb d: d: d: [A ] ( + d, z) - [A ] (, z) d: d = e [A ] (, z) d + [A ] ( + d, z) ( + d) d - [A ] ( + d, z) d - [A ] (, z) d (A ) - A d d [A ] (, z) - [A ] ( + d, z) d: d (B.2c)

3 422 Appendix B Cul, Divegence, and Gadient Combining (B.2a), (B.2b), and (B.2c), we obtain the expession o the cul o a vecto in cylindical coodinates as : A = c A z - A z da + c A z - A z da + c (A ) - A d a z a = 5 A a A a z 5 z A z (B.3) To ind the expession o the divegence, we make use o the basic deinition o the divegence o a vecto, intoduced in Section 3.6 and given by AS A ds A v: v (B.4) Evaluating the ight side o (B.4) o the box o Figue B., we obtain e [A ] + d ( + d) d dz - [A ] d dz + [A ] + d d dz - [A ] d dz + [A z ] z + dz d d - [A z ] z d d A d: d d dz d: dz: [A ] + d - d: d [A ] [A ] + d - [A ] d: d = [A z ] z + dz - [A z ] z dz: dz (A ) + A + A z z (B.5) To obtain the expession o the gadient o a scala, we ecall om Appendix A that in cylindical coodinates, and hence d = d + d + z dz = a a + = dl dl = d a + d a + dz a z a + z a zb (d a + d a + dz a z ) (B.6) (B.7)

4 Appendix B 423 Thus, = a + (B.8) Tuning now to the spheical coodinate system, we ecall om Appendix A that the ininitesimal box deined by the thee othogonal suaces intesecting at P(, u, ) and the thee othogonal suaces intesecting at Q( + d, u + du, + d) is as shown in Figue B.2. Fom the basic deinition o the cul o a vecto given by (B.), we then ind the components o : A as ollows with the aid o Figue B.2: Aabcda ( : A) A dl du: aea abcd d: a + e [A u] (, ) du + [A ] (, u + du) sin (u + du) d - [A u ] (, + d) du - [A ] (, u) sin u d du: 2 sin u du d d: [A sin u] (, u + du) - [A sin u] (, u) du: sin u du [A u ] (, ) - [A u ] (, + d) d: sin u d z a z = sin u u (A sin u) - A u sin u (B.9a) e d ( d) sin u d sin u d a P(, u, ) d c ( d) du h Q( d, u du, d) du sin (u du) d b g FIGURE B.2 Ininitesimal box omed by incementing the coodinates in the spheical coodinate system.

5 424 Appendix B Cul, Divegence, and Gadient Aadea A dl ( : A) u d: aea ade d: e [A ] (, u) sin u d + [A ] (u, + d) d - [A ] ( + d, u) ( + d) sin u d - [A ] (u, ) d d: sin u d d d: [A ] (u, + d) - [A ] (u, ) d: sin u d = [A ] (, u) - [A ] ( + d, u) d: d A sin u - (A ) (B.9b) Aagba A dl ( : A) d: aea agb du: e [A ] (u, ) d + [A u ] ( + d, ) ( + d) du - [A ] (u + du, ) d - [A u ] (, ) du d: d du du: [A u ] ( + d, ) - [A u ] (, ) d: d = [A ] (u, ) d - [A ] (u + du, ) d du: du (A u) - A u (B.9c) Combining (B.9a), (B.9b), and (B.9c), we obtain the expession o the cul o a vecto in spheical coodinates as : A = sin u c u (A sin u) - + A u da c A sin u - (A ) da u + c (A u) - A u da = 5 a 2 sin u A a u sin u u A u a sin ua 5 (B.)

6 Appendix B 425 To ind the expession o the divegence, we make use o the basic deinition o the divegence o a vecto given by (B.4) and by evaluating its ight side o the box o Figue B.2, we obtain A d: du: d: [A ] + d ( + d) 2 sin u du d - [A ] 2 sin u du d c + [A u ] u + du sin (u + du) d d - [A u ] u sin u d d + [A ] + d d du - [A ] d du [ 2 A ] + d - [ 2 A ] d: 2 d [A ] + d - [A ] d: sin u d 2 sin u d du d [A u sin u] u + du - [A u sin u] u du: sin u du s = 2 (2 A ) + sin u u (A u sin u) + A sin u (B.) To obtain the expession o the gadient o a scala, we ecall om Appendix A that in spheical coodinates, dl = d a + du a u + sin u d a (B.2) and hence d = d + u du + d Thus, = a = a + dl u a u + sin u a b (d a + du a u + sin u d a ) (B.3) = a + u a u + sin u a (B.4) REVIEW QUESTIONS B.. B.2. B.3. Biely discuss the basic deinition o the cul o a vecto. Justiy the application o the basic deinition o the cul o a vecto to detemine sepaately the individual components o the cul. How would you genealize the intepetations o the components o the cul o a vecto in tems o the lateal deivatives involving the components o the vecto to hold in cylindical and spheical coodinate systems?

7 426 Appendix B Cul, Divegence, and Gadient B.4. B.5. B.6. Biely discuss the basic deinition o the divegence o a vecto. How would you genealize the intepetation o the divegence o a vecto in tems o the longitudinal deivatives involving the components o the vecto to hold in cylindical and spheical coodinate systems? Povide geneal intepetation o the components o the gadient o a scala. PROBLEMS B.. B.2. B.3. B.4. B.5. B.6. Find the cul and the divegence o each o the ollowing vectos in cylindical coodinates: (a) cos a - sin a ; (b) ; (c). a a Find the gadient o each o the ollowing scala unctions in cylindical coodinates: (a) ; (b) sin. cos Find the expansion o the Laplacian, that is, the divegence o the gadient, o a scala in cylindical coodinates. Find the cul and the divegence o each o the ollowing vectos in spheical coodinates: (a) 2 a + sin u a u ; (b) ; (c). e - a 2 a u Find the gadient o each o the ollowing scala unctions in spheical coodinates: sin u (a) ; (b) cos u. Find the expansion o the Laplacian, that is, the divegence o the gadient, o a scala in spheical coodinates.