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- 1. - - (invariant ) (Cartesian coordinate) (spherical coordinate) (cylindrical coordinate) , (x,y,z) (r,ϕ,z ) (r , θ , ϕ ) . z DEDODA university.arabsbook.com P y x -1 - 1
- 2. (Cartesian) P , (x,y,z) (r ,θ ,ϕ ) (spherical coordinate) -2 - ϕ θ (displacement) . (r,ϕ,z ) ϕ 2
- 3. -3 - . (Coordinate) (component) r -: A = 5i + 3 j − 4k (unit vector) (i , j , k ) -: i .i = 1 j.j = 1 k .k = 1 i × i = 0 j× j = 0 k × k = 0 i .j = 0 i .k = 0 j.k = 0 i × j= k j× k = i k × i = j 3
- 4. (unit vectors) (Cartesian coordinate) (in general ) : (a1 , a 2 , a3 ) a1.a1 = 1 a2 .a2 = 1 a3.a3 = 1 a1 × a1 = 0 a2 × a2 = 0 a3 × a3 = 0 a1.a2 = 0 a1.a3 = 0 a2 .a3 = 0 a1 × a2 = a3 a2 × a3 =a1 a3 × a2 = a1 (orthogonal unit curvilinear) . (Orthogonal curvilinear coordinate) (Cartesian coordinate) . -: r l = x i +y j +z k ...........1-a r d l = dx i +dy j +dz k ...........(1-b) r (distance) (dl) d l2 = dx 2 +dy 2 + dz 2 (2) 4
- 5. z y x x = F1 (q1, q2 , q3 ) (4-a) y=F2 (q1, q2 , q3 ) (4-b) z=F3 (q1, q2 , q3 ) (4-c) (q 1 , q 2 , q 3 ) x (q 1 , q 2 , q 3 ) y (q 1 , q 2 , q 3 ) z (certain value) qs (2) (1-b) (1-a) -: z y x ∂x ∂x ∂x dx = dq1 + dq 2 + dq 3 (5-a) ∂q1 ∂q 2 ∂q 3 ∂y ∂y ∂y dy = dq1 + dq 2 + dq 3 (5-b) ∂q1 ∂q 2 ∂q 3 ∂z ∂z ∂z dz = dq1 + dq 2 + dq 3 (5-c) ∂q1 ∂q 2 ∂q 3 -: 3 ∂x dx = ∑i =1 ∂q i dq i (6 -a ) 3 ∂y dy = ∑i =1 ∂q i dq i (6 -b ) 3 ∂z dz = ∑ i =1 ∂q i dq i (6 -c ) 5
- 6. 3 ∂x ∂x dx 2 = ∑ dq i dq j (7-a) i =1 ∂q i ∂ q j 3 ∂y ∂y dy 2 = ∑ dq i dq j (7-b) i =1 ∂q i ∂ q j 3 ∂z ∂z dz 2 = ∑ dq i dq j (7-c) i =1 ∂ q i ∂q j (2) (7-c) (7-b) (7-a) -: (2) 3 ∂x ∂x ∂y ∂y ∂z ∂z dl = ∑ 2 + + dqi dq j (8) i , j =1 ∂qi ∂q j ∂qi ∂q j ∂qi ∂q j ∂x ∂ x ∂y ∂y ∂z ∂ z + + = hij ∂q ∂q ∂q i ∂q j ∂q i ∂q j i j (8) (metric coefficients) (hij ) -: 3 dl = 2 ∑ h dq dq i , j =1 ij i j (9) Kronecker delta 6
- 7. hii = hi2 ,i=j hij = hij δij = h ij = 0 ,i ≠ j (9) 3 d l2 = ∑hi2dqi2 (10) i d l2 = d l12 +d l22 +d l32 (metric coefficients) ∂x 2 ∂y 2 ∂z 2 h = 2 + + (11) ∂q i ∂q i ∂q i i (10) d l 1 = h1dq1 (12-a) d l 2 = h 2dq 2 (12-b) d l 3 = h3dq 3 (12-c) (Curvilinear coordinate) r d l = hdq1 a1+h2dq2 a2 +hdq3 a3 1 3 (13) r ∂l ∂l ∂l dl= dq1 + dq2 + dq3 (14) ∂q1 ∂q 2 ∂q3 7
- 8. (14) (13) 1 ∂l = a1 ( 1 5 -a ) h1 ∂q 1 1 ∂l = a2 (1 5 -b ) h2 ∂q 2 1 ∂l =a (1 5 -c ) h3 ∂q 3 3 1 ∂l ai = (16) hi ∂q i (metric coefficients) (11) (spherical coordinate) x = r s in θ c o s ϕ y = r sin θ sin ϕ z = r cosθ q = (q r , q θ , q ϕ ) (11) 8
- 9. ∂x ∂y ∂z 2 2 2 h r2 = + + ∂r ∂r ∂r h r = (sin θ cos ϕ ) 2 2 + (sin θ sin ϕ ) 2 + (co s θ ) 2 h r2 = sin 2 θ co s 2 ϕ + sin 2 θ sin 2 ϕ + co s 2 θ h r2 = sin 2 θ (co s 2 ϕ + sin 2 ϕ ) + co s 2 θ = 1 ∴ hr = 1 (17 ) ∂x ∂y ∂z 2 2 2 h = θ 2 + + ∂θ ∂θ ∂θ hθ2 = r 2 cos 2 θ cos 2 ϕ + r 2 cos 2 θ sin 2 ϕ + r 2 sin 2 θ hθ2 = r 2 →∴ hθ = r (18) 2 2 2 ∂x ∂y ∂z h = 2 ϕ + + ∂ϕ ∂ϕ ∂ϕ hϕ2 = r 2 sin 2 θ sin 2 ϕ + r 2 sin 2 θ cos2 ϕ hϕ2 = r 2 sin 2 θ →∴ hϕ = r sin θ (19) (cylindrical coordinate) x = r cosϕ y = r s in ϕ z = z q = q r ,q ϕ ,q z (11) 9
- 10. ∂x ∂y ∂z 2 2 2 h r2 = + + ∂r ∂r ∂r h r = cos2 2 ϕ + sin 2 ϕ = 1 ∴ h r =1 (2 0 ) 2 2 2 ∂x ∂y ∂z h = ϕ 2 + + ∂ϕ ∂ϕ ∂ϕ h ϕ2 = r 2 sin 2 ϕ + r 2 cos 2 ϕ h ϕ2 = r 2 → ∴ h ϕ = r (21) ∂x ∂y ∂z 2 2 2 h = 2 + + ∂z ∂z ∂z z hz2 = 1 →∴ hz = 1 (22) curvilinear Cartesian spherical cylindrical q1 x r r q y θ ϕ 2 q3 z ϕ z h1 1 1 1 h2 1 r r h3 1 r sinθ 1 a1 i r0 r0 a2 j θ ϕ a3 k ϕ k 10
- 11. -1 - Gradient Curl Laplacian Divergence (Orthogonal curvilinear coordinate) . Gradient r ∂φ ∂φ ∂φ ∇φ = a1 + a2 + a3 (23) ∂l1 ∂l2 ∂l3 (23) (12) ∂l r ∂φ ∂φ ∂φ ∇φ = a1 + a2 + a3 (24) h1∂q1 h2∂q2 h3∂q3 Divergence 11
- 12. (Gauss's or Divergencetheorem) r r r r Ñ ∫ S F .da = ∫ ∇.Fdτ v (25) r r ∇ .F = constant a3 DEDODA university.arabsbook.com r F a2 (q1 , q 2 , q 3 ) a1 -4 - (25) r r r r ∇. F = lim Ñ ∫ F . da (26) ∫dτ →0 ∫ dτ 4 r r Ñ ∫ F . da = φR +φL + φT + φBo + φF + φBa (27) -: R: - right: - L: - left: - T:-top: - 12
- 13. Bo:-bottom: - F:-front: - Ba:-back: - φR = F2R d l3R d l1R ∂F2 dq2 φR = F2 + ( h3R dq3h1R dq1 ) (28 − a) ∂q2 2 ∂F2 dq2 ∂h3 dq2 ∂h1 dq2 φR = F2 + h3 + h1 + dq1dq3 ∂q2 2 ∂q2 2 ∂q2 2 ∂F dq ∂h dq ∂h dq φL = − F2 − 2 2 h1 − 1 2 h3 + 1 2 dq1dq3 (28-b) ∂q2 2 ∂q2 2 ∂q2 2 (28-b) (28-a) ∂h ∂h ∂F φR + φL = F2h1 3 + F2h3 1 + h1h3 2 dq1dq2dq3 ∂q2 ∂q2 ∂q2 (28 − c ) ∂(F2h1h3 ) φR + φL = dq1dq2dq3 ∂q2 13
- 14. ∂( F2h1h2 ) φT + φBo = dq1dq 2dq 3 (28-d) ∂q 3 ∂(F1h2 h3 ) φF + φBa = dq1dq 2dq 3 (28-e) ∂q1 (27) (28-e) (28-d) (28-c) r r ∂(F h h ) ∂(F h h ) ∂(F h h ) Ñ ∫ S F . da= 2 2 3 + 2 1 3 + 3 1 2 dq1dq2dq3 (29) ∂q1 ∂q2 ∂q3 r r ∇. F (26) (29) ∂(F2h2h3 ) ∂(F2h1h3 ) ∂(F3h1h2 ) + + dq1dq2dq3 r r ∂q1 ∂q2 ∂q3 ∇. F= lim ∫dτ →0 ∫ dτ but ∫ dτ = h h h 1 2 3 dq1dq2dq3 ∂(F2h2h3 ) ∂(F2h1h3 ) ∂(F3h1h2 ) + + dq dq dq r r ∂q1 ∂q2 ∂q3 1 2 3 ∴ ∇. F= h1h2h3 dq1dq2dq3 r r 1 ∂(Fh2h3 ) ∂(F2h1h3 ) ∂(F3h1h2 ) ∇. F= 1 + + (30) h1h2h3 ∂q1 ∂q2 ∂q3 Divergence (30) (Orthogonal curvilinear coordinate) Divergence . Laplacian (The Laplacian in orthogonal curvilinear coordinate) 14
- 15. r r r2 ∇. ∇φ =∇ φ (24) r 1 ∂ h2h3 ∂φ ∂ hh3 ∂φ ∂ hh2 ∂φ ∇2φ = + 1 + 1 (31) hh2h3 ∂q1 h1 ∂q1 ∂q2 h2 ∂q2 ∂q3 h3 ∂q3 1 Laplacian The curl in orthogonal curvilinear coordinate " Stoke's Theorem" r r r r r ∫∇×F. da=ÑF. dr (32) ∫ r r ∇×F = constant q3 DEDODA h2cdq universi ty.arabsbook.com 2 c h3dq 3 d h3b dq 3 b a q2 h2dq 2 q1 15
- 16. -5 - q2 q3 curl q2 q1 q3 q1 . (32) 5 curl r r r r (∇×F )1 = lim Ñ F. dr ∫ (33) ∫da→0 ∫da r r Ñ F . dr=F2h2dq 2 + F3b h3b dq3 − F2c h2cdq 2 − F3h3dq3 ∫ r r ∂F ∂h Ñ ∫ F . dr=F2h2dq 2 + F3 + 3 dq 2 h3 + 3 dq 2 dq3 ∂q 2 ∂q 2 ∂F ∂h - F2 + 2 dq3 F2 + 2 dq3 dq2 − F3h3dq3 ∂q3 ∂q3 r r ∂(F h ) ∂(F2h2 ) →∴ Ñ F . dr = 3 3 − ∫ dq2dq3 (34) ∂q 2 ∂q3 ∫da = h h dq dq 2 3 2 3 (33) (34) r r 1 ∂(Fh ) ∂(Fh ) (∇×F)1 = 3 3 − 2 2 (35-a) h2h3 ∂q2 ∂q3 q3 q1 16
- 17. r r (∇ × F ) 2 q2 q1 r r (∇ × F )3 r r 1 ∂(Fh ) ∂(Fh ) (∇×F)2 = 1 1 − 3 3 (35-b) hh3 ∂q3 1 ∂q1 r r 1 ∂(Fh ) ∂(Fh ) (∇×F)3 = 2 2 − 1 1 (35-c) hh2 ∂q1 1 ∂q2 (35) a1h1 a 2 h2 a 3h3 r r 1 ∂ ∂ ∂ ∇× F = (36) h1h2 h3 ∂q1 ∂q 2 ∂q3 h1F1 h 2 F2 h3F3 17
- 18. ϕ θ , (displacement) z DEDODA universi ty.arabsbook.com dr rd θ r θ dθ y ϕ r sin θ dϕ x dϕ r sin θ -6 - ϕ θ 1 12 18
- 19. d lϕ d lθ 1 12 d lθ = hθ dqθ → d lθ = rd θ d lϕ = hϕdqϕ → d lϕ = r sinθd ϕ Laplacian Divergence Gradient Curl 1 (36) (31) (30) (24) (24) r ∂φ ∂φ ∂φ ∇φ = a1 + a2 + a3 h1∂q1 h2∂q2 h3∂q3 Gradient 1 ( Cartesian coordinate) Gradient r ∂φ ∂φ ∂φ ∇φ = i + j +k (37) ∂x ∂y ∂z r ∂φ ∂φ ∂φ ∇φ = r0 +θ +ϕ (38) ∂r r ∂θ r sinθ∂ϕ 19
- 20. r ∂φ ∂φ ∂φ ∇φ = r0 +ϕ +k (39) ∂r r ∂ϕ ∂z Divergence r r 1 ∂(Fh2h3 ) ∂(F2h1h3 ) ∂(F3h1h2 ) ∇. F= 1 + + h1h2h3 ∂q1 ∂q2 ∂q3 r r ∂F ∂F ∂F ∇. F= x + y + z (40) ∂x ∂y ∂z r r 1 ∂(Fr r 2 sinθ ) ∂(Fθ r sinθ ) ∂(Fϕ r ) ∇. F= + + r 2 sin θ ∂r ∂θ ∂ϕ r r 1 ∂(r 2Fr ) 1 ∂(sinθ Fθ ) 1 ∂(Fϕ ) → ∇. F= 2 + + (41) r ∂r r sin θ ∂θ r sinθ ∂ϕ r r 1 ∂(F r ) ∂(F ) ∂(F r ) ∇. F= r + ϕ + z r ∂r ∂ϕ ∂z r r 1 ∂(rFr ) 1 ∂(Fϕ ) ∂(Fz ) → ∇. F= + + (42) r ∂r r ∂ϕ ∂z Laplacian r 1 ∂ h2h3 ∂φ ∂ hh3 ∂φ ∂ hh2 ∂φ ∇2φ = + 1 + 1 hh2h3 ∂q1 h1 ∂q1 ∂q2 h2 ∂q2 ∂q3 h3 ∂q3 1 20
- 21. r2 ∂2φ ∂2φ∂2φ ∇φ = 2 + 2 + 2 (43) ∂x ∂y ∂z r 1 ∂ 2 ∂φ ∂ ∂φ ∂ 1 ∂φ ∇2φ = 2 r sinθ + sinθ + (44) r sinθ ∂r ∂r ∂θ ∂θ ∂ϕ sinθ ∂ϕ r2 1 ∂ ∂φ ∂ 1 ∂φ ∂ ∂φ ∇φ = r + + r (45) r ∂r ∂r ∂ϕ r ∂ϕ ∂z ∂z Curl .1 . 21