tailieunhanh - Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae for Vector Analysis李国华

Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae for Vector Analysis 李国华 （Kok-Wah LEE） @ 08 May 2009 (Version ) No. 4, Jalan Bukit Beruang 5, Taman Bukit Beruang, 75450 Bukit Beruang, Melaka, Malaysia. Email: contact@; E96LKW@ Tel.: +6013-6134998; +606-2312594; +605-4664998 All rights reserved. Vector: Scalar: Nabla: | Formulae involving V Vector Identities with Proofs Nabla Formulae for Vector Analysis a Kok-Wah LEE @ 08 May 2009 Version No. 4 Jalan Bukit Beruang 5 Taman Bukit Beruang 75450 Bukit Beruang Melaka Malaysia. Email contact@ E96LKW@ Tel. 6013-6134998 606-2312594 605-4664998 All rights reserved. Vector A Al i A2 j A3 k B Bl i B2 j B3 k Scalar ộ ộ x y z Y Y x y z Nabla V i - j-d k Ậ dx dy dz C Cl i C2 j C3 k 1 A x B .C B x C .A C x A .B 2 A x B x C B - C 3 Prove V ộ Y Vộ Vy p i ịj k ộ v dộ Yi dY Y j . ộ Yk t dx dy d z J dx dy dz dộ . a dộ . . Àỵ dộ du -r-i i j j k k dx dx dy dy dz dz j Ộ1 Ì- Y jY k 1 t dx dy d z J t dx dy dz J d .d . . d L d . . d . . d Ỵ . I i j k ộ I i j k w t dx dy dz t dx dy dz J V ộ Y Vộ Vw 4 Prove V ộ Y ộ V Y YVộ fv i T- j 3 k ỘY t dx dy dz J d ộY . d 0Y .d 0Y k i j k x y z 1 d dộ . . r ụ . _ dộ . . r ụ _ dộ Á33- i f i 7 j f j Á 33- k f k dx dx dx dx dx dx Á If i lf j k i j k ì V dx ơy ỏz V dx ơy ỏz J_I 7 Á 77i 77 j - k f ỳ -f- ì 33 j 33 k Á V dx dy dz V dx dy dz VÁ ỳ Á Vf f V0 5 Prove V. A B V- A B I 3-i j 3-k A1 B1 i A2 B2 j A3 B3 k V dx dy dz d Ai B1 d A B2 a A3 B3 x sT d LHS I i j k AẠ A2 j Ajk 1 i j k Bii B2 j B3k 3x dy dz 3x dy dz dA 3A dA 3B dB 3B T 1 -z -F3 . 2 -3 dx dy dz dx dy dz a Ai B1 a A2 B2 a A3 B3 7 a - RHS LHS RHS A V. A B 6 Prove Vx A B VxA VxB Vx A B I i j k Ix A1 B1 i A2 B2 j A3 B3 k dx dy dz i _d_ dx A1 B1 j d_ dy a2 b2 k _d_ dz A3 B3 d A3 B3 - d A2 B2 ìi _ Id A3 B3 _ d A1 B1 Ị . I d A2 B2 _ d A1 B1 dy dz V dx dz dx dy 2 d j IÁ-Ml f ặ- B - f dB--M f B-M dz V dx dy J _ _l dy dz JI dx dz J V dx dy J dx dy dz Á1 Á2 Á3 i j k d d d dx dy dz B1 B2 B3 i j k d d d Vx A B VxA VxB 7 Prove V. 0A V0 .A 0 V0Á I Jci j k . 0A 0 j 0Ák V dx dy dz J d0Á d 0Á2 d 0Á3 LHS dx dy dz V0 .A ộ I r 0 i d0 j l k . A1i Á j A3k 0 V dx dy dz J f d . d M 4 y 1 V i j ir k r Aii Áj Ák V ax dy dz J ỈÉ A2 dt A3 d0 À ả IA1 dA dx dy dz J V dx dy dz it 0 f Ad0 .