Question

Using Cartesian tensor notation and the fact that we can write the curl operator in the form: (vxA), = eijk BKI where Eijk is the Levi-Civita tensor, and the relationship: Ekijeklm = dilim - Sim Oil prove Greens' vector identity: 02(A. B) = A -12B – B-02Ã+20. [(B.V)X + B ® (V x A)] where A and B are vector functions. [8 marks]

Answer 1

Answer:

where pm and qm are two arbitrary twice continuously differentiable scalar fields. This identity is of great importance in physics because continuity equations can thus be established for scalar fields such as mass or energy.

for example,

Green's theorem vs. Green's theorems JOΣ Although, generalization to higher dimension of GT is called (Kelvin-)Stokes theorem (StT), Ā.dř= // (V x A) · dà • where V = (a/ax, alay, a/az) should be understood as a symbolic vector operator • in electrodynamics books one will find electrodynamic Green's theorem' (EGT), dx = | (@Vy – Vºdy LOV 06 -2 2 AJT Uan Soo [o an - Van Wednesday, January 23. 13

In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. ... Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics.

x e3 X,e, x23 X

The Levi-Civita tesnor is totally antisymmetric tensor of rank n. The Levi-Civita symbol is also called permutation symbol or antisymmetric symbol. It is named after the Italian mathematician and Physicist Tullio Levi-Civita [1-3]. The indices i, j, and k run from 1, 2, and 3.

Greens' Vector Identity. е (А - ) . . ** - В.a® + 27. [(в с)+ Вх (9) Інс : - Сб.в) - о? (е; B: ) - 7; 9; te; B: ) Ән: . әB х + 29 28i re: 'в эхэх ЭХ - gi o'a: a po se, to: *в: Әxi o" - во в те , 2 әрі эG ә, эх; Now take 27. [B x D x A)] - 2 с; (вх () ):

= 27. ( са је В; (об) ) - 27; (Eijk B; Ekem Te Am ) - аа: (tit Etem Bi Pam) 2 (die Sion - Sim Sje) o t (Bj Dam ) . . . . km - Sim 35 ) эе, эв . , ...) Әxt Эх А4 IBM 2AM Am әХі әкі + Bm Am әee Pm Әx, 2 - В. *Am axmaxe әХm əxe 2Bm 2 Am . ав аа, аве Р 21 ч. Хі əxt әве Am ч эх әке Now за (во) ] - 2 » [в, о, у, ] аа: Әx; əxj Exeixe

Thus R.H.S. Ã 3²B - B + 2 dBi ani IDX axj B. RAY & 21 Bm Jei daj OX; Am & a B. o axi - 2B; . 2.08 AM 2m xe ao da ... Ã. 28+ Boata a Bm BAM 2x axi .. Theodore L. H.S= R. 'H.S. ... B) e n. 220 - Balitat [ 6 ) Ã + xloral Hence proved.