Several important vector calculus identities can be proved with the aid of the following two properties...

 

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Several important vector calculus identities can be proved with the aid of the following two properties of the Levi-Civita symbol Eijk: Eijkeilm = 8; 8km - 8; mSk ² = = -Ekji Eijk Ekij Ejki = -Eikj = - jik (Check for yourself that these are true!) Here & is the Kronecker delta. Prove the following: (a) á × (b × c) = b(a · c) — c(a · b) . . (b) For a scalar function () with continuous first and second derivatives, curl grad = V x (Vo) = 0. Σ; implied (c) For a vector function A(7) with continuous first and second derivatives, div curl A = V. (V x A) = 0. Several important vector calculus identities can be proved with the aid of the following two properties of the Levi-Civita symbol Eijk: Eijkeilm = 8; 8km - 8; mSk ² = = -Ekji Eijk Ekij Ejki = -Eikj = - jik (Check for yourself that these are true!) Here & is the Kronecker delta. Prove the following: (a) á × (b × c) = b(a · c) — c(a · b) . . (b) For a scalar function () with continuous first and second derivatives, curl grad = V x (Vo) = 0. Σ; implied (c) For a vector function A(7) with continuous first and second derivatives, div curl A = V. (V x A) = 0.