In lecture we introduced the Kronecker delta dij and the Levi-Civita symbol Eik which let us express the dot product and cross product of vectors in our usual three-dimensional space in terms of components. You should use Einstein summation convention (any re- peated index is summed over) throughout this problem. 1.01 Derive the vector triple product identity Āx (B x T) = (Ā. C)B – (A: BC us- ing the Kronecker delta and Levi-Cevita symbol. Note we have used the 'dot notation, e. g. (Ā• C) = (A, C). Carefully explain and/or justify each step you take (e. g. "by the an- tisymmetry of Eijk...")! Hint: Start with the left-hand side. Since both sides are vectors, we really want to show the component-version of the identity, (A ~ (B x ©))' = (A.Č)Bİ – (Ā• B)C'. 1.02 Using index notation, show Lagrange's identity, (A x B) · (A x B) = (Õ Ã)(B • B) – (A:B)2. We can treat the nabla/del operator in components as: Y H ai = o. Using this, the gradient, divergence, and curl can be expressed in index notation: Gradient: Divergence: Curl: ✓ f)' = d'f $ = 8;0') (ỹ xv)' = e' italok 1.03 Write out the Laplacian of a scalar function v2f = V . ïf in index notation and then carry out the sum. 1.04 Prove that the curl of the gradient is zero: V 1.05 Prove that the curl of the curl is given by ✓ x (ỹ x A) = 7 (7 • Ā) – V2Ā.