Electromagnetism and Relativity 2014/15 Tutorial Sheet 1: Index notation and vector calculus 1. Revision (if you need it): by writing out the components explicitly, show that a × (b × c) = (a · c)b − (a · b)c , where a, b and c are any three vectors. 2. To what quantities do the following expressions in index notation (with summation convention) correspond? δii , δij ai aj , δij δij , iji , ijk δij , bi ijk ak cj , ijk a3i a1k a2j . 3. Verify the identity ijk klm = δil δjm − δim δjl , for the cases (i) i = 1, j = 2, l = 1, m = 2 (ii) i = 1, j = 2, l = 2, m = 1 (iii) i = 1, j = 2 4. (i) Write in index notation the vector equation a × b + c = (a · b)b − d . (ii) If ui (i = 1, 2, 3) are three independent variables show that ∂ui = δij . ∂uj (iii) Show both by using the identity ijk klm = δil δjm − δim δjl and also by writing the indices explicitly that ijk ijl = 2δkl , ijk ijk = 6 . 5. Evaluate ijk δjk . Solve the following equation for vk , f , pij in terms of kij : ijk vk + δij f + pij = kij , where pij = pji and pii = 0. (PTO) 1 6. Using index notation, evaluate n (i) ∇ c · r (ii) ∇rn (iii) ∇ · rn r (iv) ∇ × rn r (v) ∇ · c × r (vi) ∇ × c × r (vii) c · ∇ r where c is a constant vector. 7. If a and b are vector fields, establish the following identities using index notation (i) ∇ · (∇ × a) = 0 (ii) ∇ · (a × b) = b · (∇ × a) − a · (∇ × b) (iii) ∇ × a × b = a ∇ · b − b ∇ · a + b · ∇ a − a · ∇ b 2
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