1. (a) Using index notation, prove the identity where A is a general vector field. (b) Using index notation, prove the identity F V. V.VX A = 0, grad g = (39 where F and g are general vector and scalar fields respectively. (c) By setting F= ▼ x A, use the identities in parts (a) and (b) to find an expression for div ((curl A) /g) that involves no second derivatives. = (d) Given that general expressions for grad and curl are given, in cylindrical polar basis, by curl B = eR + V.F-F.Vg g² R - find the curl of the vector field B to evaluate the divergence of the field 1 ag R 06 eo + 2e₂ R+z eR Re ez ə ə ə ƏR аф Əz BR RB B₂ Reo+z²e and hence use the result of part (c) RE-X ag əz 2 ezi

Find the divergence of the vector field F = div F = Calculator Check Answer Ei 1 (99+)

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For each of the following vector fields, find its curl and determine if it is a gradient field. (a) ♬ = 5yz i + (5xz + z²) j + (5xy+2yz) k curl F F? = (b) Ğ = (10xy + x³) i+ (5x² + z²) j + (2yz – 4z) k curl G G ? = (c) H = 5yzi+ (z² – 5xz) j + (5xy + 2yz) k: curl H H? =