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18.022 Recitation Handout (with solutions) 22 October 2014 1. Verify that divergence, curl, and gradient are linear operators. Solution. For divergence, we want to show that for all vector fields F and G and scalars α and β, we have ∇ · (αF + βG) = α∇ · F + β∇ · G. The left-hand side is ∂ ∂ ∂ ∂F 1 ∂G 1 ∂F 2 ∂G 2 ∂F 3 ∂G 3 (αF 1 + βG 1 ) + (αF 2 + βG 2 ) + (αF 3 + βG 3 ) = α +β +α +β +α +β , ∂x ∂y ∂z ∂x ∂x ∂y ∂y ∂z ∂z which equals the right-hand side. Calculations for gradient and curl are similar. 2. Let F(x, y, z) = (3x 2 + 21 y 2 + e z , x y + z, f (x, y, z)). Find all f such that F is curl-free. Solution. The third component of ∇ × F is y − (1/2)(2y) = 0, as desired. For the first component to be zero, we must have f y = 1, and for the second component to be zero we must have f x = e z . Integrating these two equations tells us that f (x, y, z) = y + C 1 (x, z) and f (x, y, z) = xe z + C 2 (y, z) for functions C 1 and C 2 which do not depend on y or x respectively. Putting these two together, we see that f (x, y, z) = xe z + y +C (z) for any differentiable function C . 3. Confirm that for a vector field F : R3 → R3 , we have ∇ × (∇ × F) = ∇(∇ · F) − ∇2 F, where ∇2 F is defined to mean “take the Laplacian of each component of F.” Is it possible to derive this identity from a × (b × c) = (a · c)b − (a · b)c? Solution. [Omitted] 4. Let F be a C 2 vector field on R3 . Show that ∇ × F is incompressible. Solution. We calculate ∇ · (∇ × F) = ∂2 F 3 ∂2 F 2 ∂2 F 1 ∂2 F 3 ∂2 F 2 ∂2 F 1 − + − + − = 0, ∂x∂y ∂x∂z ∂y∂z ∂y∂x ∂z∂x ∂z∂y since the mixed partials don’t depend on the order of differentiation, as F is C 2 .