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 .