MATH 241, SECTION B1 CLASS NOTES

1. 16.5: Curl and Divergence The curl and divergence are two operators that allow us to transform a vector field F into either a new vector field (in the case of the curl) or into a function (in the case of the divergence). In either case, we will use the concept of a vector differential operator: we use the “vector”  ∂ ∂ ∂  ∇ = , , ∂x ∂y ∂z to act upon vector fields.

The curl:

Let F = hP, Q, Ri be a vector field. Then the curl of F, denoted curl F, is given by applying the differential operator ∇ via the cross product; i.e., i j k ∂R ∂Q ∂R ∂P  ∂Q ∂P  curl F = ∇ × F = ∂ ∂ ∂ = − i − − j + − k ∂x ∂y ∂z ∂x ∂z ∂x ∂z ∂x ∂z PQR Notice this is a vector field. Example 1.1. For F = hsin(y), x, ezi, i j k ∂ ∂ ∂ ∇ × F = ∂x ∂y ∂z = 0i − 0j + (1 − cos(y))k = h0, 0, 1 − cos(y)i sin(y) x ez Notice: when F has no z-component (i.e., when R = 0), F can represent a vector field in the xy-plane. When this is the case, notice that  ∂P ∂Q curl F = 0, 0, − ∂x ∂y Remember from way back two to three weeks ago, we discussed how, if F = ∇f, Clairaut’s Theorem tells ∂P ∂Q us that ∂x − ∂y = 0. We can now upgrade this neatly using the curl: Theorem 1.2. Suppose that F = ∇f and that f has continuous partial second derivatives. Then curl F = 0. ∂f ∂f ∂f  Proof. If F = ∇f, then F = , , . So ∂x ∂y ∂z

i j k  2 2   2 2   2 2  ∂ ∂ ∂ ∂ f ∂ f ∂ f ∂ f ∂ f ∂ f curl F = ∂x ∂y ∂z = − i − − j + − k = 0 ∂f ∂f ∂f ∂y∂z ∂z∂y ∂x∂z ∂z∂x ∂x∂y ∂y∂x ∂x ∂y ∂z where Clairaut’s theorem is telling us each component is zero.  As in the case of the plane, we have a partial inverse to this theorem with a somewhat heavy caveat: Theorem 1.3. If F is a vector field such that its components have continuous partial derivatives and F is defined on all of R3 and curl F = 0, then F = ∇f for some function f. As you can imagine, this is a useful tool:

Date: July 28th. 1 Example 1.4. Show that F = hx2, y2, z2i is conservative.

∇ × F = 0, and since F is defined on all of R3, F is conservative.

Divergence:

For the curl, we used the vector differential operator ∇ together with the cross product. For the divergence of a vector field we use ∇ with the dot product. The divergence of a vector field F = hP, Q, Ri is defined as  ∂ ∂ ∂  ∂P ∂Q ∂R div F = ∇ · F = , , · hP, Q, Ri = + + ∂x ∂y ∂z ∂x ∂y ∂z Notice this is a function. Example 1.5. For F = hx2 + sin(y), y2 − xz2, z2 + exyi, div F = 2x + 2y + 2z. The divergence and curl together give an interesting fact: Theorem 1.6. For any vector field F = hP, Q, Ri for which the components of F have continuous second partial derivatives, div curl F = 0.

Proof. Exercise (use Clairaut’s theorem).  This fact will be vital later. Example 1.7. Show that F = hx2, y2, z2i is not the curl of another vector field G.

Since div F = 2x + 2y + 2z 6= 0, F cannot be the curl of another vector field (as, if F = curl G, then we would need to have that div F = div curl G = 0).

Physical interpretations of divergence and curl (see slides)

Green’s Theorem reformulation/Stokes’ Theorem teaser: