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11.5 Divergence and Curl 363

3. The cross product of ∇ with F is the curl of F:

i j k

∇× F = ∂/∂x ∂/∂y ∂/∂z

f g h
∂h ∂g ∂ f ∂h ∂g ∂ f

= − i + − j + − k = curl of F.
∂y ∂z ∂z ∂x ∂x ∂y
The del (or nabla) operator is part of the MAPLE set of routines collected under the Vec-
torCalculus designation. Using this and the operations of scalar multiplication, dot product
(DotProduct) and cross product (CrossProduct), we can carry out computations with vector fields.
This package can also be used to compute divergence and curl in other coordinate systems, such
as cylindrical and spherical coordinates.
There are two relationships between gradient, divergence and curl that are fundamental to
vector analysis: the curl of a gradient is the zero vector, and the divergence of a curl is (the
number) zero.

THEOREM 11.3

Let F be a continuous vector field whose components have continuous first and second par-
tial derivatives and let ϕ be a continuous scalar field with continuous first and second partial
derivatives. Then
1.
∇× (∇ϕ) = O.

2.
∇· (∇× F) = 0.

These conclusions may be paraphrased:

curl grad = O, div curl = 0.
Both of these identities can be verified by direct computation, using the equality of mixed second
partial derivatives with respect to the same two variables. For example, for conclusion (1),
∂ϕ ∂ϕ ∂ϕ

∇× (∇ϕ) =∇ × i + j + k
∂x ∂y ∂z

i j k

= ∂/∂x ∂/∂y ∂/∂z

∂ϕ/∂x ∂ϕ/∂y ∂ϕ/∂z

∂ ϕ ∂ ϕ ∂ ϕ ∂ ϕ ∂ ϕ ∂ ϕ
2 2 2 2 2 2
= − i + − j + − k
∂y∂z ∂z∂y ∂z∂x ∂x∂z ∂x∂y ∂y∂x
= O
because the mixed partials cancel in pairs in the components of ∇× (∇ϕ).
Operator notation with ∇ can simplify such calculations. In this notation, ∇× (∇ϕ) = O is
immediate because ∇× ∇ is the cross product of a "vector" with itself, which is always zero.
Similarly, for conclusion (2), ∇× F is orthogonal to ∇, so its dot product with ∇ is zero.

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October 15, 2010 16:11 THM/NEIL Page-363 27410_11_ch11_p343-366