6Div, Grad, Curl and ∇
IA Vector Calculus
6.2 Second-order derivatives
We have
Proposition.
∇ × (∇f) = 0
∇ · (∇ × F) = 0
Proof. Expand out using suffix notation, noting that
ε
ijk
∂
2
f
∂x
i
∂x
j
= 0.
since if, say, k = 3, then
ε
ijk
∂
2
f
∂x
i
∂x
j
=
∂
2
f
∂x
1
∂x
2
−
∂
2
f
∂x
2
∂x
1
= 0.
The converse of each result holds for fields defined in all of R
3
:
Proposition. If F is defined in all of R
3
, then
∇ × F = 0 ⇒ F = ∇f
for some f .
Definition
(Conservative/irrotational field and scalar potential)
.
If
F
=
∇f
,
then f is the scalar potential. We say F is conservative or irrotational.
Similarly,
Proposition.
If
H
is defined over all of
R
3
and
∇ · H
= 0, then
H
=
∇ × A
for some A.
Definition
(Solenoidal field and vector potential)
.
If
H
=
∇ × A
,
A
is the
vector potential and H is said to be solenoidal.
Not that is is true only if F or H is defined on all of R
3
.
Definition (Laplacian operator). The Laplacian operator is defined by
∇
2
= ∇ · ∇ =
∂
2
∂x
i
∂x
i
=
∂
2
∂x
2
1
+
∂
2
∂x
2
2
+
∂
2
∂x
3
3
.
This operation is defined on both scalar and vector fields — on a scalar field,
∇
2
f = ∇·(∇f),
whereas on a vector field,
∇
2
A = ∇(∇ · A) − ∇ × (∇ × A).