Introduction to vector and tensor analysis

The full physical and geometrical meaning of the divergence is discussed in next
section. Clearly (∇ · a)(r) is a scalar field. Now if some vector field a is itself
derived from a scalar field via a = ∇φ then ∇ · a has the form ∇ · ∇φ or, as it
is usually written ∇ 2 φ where
∇ 2 =
i
∂ 2 ii
=
∇2φ is called the laplacian of φ and is typically encountered in electrostatic
problems or in diffusion equations of physical scalar field such as a temperature
or density distribution. The laplacian can also act on vector through the
components
∇ 2 a =
∂ 2 iia =
(∂ 2 iiaj)ej
Curl of a vector field
i
The curl of a vector field a(r) is defined by
curl (a) = ∇ × a =
∂2a3 − ∂3a2 e1 +
∂3a1 − ∂1a3 e2 +
∂1a2 − ∂2a1 e3,
In analogy to the definition of cross-product between two ordinary vectors we
can express the definition symbolically
⎛
⎞
∇ × a = det
j
⎝ e1 e2 e3
∂1 ∂2 ∂3
i
i
a1 a2 a3
where it is understood that, on expanding the determinant, the partial derivatives
act on the components of a. The tensor notation of this expression (for
the i’th component) is even more compact
[∇ × a]i =
jk
ɛijk∂jak
∂ 2
∂x 2 i
⎠ ,
(2.47)
Clearly, ∇ × a is itself a vector field.
For a vector field v(x) describing the local velocity at any point in a fluid,
∇ × v is a measure of the angular velocity of the fluid in the neighbourhood of
that point. If a small paddle wheel were placed at various points in the fluid
then it would tend to rotate in regfions where ∇ × v = 0, while it would not
rotate in regions where ∇ × v = 0.
Another insight into the physical interpretation of the curl operator is gained
by considering the vector field v describing the velocity at any point in a regied
body rotation about some axis with angular velocity ω. If r is the position
vector of the point with respect to some origin on the axis of rotation then the
velocity field would be given by v(x) = ω × r(x). The curl of the vector field is
then found to be ∇ × v = 2ω.
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