16.8 The gradient of a scalar field 467
defines a vector associated with each point r. An example of a vector field is the
velocity field used to describe fluid flow in hydrodynamics; a velocity is associated
with every point. Electric and magnetic fields are vector fields.
The theory of vector fields is an essential tool in hydrodynamics and electromag-
netism. A basic feature of vector field theory is the use of vector differential operators
to describe how a field changes from point to point in space. The concept of the
gradient of a scalar field, with some applications, is discussed in the following section.
The divergence and the curl of a vector field are introduced in Section 16.9. These
quantities are used in more advanced applications of the vector calculus in the theory
of fields, and only a very brief description is given here.
16.8 The gradient of a scalar field
The gradientof a scalar function of positionf 1 = 1 f(x, y, z)is defined as the vector
(16.66)
This quantity can be interpreted as the result of operating on the functionfwith the
vector differential operator
(16.67)
(read as ‘del’ or ‘nabla’), so that
(16.68)
(i, j, and kare assumed to be constant vectors).
5EXAMPLE 16.19Find grad VforV 1 = 1 x 1 + 12 yz 1 + 13 z
2.
We have and Therefore∇V 1 = 1 i 1 + 12 zj 1 + 1 (2y 1 + 16 z)k.
∂
∂
=+
V
z
26.yz
∂
∂
=,
∂
∂
=
V
x
V
y
12 ,z
=
∂
∂
∂
∂
∂
∂
f
x
f
y
f
z
ijk
gradff
xy z
=∇ = f
∂
∂
∂
∂
∂
∂
ijk
∇=
∂
∂
∂
∂
∂
xy z∂
ijk
gradf
f
x
f
y
f
z
=
∂
∂
∂
∂
∂
∂
ijk
5The symbol ∇was introduced by Hamilton, and (possibly) called nabla after a harp-like musical instrument
used in Palestine in Biblical times.