Introduction to vector and tensor analysis

where ci(u)du is the indefinite integral of an ordinary scalar function. If s(u)
is a vector satisfying c(u) = d
dus(u) then
d
c(u)du = (s(u)) du = s(u) + k
du
where k is an arbitrary constant vector independent of u. The definite integral
between two limits u = u0 and u = u1 can in such case be written
u1
u1
d
c(u)du = (s(u))du = [s(u) + k]u1 u0 du = s(u1) − s(u0).
u0
u0
This integral can also be defined as a limit of a sum in a manner analogous to
that of elementary integral calculus.
2.6 Fields
2.6.1 Definition
In continuous systems the basic physical variables are distributed over space.
A function of space is known as a field. Let an arbitrary coordinate system be
given.
Scalar field. If to each position x = (x1, x2, x3) of a region in space the
corresponds a number or scalar φ(x1, x2, x3), then φ is called a scalar function of
position or scalar field. Physical examples of scalar fields are the mass or charge
density distribution of an object, the temperature or the pressure distribution
at a given time in a fluid.
Vector field. If to each position x = (x1, x2, x3) of a region in space there
corresponds a vector a(x1, x2, x3) then a is called a vector function of position
or a vector field. Physical examples of vector fields are the gravitational field
around the earth, the velocity field of a moving fluid, electro-magnetic field of
charged particle systems.
In the following we shall assume the choise of a rectangular coordinate system
C = (O, R). Introducing the position vector r(x) =
i xiei the expressions φ(r)
and a(r) is taken to have the same meaning as φ(x) and a(x), respectively.
2.6.2 Partial derivatives
Let a be a vector field a(r) = a(x1, x2, x3). We can define the partial derivative
in the same fashion as in Eq.
notation:
(2.26). Here we will use the short hand
∂a
∂xi
∂i = ∂
∂xi
∂ 2 ij
= ∂2
∂xi∂xj
In complete analogy to the usual definition of partial derivatives of a scalar
function, the partial derivative of a vector field with respect to xi (i = 1, 2, 3) is
a(r + ∆xiei) − a(r)
(∂ia)(r) = lim
∆xi→0 ∆xi
15
(2.32)