7/29/2019 Vector calculus.pdf
1/2
Vector calculusIn SectionA.9 of Appendix A we review the algebra
of vectors, and in Chapter 1 weconsidered how to transform one
vector into another using a linear operator. In thischapter and the
next we discuss the calculus of vectors, i.e. the differentiation
andintegration both of vectors describing particular bodies, such
as the velocity of a particle,and of vector fields, in which a
vector is defined as a function of the coordinates
throughout some volume (one-, two- or three-dimensional). Since
the aim of this chapteris to develop methods for handling
multi-dimensional physical situations, we will assumethroughout
that the functions with which we have to deal have sufficiently
amenablemathematical properties, in particular that they are
continuous and differentiable.
2.1 Differentiation of vectors
Let us consider a vectora that is a function of a scalar
variable u. By this we mean that
with each value ofuwe associate a vectora(u). For example, in
Cartesian coordinates
a(u) = ax(u)i + ay(u)j + az(u)k, where ax(u), ay(u) and az(u)
are scalar functions ofu
and are the components of the vectora(u) in the x-, y- and
z-directions respectively. We
note that ifa(u) is continuous at some point u= u0 then this
implies that each of the
Cartesian components ax(u), ay(u) and az(u) is also continuous
there.Let us consider the derivative of the vector function a(u)
with respect to u. The derivative
of a vector function is defined in a similar manner to the
ordinary derivative of a scalar
function f(x). The small change in the vectora(u) resulting from
a small change uin
the value ofuis given by a = a(u+ u) a(u) (see Figure 2.1). The
derivative ofa(u)
with respect to uis defined to be
da
du
= lim
u0
a(u+ u) a(u)
u
, (2.1)
assuming that the limit exists, in which case a(u) is said to be
differentiable at that point.
Note that da/duis also a vector, which is not, in general,
parallel to a(u). In Cartesian
coordinates, the derivative of the vectora(u) = axi + ayj +
azkis given by
da
du
= dax
du
i + day
du
j + daz
du
k.
Perhaps the simplest application of the above is to finding the
velocity and accelerationof a particle in classical mechanics. If
the time-dependent position vector of the particle
with respect to the origin in Cartesian coordinates is given by
r(t) = x(t)i + y(t)j + z(t)k
