Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.6 SCALAR AND VECTOR FIELDS

A normalnto the surface at this point is then given by

n=

∂r

∂θ

×

∂r

∂φ

=

∣∣

∣

∣

∣∣

∣

ijk

acosθcosφacosθsinφ −asinθ

−asinθsinφasinθcosφ 0

∣∣

∣

∣

∣∣

∣

=a^2 sinθ(sinθcosφi+sinθsinφj+cosθk),

which has a magnitude ofa^2 sinθ. Therefore, the element of area atPis, from (10.23),

dS=a^2 sinθdθdφ,

and the total surface area of the sphere is given by

A=

∫π

0

dθ

∫ 2 π

0

dφ a^2 sinθ=4πa^2.

This familiar result can, of course, be proved by much simpler methods!

10.6 Scalar and vector fields

We now turn to the case where a particular scalar or vector quantity is defined

not just at a point in space but continuously as afieldthroughout some region

of spaceR(which is often the whole space). Although the concept of a field is

valid for spaces with an arbitrary number of dimensions, in the remainder of this

chapter we will restrict our attention to the familiar three-dimensional case. A

scalar fieldφ(x, y, z) associates a scalar with each point inR, while avector field

a(x, y, z) associates a vector with each point. In what follows, we will assume that

the variation in the scalar or vector field from point to point is both continuous

and differentiable inR.

Simple examples of scalar fields include the pressure at each point in a fluid

and the electrostatic potential at each point in space in the presence of an electric

charge. Vector fields relating to the same physical systems are the velocity vector

in a fluid (giving the local speed and direction of the flow) and the electric field.

With the study of continuously varying scalar and vector fields there arises the

need to consider their derivatives and also the integration of field quantities along

lines, over surfaces and throughout volumes in the field. We defer the discussion

of line, surface and volume integrals until the next chapter, and in the remainder

of this chapter we concentrate on the definition of vector differential operators

and their properties.

10.7 Vector operators

Certain differential operations may be performed on scalar and vector fields

and have wide-ranging applications in the physical sciences. The most important

operations are those of finding thegradientof a scalar field and thedivergence

andcurlof a vector field. It is usual to define these operators from a strictly