Page 167 - Schaum's Outline of Theory and Problems of Advanced Calculus

158 VECTORS [CHAP. 7

from which we see that the magnitude of v is v ¼ ds=dt. Similarly,
2
d r
¼ a
dt 2 ð10Þ
is the acceleration with which the terminal point of r describes the curve. These concepts have important
applications in mechanics and differential geometry.
A primary objective of vector calculus is to express concepts in an intuitive and compact form.
Success is nowhere more apparent than in applications involving the partial differentiation of scalar and
vector fields. [Illustrations of such fields include implicit surface representation, fx; y; zðx; yÞ¼ 0, the
electromagnetic potential function ðx; y; zÞ, and the electromagnetic vector field Fðx; y; zÞ.] To give
mathematics the capability of addressing theories involving such functions, William Rowen Hamilton
and others of the nineteenth century introduced derivative concepts called gradient, divergence, and curl,
and then developed an analytic structure around them.
An intuitive understanding of these entities begins with examination of the differential of a scalar
field, i.e.,
@ @ @
dz
@x @y @z
d ¼ dx þ dy þ
Now suppose the function is constant on a surface S and that C; x ¼ f 1 ðtÞ; y ¼ f ðtÞ; z ¼ f 3 ðtÞ is a
2
dr dx dy dz
curve on S. Atany point of this curve ¼ i þ j þ k lies in the tangent plane to the surface.
dt dt dt dt
Since this statement is true for every surface curve through a given point, the differential dr spans the
@ @ @
tangent plane. Thus, the triple , , represents a vector perpendicualr to S. With this special
@x @y @z
geometric characteristic in mind we define
@ @ @
k
@x @y @z
r ¼ i þ j þ
to be the gradient of the scalar field .
Furthermore, we give the symbol r a special significance by naming it del.

EXAMPLE 1. If ðx; y; zÞ¼ 0isanimplicity defined surface, then, because the function always has the value zero
for points on it, the condition of constancy is satisfied and r is normal to the surface at any of its points. This
allows us to form an equation for the tangent plane to the surface at any one of its points. See Problem 7.36.

EXAMPLE 2. For certain purposes, surfaces on which is constant are called level surfaces. Inmeteorology,
surfaces of equal temperature or of equal atmospheric pressure fall into this category. From the previous devel-
opment, we see that r is perpendicular to the level surface at any one of its points and hence has the direction of
maximum change at that point.
The introduction of the vector operator r and the interaction of it with the multiplicative properties
of dot and cross come to mind. Indeed, this line of thought does lead to new concepts called divergence
and curl.A summary follows.

GRADIENT, DIVERGENCE, AND CURL
Consider the vector operator r (del)defined by
@ @ @
r i þ j þ k ð11Þ
@x @y @z
Then if ðx; y; zÞ and Aðx; y; zÞ have continuous first partial derivatives in a region (a condition which is
in many cases stronger than necessary), we can define the following.