DIFFERENTIAL LENGTH AREA AND VOLUME Differential Elements In Length

differential length, area, and volume differential elements in length, area, and volume are useful in vector calculus. they are defined

DIFFERENTIAL LENGTH, AREA, AND VOLUME
Differential elements in length, area, and volume are useful in vector
calculus. They are
defined in the Cartesian, cylindrical, and spherical coordinate
systems.
A. Cartesian Coordinates
From Figure 3.1, we notice that
(1) Differential displacement is given by
dL= dx ax + dy ay + dz az



These differential elements are very important as they will be
referred to again and
again throughout the book. The student is encouraged not to memorize
them, however, but
to learn to derive them from Figure 3.1. Notice from eqs. (3.1) to
(3.3) that d\ and dS are
vectors whereas dv is a scalar. Observe from Figure 3.1 that if we
move from point P to Q
(or Q to P), for example, d\ = dy ay because we are moving in the
y-direction and if we
move from Q to S (or S to Q), dL = dy ay + dz az because we have to
move dy along y, dz
along z, and dx = 0 (no movement along x). Similarly, to move from D
to Q would mean
that dl = dxax + dyay + dz az.
The way dS is denned is important. The differential surface (or area)
element dS may
generally be defined as
dS = dSan (3.4)
where dS is the area of the surface element and an is a unit vector
normal to the surface dS
(and directed away from the volume if dS is part of the surface
describing a volume). If we
consider surface ABCD in Figure 3.1, for example, dS = dydzax whereas
for surface
PQRS, dS = -dy dz ax because an = -ax is normal to PQRS.
What we have to remember at all times about differential elements is
d\ and how to get dS and dv from it. Once d\ is remembered, dS and dv
can easily be found. For example,
dS along ax can be obtained from d\ in eq. (3.1) by multiplying the
components of d\ along a^, and az; that is, dy dz ax. Similarly, dS
along az is the product of the components of d\
along ax and ay; that is dx dy az. Also, dv can be obtained from dL as
the product of the three
components of dl; that is, dx dy dz. The idea developed here for
Cartesian coordinates will
now be extended to other coordinate systems.
B. Cylindrical Coordinates
Notice from Figure 3.3 that in cylindrical coordinates, differential
elements can be found as follows:


As mentioned in the previous section on Cartesian coordinates, we only
need to remember dL; dS and dv can easily be obtained from dl. For
example, dS along az is the
product of the components of dl along ap and a^; that is, dp p dø az.
Also dv is the product of the three components of