Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

even be replaced by differences, whenever necessary, to obtain approximate

results. The widely used finite difference numerical method is based on this

simple principle.

Partial Differentials

Now consider a function that depends on two (or more) variables, such as
zz(x,y). This time the value of zdepends on both xand y. It is sometimes
desirable to examine the dependence of zon only one of the variables. This
is done by allowing one variable to change while holding the others constant
and observing the change in the function. The variation of z(x,y) with x
when yis held constant is called the partial derivativeof zwith respect to
x, and it is expressed as

(12–2)

This is illustrated in Fig. 12–3. The symbol  represents differential
changes, just like the symbol d. They differ in that the symbol drepresents
the totaldifferential change of a function and reflects the influence of all
variables, whereas represents the partialdifferential change due to the
variation of a single variable.
Note that the changes indicated by dand are identical for independent
variables, but not for dependent variables. For example, (x)ydxbut (z)y
dz. [In our case,dz(z)x(z)y.] Also note that the value of the par-
tial derivative (z/x)y, in general, is different at different yvalues.
To obtain a relation for the total differential change in z(x,y) for simulta-
neous changes in xand y, consider a small portion of the surface z(x,y)
shown in Fig. 12–4. When the independent variables xand ychange by
x
and
y, respectively, the dependent variable zchanges by
z, which can be
expressed as

Adding and subtracting z(x,y
y), we get

or

Taking the limits as
x→0 and
y→0 and using the definitions of partial
derivatives, we obtain

(12–3)

Equation 12–3 is the fundamental relation for the total differentialof a
dependent variable in terms of its partial derivatives with respect to the
independent variables. This relation can easily be extended to include more
independent variables.

dza

0 z

0 x

b

y

dxa

0 z

0 y

b

x

dy

¢z

z 1 x¢x, y¢y 2 z 1 x, y¢y 2

¢x

¢x

z 1 x, y¢y 2 z 1 x, y 2

¢y

¢y

¢zz 1 x¢x, y¢y 2 z 1 x, y¢y 2 z 1 x, y¢y 2 z 1 x, y 2

¢zz 1 x¢x, y¢y 2 z 1 x, y 2

a

0 z

0 x

b

y

¢limxS 0 ¬a

¢z

¢x

b

y

¢limxS 0 ¬

z 1 x¢x, y 2 z 1 x, y 2

¢x

Chapter 12 | 653

x y

z

z(x + ∆x, y + ∆y)

x + ∆x, y + ∆y

x, y + ∆y

x + ∆x, y

z(x, y)

FIGURE 12–4

Geometric representation of total

derivative dzfor a function z(x,y).

z

x

y

––∂z

( )∂x (^) y
FIGURE 12–3
Geometric representation of partial
derivative (z/x)y.