Understanding Engineering Mathematics

dzis called thetotal differentialofz.Thedx,dy,dzare not actually numerical quantities,
but simply symbolise quantities that can be taken as small as we wish, but never zero.
They really only serve to define derivatives. Thus, ifx,y, are both functions of a parameter
t(e.g. in dynamicst=time) we can formally ‘divide bydt’ and write:

dz

dt

=

∂z

∂x

dx

dt

+

∂z

∂y

dy

dt

and this gives the total rate of change ofzwithtgiven the rate of change ofxandywith
t.dz/dtis called thetotal derivative ofzwith respect tot. Both the formula fordzand
dz/dtextend in an obvious way to functions of greater than two variables, although the
geometrical significance is not so easy to visualize.

Problem 16.4

Finddzat the point (1, 2, 5) forz=x^2 Yy^2.

Essentially, all we need are the first partial derivatives. We have

∂z

∂x

= 2 x,

∂z

∂y

= 2 y

The total differential is therefore given by

dz=

∂z

∂x

dx+

∂z

∂y

dy

= 2 xdx+ 2 ydy

Soat(1,2,5)thisgives

dz= 2 ( 1 )dx+ 2 ( 2 )dy= 2 dx+ 4 dy

This is a formal relation betweendx,dy,dz, which can for example be used to evaluate
the total derivative ofzat the point (1, 2, 5) as

dz

dt

= 2

dx

dt

+ 4

dy

dt

This gives us the rate of change ofzat (1, 2, 5) in terms of the rates of change ofxand
yat that point. However, the above relation between differentials can also be regarded as
an approximate relation between small incrementsδx,δy,δzto give

δz 2 δx+ 4 δy

This gives us the approximate change inzat (1, 2, 5) ifxandyare changed by small
amountsδx,δyrespectively.
As an example of the use of the total differential in approximations consider the problem
of finding the percentage error in functions of the form

z=xαyβ

due to given percentage errors inxandy.