Understanding Engineering Mathematics

z

x

(x,y)

(x+dx,y)

(x+dx, y+dy)

0 y

dz 2 = dy

dz 1 =∂∂xzdx

∂z

∂x

Figure 16.6The total derivative.

This is exact. However, if we neglectδxδythen we get anapproximation:

δzyδx+xδy

Notice that sincezx=yandzy=xthis may be written

δzzxδx+zyδy

(iii)δz=(x+δx)^2 +(y+δy)^2 −x^2 −y^2

= 2 xδx+ 2 yδy+(δx)^2 +(δy)^2

 2 xδx+ 2 yδy

if we neglect theδproducts. Again, notice that this is

δzzxδx+zyδy

These examples illustrate the result

δzzxδx+zyδy

stated above.
Note that this is anapproximateformula betweenincrementsδx,δy,δz.Itisuseful
todefine differentialsdx,dy,dzwhich satisfy:

dz=

∂z

∂x

dx+

∂z

∂y

dy