8—Multivariable Calculus 183∆x
Is this result reasonable? Look at what happens toywhen you changexby ∆y
a little bit. Constantris a circle, and ifφputs the position over near the right
side (ten or twenty degrees), a little change inxcauses a big change inyas shown
by the rectangle. As drawn,∆y/∆xis big and negative, sort of like the (negative)
cotangent ofφas in Eq. (8.8).
8.3 Differentials
For a function of a single variable you can write
df=
df
dx
dx (8.11)
and read (sort of) that the infinitesimal change in the functionf is the slope times the infinitesimal
change inx. Does this really make any sense? What is an infinitesimal change? Is it zero? Isdxa
number or isn’t it? What’s going on?
Itispossible to translate this intuitive idea into something fairly simple and that makes perfectly
good sense. Once you understand what it really means you’ll be able to use the intuitive idea and its
notation with more security.
Letgbe a function oftwovariables,xandh.
g(x,h) =
df(x)
dx
h has the property that
1
h
∣
∣f(x+h)−f(x)−g(x,h)
∣
∣−→ 0 ash→ 0
That is, the functiong(x,h)approximatesvery wellthe change infas you go fromxtox+h. The
difference betweengand∆f=f(x+h)−f(x)goes to zero so fast that even after you’ve divided by
hthe difference goes to zero.
The usual notation is to use the symboldxinstead ofhand to call the functiondfinstead* of
g.
df(x,dx) =f′(x)dx has the property that
1
dx
∣
∣f(x+dx)−f(x)−df(x,dx)
∣
∣−→ 0 asdx→ 0
(8.12)
In this languagedxis just another variable that can go from−∞to+∞anddf is just a specified
function of two variables. The point is that this function is useful because when the variabledxis small
dfprovides a very good approximation to the increment∆finf.
What is the volume of the peel on an orange? The volume of a sphere isV = 4πr^3 / 3 , so its
differential isdV= 4πr^2 dr. If the radius of the orange is 3 cm and the thickness of the peel is 2 mm,
the volume of the peel is
dV= 4πr^2 dr= 4π(3cm)^2 (0. 2 cm) = 23cm^3
The whole volume of the orange is^43 π(3cm)^3 = 113cm^3 , so this peel is about 20% of the volume.
Differentials in Several Variables
The analog of Eq. (8.11) for several variables is
df=df(x,y,dx,dy) =
(
∂f
∂x
)
ydx+
(
∂f
∂y
)
xdy (8.13)
* Who says that a variable in algebra must be a single letter? You would never write a computer