Mathematical Tools for Physics

8—Multivariable Calculus 210

and read (sort of) that the infinitesimal change in the functionfis 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?
Itis possible 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∆fgoes to zero so fast that even after you’ve divided byhthe difference goes to zero.
The usual notation is to use the symboldxinstead ofhand to call the functiondfinstead* ofg.

df(x,dx) =f′(x)dx has the property that

1

dx

∣

∣f(x+dx)−f(x)−df(x,dx)

∣

∣−→ 0 asdx→ 0

(4)

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 smalldfprovides a very
good approximation to the increment∆finf.

Differentials in Several Variables
The analog of Eq. ( 3 ) for several variables is

df=df(x,y,dx,dy) =

(

∂f

∂x

)

y

dx+

(

∂f

∂y

)

x

dy (5)

Roughly speaking, near a point in thex-yplane, the value of the functionfchanges as a linear function of the
coordinates as you move a (little) distance away. This functiondfdescribes this change to high accuracy.

• Who says that a variable in algebra must be only a single letter? You would never write a computer program
  that way.dFred^2

/

dFred= 2Fredis perfect sensible.