8—Multivariable Calculus 184Roughly 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. It bears the same relation to Eq. (8.4) that (8.11) bears to Eq. (8.3).
For example, take the functionf(x,y) =x^2 +y^2. At the point(x,y) = (1,2), the differential
is
df(1, 2 ,dx,dy) = (2x)
∣∣
∣∣
(1,2)dx+ (2y)
∣∣
∣∣
(1,2)dy= 2dx+ 4dy
so that
f(1. 01 , 1 .99)≈f(1,2) +df(1, 2 ,. 01 ,−.01) = 1^2 + 2^2 + 2(.01) + 4(−.01) = 4. 98
compared to the exact answer, 4. 9802.
The equation analogous to (8.12) isdf(x,y,dx,dy) has the property that
1
dr
∣
∣f(x+dx,y+dy)−f(x,y)−df(x,y,dx,dy)
∣
∣−→ 0 asdr→ 0 (8.14)
wheredr=
√
dx^2 +dy^2 is the distance to(x,y). It’s not that you will be able to do a lot more with
this precise definition than you could with the intuitive idea. You will however be able to work with a
better understanding of you’re actions. When you say that “dxis an infinitesimal” you can understand
that this means simply thatdxisanynumber but that the equations using it are useful only for very
small values of that number.
You can’t use this notation for everything as the notation for the derivative demonstrates. The
symbol “df/dx” does not mean to divide a function by a length; it refers to a well-defined limiting
process. This notation is however constructed so that it provides an intuitive guide, and even if youdo
think of it as the functiondfdivided by the variabledx, you get the right answer.
Why should such a thing as a differential exist? It’s essentially the first terms after the constant
in the power series representation of the original function: section2.5. But how to tell if such a series
works anyway? I’ve been notably cavalier about proofs. The answer is that there is a proper theorem
guaranteeing Eq. (8.14) works. It is that if both partial derivatives exist in the neighborhood of the
expansion point and if these derivatives are continuous there, then the differential exists and has the
value that I stated in Eq. (8.13). It has the properties stated in Eq. (8.14). For all this refer to one of
many advanced calculus texts, such as Apostol’s.*
8.4 Geometric Interpretation
For one variable, the picture of the differential is simple. Start with a graph of the function and at a
point(x,y) = (x,f(x)), find the straight line that best approximates the function in the immediate
neighborhood of that point. Now set up a new coordinate system with origin at this(x,y)and call the
new coordinatesdxanddy. In this coordinate system the straight line passes through the origin and
the slope is the derivativedf(x)/dx. The equation for the straight line is then Eq. (8.11), describing
the differential.
dy=
df(x)
dx
dx
* Mathematical Analysis, Addison-Wesley