Mathematical Tools for Physics

8—Multivariable Calculus 211

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 ( 4 ) is

df(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 (6)

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 thatdx
isanynumber 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 youdothink of it as the functiondf
divided by the variabledx, you get the right answer.

8.3 Chain Rule
If the coordinatesxandyare themselves functions of another variable, perhaps time, then how does does the
functionfvary as a function oft? Just use the idea of the differential to find out. At timetthe differentials of
xandyare, from Eq. ( 3 ),

dx=

dx

dt

dt, and dy=

dy

dt

dt

Notational confusion reminder: Thedxon the left is the function of two variables: dx(t,dt). Thedxin the
numerator on the right is a part of the notation for a derivative and isn’t defined independently ofdx/dt. But,