Mathematical Tools for Physics

8—Multivariable Calculus 214

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 coordinatesdxand
dy. 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. ( 3 ), describing the differential.

dy=

df(x)

dx

dx

y

x

dy

dx

For two variables, the picture parallels this one. At a point(x,y,z) = (x,y,f(x,y))find theplanethat
best approximates the function in the immediate neighborhood of that point. Set up a new coordinate system
with origin at this(x,y,z)and call the new coordinatesdx,dy, anddz. The equation for a plane that passes
through this origin isαdx+β dy+γ dz= 0, and for this best approximating plane, the equation is nothing more
than the equation for the differential, Eq. ( 5 ).

dz=

(

∂f(x,y)

∂x

)

y

dx+

(

∂f(x,y)

∂y

)

x

dy

dx

dy

dz

The picture is a bit harder to draw, but with a little practice you can do it. (I didn’t say that I could.)