Mathematical Tools for Physics

Multivariable Calculus

The world is not one-dimensional, and calculus doesn’t stop with a single independent variable. The ideas of
partial derivatives and multiple integrals are not too different from their single-variable counterparts, but some of
the details about manipulating them are not so obvious. Some are downright tricky.

8.1 Partial Derivatives
The basic idea of derivatives and of integrals in two, three, or more dimensions follows the same pattern as for
one dimension. They’re just more complicated.
The derivative of a function of one variable is defined as
df(x)
dx

= lim

∆x→ 0

f(x+ ∆x)−f(x)

∆x

(1)

You would think that the definition of a derivative of a function ofxandywould then be defined as

∂f(x,y)

∂x

= lim

∆x→ 0

f(x+ ∆x,y)−f(x,y)

∆x

(2)

and more-or-less it is. The∂notation instead ofdis a reminder that there are other coordinates floating around
that are temporarily being treated as constants.
In order to see why I used the phrase “more-or-less” I’ll take a very simple example:f(x,y) =y. Use the
preceding definition, and becauseyis being held constant, the derivative∂f/∂x= 0. What could be easier?
I don’t like these variables so I’ll switch to a different set of coordinates,x′andy′:

y′=x+y and x′=x

What is∂f/∂x′now?
f(x,y) =y=y′−x=y′−x′

Now the derivative off with respect tox′is− 1 , because I’m keeping the other coordinate fixed. Or is the
derivative still zero becausex′=xand I’m taking∂f/∂xand why should that change just because I’m using a
different coordinate system?

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