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Functions of Several Variables

Suppose we go beyond f(x) and talk about F(x,y,z) - e.g. a function of the exact position in space. This is just an example, of course; the abstract idea of a function of several variables can have "several" be as many as you like and "variables" be anything you choose. Another place where we encounter lots of functions of "several" variables is in THERMODYNAMICS, but for the time being we will focus our attention on the three spatial variables x (left-right), y (back-forth) and z (up-down).

How can we tackle derivatives of this function?

Partial Derivatives

Well, we do the obvious: we say, "Hold all the other variables fixed except [for instance] x and then treat F(x,y,z) as a function only of x, with y and z as fixed parameters." Then we know just how to define the derivative with respect to x. The short name for this derivative is the PARTIAL DERIVATIVE with respect to x, written symbolically

$${\partial F \over \partial x}$$

where the fact that there are other variables being held fixed is implied by the use of the symbol $\partial$ instead of just d.

Similarly for $${\partial F \over \partial y}$$ and $${\partial F \over \partial z}$$.