The Partial Derivative

The partial derivative is what we typically use when we have a function of multiple coordinates. Suppose we have $f(x,y,z)$ , but wish to see how this function varies when we vary only $x$ , holding the other variables constant. This defines the partial derivative:

$$\partialdiv{f}{x} = \lim_{\Delta t \to 0} \frac{f(x + \Delta x,y,z) - f(x,y,z)}{\Delta x}$$

Note that this is just taking the ordinary scalar derivative, while treating the other variables as constants. Indeed, our scalar derivative above is also a partial derivative in the case where there are no other variables!

Forming the total differential, however, now requires us to consider what happens when we vary all three coordinates:

$$df = \left(\partialdiv{f}{x}\right) dx + \left(\partialdiv{f}{y}\right) dy + \left(\partialdiv{f}{z}\right) dz$$

These are not necessarily spatial variations - we could throw time in there as well, but for the moment we will consider time an independent variable that we need consider only via the chain rule. We can write this as a dot product:

$$df = \left\{\left(\partialdiv{f}{x}\right) \hx + \left(\partialdi... ...tialdiv{f}{z}\right) \hz \right\} \cdot \left\{ dx\hx + dy\hy + dz \hz\right\}$$

which we write as:

$$df = (\grad f) \cdot d\vell$$

where we have implicitly defined $\grad f$ and $d\vell$ .