Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/220

In the same way, if

$u=f(x,\ y,\ z),$

and $x$, $y$, $z$ are all functions of $t$, we get

(52)	${\frac {du}{dt}}={\frac {\partial u}{\partial x}}{\frac {dx}{dt}}+{\frac {\partial u}{\partial y}}{\frac {dy}{dt}}+{\frac {\partial u}{\partial z}}{\frac {dz}{dt}},$

and so on for any number of variables.^[1]

In (51) we may suppose $t=x$; then $y$ is a function of $x$, and $u$ is really a function of the one variable $x$, giving

(53)	${\frac {du}{dx}}={\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial y}}\cdot {\frac {dy}{dx}}.$

In the same way, from (52) we have

(54)	${\frac {du}{dx}}={\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial y}}\cdot {\frac {dy}{dx}}+{\frac {\partial u}{\partial z}}\cdot {\frac {dz}{dx}}.$

The student should observe that ${\tfrac {\partial u}{\partial x}}$ and ${\tfrac {du}{dx}}$ have quite different meamngs. The partial derivative ${\tfrac {\partial u}{\partial x}}$ is formed on the supposition that the particular variable x alone varies, while

${\frac {du}{dx}}=\lim _{\Delta x\to 0}\left({\frac {\Delta u}{\Delta x}}\right),$

where $\Delta u$ is the total increment of $u$ caused by changes in all the variables, these increments being due to the change $\Delta x$ in the independent variable. In contradistinction to partial derivatives, ${\tfrac {du}{dt}},{\tfrac {du}{dx}}$ are called total derivatives with respect to $t$ and $x$ respectively.^[2]

1. This is really only a special case of a general theorem which may be stated as follows: If $u$ is a function of the independent variables $x$, $y$, $z$,..., each of these in turn being a function of the independent variables $r$, $s$, $t$, ..., then (with certain assumptions as to continuity)
   ${\frac {\partial u}{\partial r}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}}+{\frac {\partial u}{\partial z}}{\frac {\partial z}{\partial r}}+\cdots ,$

   and similar expressions hold for ${\tfrac {\partial u}{\partial s}},{\tfrac {\partial u}{\partial t}}$ etc.

2. It should be observed that ${\tfrac {\partial u}{\partial x}}$ has a perfectly definite value for any point $(x,y)$, while ${\tfrac {du}{dx}}$ depends not only on the point $(x,y)$, but also on the particular direction chosen to reach that point. Hence

   ${\tfrac {\partial u}{\partial x}}$ is called a point function; while

   ${\tfrac {du}{dx}}$ is not called a point function unless it is agreed to approach the point from some particular direction.