Page 140 - Foundations Of Differential Calculus

7

On the Differentiation of Functions
of Two or More Variables

208. If two or more variable quantities x, y, z are independent of each
other, it can happen that while one of the variables increases or decreases,
the other variables remain constant. Since we have supposed that there is
no connection between the variables, a change in one does not affect the
others. Neither do the differentials of the quantities y and z depend on the
differential of x, with the result that when x is increased by its differential
dx, the quantities y and z can either remain the same, or they can change in
any desired way. Hence, if the differential of x is dx, the differentials of the
remaining quantities, dy and dz, remain indeterminate and by our arbitrary
choice will be presumed to be either practically nothing or infinitely small
when compared to dx.
209. However, frequently the letters y and z are wont to signify functions
of x that are either unknown or whose relationship to x is not considered.
In this case the differentials dy and dz do have a certain relationship to dx.
Whether or not y and z depend on x, the method of differentiation that we
now consider is the same. We look for the differential of a function that is
formed in any way from the several variables x, y, and z that the function
receives when each variable x, y, z increases by its respective differential
dx, dy,or dz. In order to find this for the given function, for each of the
variables x, y, and z we write x + dx, y + dy, and z + dz, and from this
expression we subtract the given function. The remainder is the desired
differential. This should be clear from the nature of differentials.