Page 141 - Foundations Of Differential Calculus

124 7. On the Differentiation of Functions of Two or More Variables
210. Let X be a function of x, and let its differential, or increase, be
equal to Pdx, when x increases by dx. Then let Y be a function of y and
let its differential be equal to Qdy, which is the augmentation Y receives
when y is increased to y + dy. Finally, let Z be a function or z, and let its
differential be equal to Rdz. These differentials Pdx, Qdy, and Rdz can
be found from the nature of the functions X, Y , and Z by means of the
rules we have given above. Suppose the given function is X +Y +Z, which
is a function of the three variables x, y, and z, and its differential is equal
to Pdx + Qdy + Rdz. Whether these three differentials are homogeneous
or not need not concern us. Terms that contain powers of dx will vanish in
the presence of Pdx, as if the other members Qdy and Rdz were absent.
For a similar reason we neglect terms in the differentiation of the functions
Y and Z.

211. We keep the same description of X, Y , and Z and let the given
function be XY Z, which is a function of x, y, and z. We investigate the
differential of this function. If we replace x by x+dx, y by y +dy, and z by
z + dz, then X becomes X + Pdx, Y becomes Y + Qdy, and Z becomes
Z + Rdz, so that the given function XY Z becomes

(X + Pdx)(Y + Qdy)(Z + Rdz)
= XY Z + YZP dx + XZQ dy + XY R dz
+ ZPQdxdy + YPR dx dz + XQR dy dz + PQR dx dy dz.

Since dx, dy, and dz are infinitely small, whether they are mutually homo-
geneous or not, the last term will vanish in the presence of any one of the
preceding terms. Then the term ZPQdxdy will vanish in the presence of ei-
ther YZP dx or XZQ dy. For the same reason YPR dx dz and XQR dy dz
will vanish. When we subtract the given function, the remainder is the
differential

YZP dx + XZQ dy + XY R dz.

212. These examples of functions of three variables x, y, and z, to which
we could, if desired, add more, should be sufficient to show that for any
proposed function of three variables x, y, and z, howsoever these variables
may be combined, the differential will always have this same form pdx +
qdy +rdz. These functions p, q and r will be individual functions of either
all three variables x, y, and z or of two variables, or even of only one,
depending of how the given function is formed from the three variables and
constants. In a similar way, if the given function depends on four or more
variables, say x, y, z, and v, then its differential will have the form

pdx + qdy + rdz + sdv.