An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

92

Functions of more than one independent variable. Bk. I. ch. IX.

___xx, Y+Ay) - fx_-__Ax_, y) f(x,y+Ay) —f(xy) 7
abs [r(x x + ~ i y+ ~y) - (. + a, y) f, t n y -j f(X,) ] must be < 6o,
Ay A/ ' '
therefore for A y -, a f(x +A x,y) f(x, ) must also be < hk.
ay ay
Accordingly the result is:
Provided at the point x, y, at which f is continuous, the quotient
of differences:
f(x +- Ax, y + Ay) - f(x, y -- Ay
Ax
is a uniformly continuous function of Ax and y, then xf is a continuous
a /'
function of the variable y, -y a continuous function of the variable x,
and we have bfr all values of dy: dx the Total Differential Quotient
with regard to x equal to
dz af f dy
dlx -- x y ydx'
or written more symmetrically, the Total Differential equal to
d - - dx + - fdy,
)x ay
that is, equal to the sum of the partial differentials.
The differential equation contrasts by its symmetry with the equation of differential quotients; but since there are vanishing (infinitely
small) quantities on both its sides, it derives a meaning only from the
fact that a quotient equation can always be formed from it.
In most cases of calculation it is enough to replace the condition
of this Theorem of the Total Differential by the narrower one: When
the progressive and regressive differential quotient 'f is a continuous
function of both variables x and y in the neighbourhood of a point,
and af has a determinate value, we have also
dz f - dx + a dy.
CYx y
For then we can replace the first quotient in the equation
f(x +/ x, y +A -- f(x, y) f(x+ Ax, y + A ) - f(x, y + A y)
Ax Ax
+ f(x,y+Ay)-f(x,y) A
Ay Ax
by the mean value:
f/'(x + OAx, y j Ay)
w bf3x
which becomes for Ax ==, Ay = 0.
ax