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.

~ 53.

Theorem of the Total Differential.

93

Examples.
1. The function z =/ /x2 + y' is unique and continuous even at
the point x = 0, y= 0, but its first deriveds functions
_z x __z y
ex VX y2' / Vx2 ~Y
have there no determinate values; therefore the theorem of the total
differential does not apply in this case, in the absence of further conventions respecting the partial derived functions.
2. If we replace in the function z = (3x + 3) + y all values for
y =0 by the values 6x, the function so formed is continuous in the
neighbourhood of the point x= 1, y - 0; but af(' ) 6, a(') — 3
i. e. the partial differential quotient in regard to x is not a continuous
af(t, 0) at-j /+Ax, 0) -
function of y; in like manner af(y, ) af(i +- +; thereay - ' y
fore though the function is continuous at the point yet the theorem
of the total differential does not there hold for it,
3. The function: = x sin (4 tan-1' with the convention that
for all values of y including y = 0, whenever x = 0 we have always
also == 0; offers an example in which f is continuous in both
variables and f- a continuous function of y, without the theorem of
ax
the total differential holding. This function is continuous in the neighbourhood of x = 0, y = 0. Therefore
Ax sin 4 tan-1 — )y
af(o, ) Lim sill (t )
ax Ax
is now a continuous function of y. On the other hand
x sin (4 tan-1 Ly)
'v === Lim - == 4,
ay Ay
as long as x is different from 0; while f(o,0 = 0. The theorem of
ay
the total differential does not hold in this case.
The condition under which the theorem of the total differential
holds, is the condition that the function can be represented by a surface. Just as we say of a function of one variable: it can be represented at a point by a curve, when the lines joining this point to
neighbouring points converge to a fixed limiting position, so we say
of a function of two variables: it can be represented at a point as a
surface, when every plane, which can be drawn through that point
and through any two other points belonging to the function, converges
to the same fixed limiting position, in whatever manner these other
two points close up to the original one. It is a singularity in a
surface when it behaves at a point as a cone does at its vertex;