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.

Ninth Chapter.
Functions of more than one independent variable.
51. When the value of a variable S is determined by the values
of two independent variables x and y in such a way, that to each
value of x in the interval from a to b and to each value of y in the
interval from a to 3 belong one or more values of a, then z is said to be
a function of the two independent variables x and y. Here we can
also classify functions after the nature of their analytical expression
into algebraic and transcendental, and the form in which the function
is presented may be explicit: = f(x, y), or implicit: f(x, y, a)
=, or again it may be brought about by two variable parameters:
x = c9(u, v), y = (ut, v), = x%(u, v).
The total course of the function is exhibited to intuition by the aid
of a system of Cartesian coordinates in space - most simply by a
rectangular one - each system of values x and y is represented by
a point in the plane of x y, and from this the corresponding value
of S is erected perpendicularly to the plane, towards one side or the
other according as it is positive or negative. The extremity of this
perpendicular represents the simultaneous system of values, x, y, z. The
interval from x =a to b, y = a to 3 determines in the plane xy a
rectangle, the domain, for which the function is defined, and the
points constructed lie above and below this. If x and y go through
all values from - co to + oo, these points spread over the entire plane.
A general view of the distribution of the points is arrived at, by beginning
with a fixed value of one variable ex. gr. x == a, and giving to y
different values between a and j; connecting the points thus constructed,
a polygon arises in space, of which the right line x = a is the
projection in the plane xy. As the value x is altered, different polygons
are obtained for these values of y; if we conceive the points for
which y has the same value to be connected, there arises a net whose
quadrangular meshes are more and more diminished by interpolating
further points and such that it can have as its limit a determinate
surface. This surface is accordingly the complete image of the function,
its intersections with planes parallel to those of yz or zx art curves
which form the limits of the polygons first constructed.