Page 125 - Schaum's Outline of Theory and Problems of Advanced Calculus

Partial Derivatives

FUNCTIONS OF TWO OR MORE VARIABLES
The definition of a function was given in Chapter 3 (page 39). For us the distinction for functions of
two or more variables is that the domain is a set of n-tuples of numbers. The range remains one
dimensional and is referred to an interval of numbers. If n ¼ 2, the domain is pictured as a two-
dimensional region. The region is referred to a rectangular Cartesian coordinate system described
through number pairs ðx; yÞ, and the range variable is usually denoted by z. The domain variables are
independent while the range variable is dependent.
We use the notation f ðx; yÞ, Fðx; yÞ, etc., to denote the value of the function at ðx; yÞ and write
z ¼ f ðx; yÞ, z ¼ Fðx; yÞ, etc. We shall also sometimes use the notation z ¼ zðx; yÞ although it should be
understood that in this case z is used in two senses, namely as a function and as a variable.

3
2
3
2
EXAMPLE. If f ðx; yÞ¼ x þ 2y ,then f ð3; 1Þ¼ ð3Þ þ 2ð 1Þ ¼ 7:
The concept is easily extended. Thus w ¼ Fðx; y; zÞ denotes the value of a function at ðx; y; zÞ [a
point in three-dimensional space], etc.
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
EXAMPLE. If z ¼ 1 ðx þ y Þ,the domain for which z is real consists of the set of points ðx; yÞ such that
2
2
x þ y @ 1, i.e., the set of points inside and on a circle in the xy plane having center at ð0; 0Þ and radius 1.
THREE-DIMENSIONAL RECTANGULAR COORDINATE SYSTEMS
A three-dimensional rectangular coordinate system, as referred to in the previous paragraph,
obtained by constructing three mutually perpendicular axes (the x-, y-, and z-axes) intersecting in
point O (the origin). It forms a natural extension of the usual xy plane for representing functions of
two variables graphically. A point in three dimensions is represented by the triplet ðx; y; zÞ called
coordinates of the point. In this coordinate system z ¼ f ðx; yÞ [or Fðx; y; zÞ¼ 0] represents a surface,
in general.

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
EXAMPLE. The set of points ðx; y; zÞ such that z ¼ 1 ðx þ y Þ comprises the surface of a hemisphere of radius
1 and center at ð0; 0; 0Þ.
For functions of more than two variables such geometric interpretation fails, although the termi-
nology is still employed. For example, ðx; y; z; wÞ is a point in four-dimensional space, and w ¼ f ðx; y; zÞ
2
2
2
2
2
[or Fðx; y; z; wÞ¼ 0] represents a hypersurface in four dimensions; thus x þ y þ z þ w ¼ a represents
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
a ðx þ y þ z Þ,
a hypersphere in four dimensions with radius a > 0 and center at ð0; 0; 0; 0Þ. w ¼
2 2 2 2
x þ y þ z @ a describes a function generated from the hypersphere.
116
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