Partial Dervitives

FUNCTIONS OF TWO OR MORE VARIABLES
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 ًx; yق, 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 ًx; yق, Fًx; yق, etc., to denote the value of the function at ًx; yق and write z ¼ f ًx; yق, z ¼ Fًx; yق, etc. We shall also sometimes use the notation z ¼ zًx; yق although it should be understood that in this case z is used in two senses, namely as a function and as a variable.
EXAMPLE. If f ًx; yق ¼ x2 ‏ 2y3, then f ً3;_1ق ¼ ً3ق2 ‏ 2ً_1ق3 ¼ 7:
The concept is easily extended. Thus w ¼ Fًx; y; zق denotes the value of a function at [a point in three-dimensional space], etc., the domain for which z is real consists of the set of points ًx; yق such that x2 ‏ y2 @ 1, i.e., the set of points inside and on a circle in the xy plane having center at ً0; 0ق 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 ًx; y; zق called coordinates of the point. In this coordinate system z ¼ f ًx; yق [or Fًx; y; zق ¼ 0] represents a surface, in general.