University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 2. Multivariable Function

2.1  Definition

a.  Let be a set at . A function is a given rule fitting each point of fits a one and only number . It is denoted or .

Set is the domain of the function.

b.  The set of all values is called the range of the function.

c.  The geometric description of a function of two-variables is a surface at (see III.8 for examples of such surfaces).

2.2  Level Curve and Level Surface

a.  Level curve: Curve at plane is called a level curve or a level set of function , fitting constant , if the value of the function on equals . That is, is the locus of points which hold .

The intersection between plane and surface is curve .

Figure 1 shows level curves of function , which are circles .

b.  Surface is called a level surface fitting constant , of function , if it is defined the equation .

Figure 2 shows level surfaces of function , when , , .

  Figure 1    Figure 2