Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 77Figure 2.2Function of two variables.Figure 2.3Level curvesfor various choices of the constants c,, c2. . . . ; each locus f (x, y) = c is called alevel curve of f (x. y); it may actually be formed of several distinct curves. This isillustrated in Fig. 2.3, in which f (x, y) = x2y + x2 + 2y2; the value of c is shownon each curve. The level curves often provide a better understanding of the functionthan a sketch of the surface z = f (x , y ).The method just described is available in principle for functions of three variables;here one draws the level surfaces: f (x, y , z) = cl , f (x, y , z) = c2, . . . forappropriate choices of c,. c2, . . . . The surfaces of constant gravitational potential(approximately spheres) about the earth illustrate this. The surfaces of constant temperatureor pressure are of importance in meteorology.One can also represent a function of three variables with the aid of level curves;if, for example, z is fixed, then f (x, y, z) becomes a function of x and y and canbe represented by its level curves in an xy-plane as earlier. If this is done for severalvalues of z, one obtains a corresponding number of diagrams that together representthe function. This is common practice in meteorology, in which lines of constantpressure (isobars) are plotted for various altitudes.For functions of four or more variables the loci, f = const, are "hypersurfaces"in a space of four or more dimensions. These level hypersurfaces are mainly oftheoretical interest. To represent the function graphically, one is forced to fix one or