Advanced Calculus fi..

76 Advanced Calculus, Fifth EditionBoundaryFigure 2.1Set concepts.The other concepts, such as open set and closed set, are defined in terms of neighborhoodsas before.The definitions can also be adapted to the case of one dimension. A neighborhoodof a point xl of the x axis is an interval: xl - 6 < x < xl + 6. A dornain on the x axisis one of the following four types of sets: (1) an open interval: a < .r < b; (2) aninfinite open interval: a < x; (3) an infinite open interval: x < b; (4) the entire x axis.A bounded closed region on the x axis is a closed interval: a 5 x 5 b.Occasionally it is convenient to use the term neighborhood of a point to meanan open set containing the point..2.3 FUNCTIONAL NOTATIONAND LEVEL SURFACESLEVEL CURVESMost of the functions to be considered will be defined in a domain or, occasionally,in a closed region. The notation "z = f (x, y) in domain D" will mean that z isgiven as a function of x and y for all points in a domain D of the xy-plane. Thevariables x and y are called independent variables, while z is dependent. Similarly,one writes: "u = f (x, y, z). in domain D" or "w = f (x, y, z, u) in domain D"for functions of more than two variables. As with functions of one variable, thefunctional notation also serves to indicate corresponding values for a given function.in the domainThus if z = f (.r. y) is defined by the equation z = Jw.x2+y' < l,then f(O.O)= 1, f(j, ;)=&,andsoon.The functional relationship z = f (x, y) is sometimes written thus: z = z(x, y).,For functions of three or more variables one writes, similarly, u = u(x, y, z), w =W(X, y. Z. u), y = y(xl ..., ~ n ) .A function of two variables can be represented graphically by a surface in3-dimensional space, as shown in Fig. 2.2. For functions of three or more variablesthis representation is not available.Another method for representing functions of two variables is that of level curvesor contour lines. This is the method used in making a contour map or topographicalmap. One plots the loci,