PARTIAL DERIVATIVES AND INTEGRALS, Study notes of Mathematics

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NCTIONS OF SE 2 FURIABLES VERAL 9.1 Introduction This chapter is devoted to a study of functions depending on more than one independent variable. A real function z = f(z, y) of two independent variables x and y, can be thought to represent a surface in the three-dimensional space referred to a set of co- ordinate axes X,Y, Z. A simple example of a function of two independent variables x and y is z = zy, which represents the area of a rectangle whose sides are x and y. Continuity of a function of two variables: Definition: A function z = f(x,y) is said to be continuous at the point (9, yo) provided that a small change in the values of x and y produces a corresponding (small) change in the value of z. More precisely, if the value of z = f(x,y) at (xo, yo) is zo, then the continuity of the function at the point (xo, yo) means that lim f(z,y) = f (xo, yo) = 20 t+20,yyo If a so function is continuous at all points of some region R in the