微积分英文课件：chapter14 Partial Derivatives.ppt

So far we have dealt with the calculus of functions of a single variable. But, in the real world, physical quantities often dependon two or more variables, so in this chapter we turn our attention to functions of several variables and extend the basic ideas of differential calculus to such functions. 14.1 Functions of Several Variables In this section we study functions of two or more variables from four points of view: Verbally (by a description in words) Numerically (by a table of values) Algebraically (by an explicit formula) Visually (by a graph or level curves) Functions of Two Variables Definition A function of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number denoted by f(x,y). The set D is the domain of f and its range is the set of values that f takes on, that is , {f(x,y)| (x,y) D}. We often write z= f(x,y) to make explicit the value taken on by f at the general point (x,y) . The variables x and y are independent variables and z is the dependent variable. Example 1 Find the domains of the following functions and evaluate f(3,2). (a) (b) Solution (a) (b) Graphs Another way of visualizing the behavior of a function of two variables is to consider its graph. Definition If f is a function of two variables with domain D, then the graph of f is the set of all points (x,y,z) in such that z=f(x,y) and (x,y) is in D. Functions of Three or More Variables A function of three variables, f, is a rule that assigns to each ordered triple (x, y, z) in a domain D a unique real number denoted by f(x,y,z). A function of n variables, f, is a rule that assigns a number to an n-tuple of real number. We denoted by the set of all such n-tuples. 14.2 Limits and Continuity Definition Let f be a function of two variables whose domain D includes points