Section 14.3 Partial derivatives with two variables（14.3节和两个变量的偏导数）.pdfVIP

Section 14.3 (3/23/08) Partial derivatives with two variables Overview: In this section we begin our study of the calculus of functions with two variables. Their derivatives are called partial derivatives and are obtained by differentiating with respect to one variable while holding the other variable constant. We describe the geometric interpretations of partial derivatives, show how formulas for them can be found with differentiation formulas with one variable, and demonstrate how they can be estimated from tables and level curves. Topics: • Limits of functions with two variables • Continuity of functions with two variables • Partial derivatives • A geometric interpretation of partial derivatives • Estimating partial derivatives from tables • Estimating partial derivatives from level curves Limits of functions with two variables In studying functions of one variable we used one- and two-sided limits. We cannot talk of two-sided or one-sided limits of functions of two variables. Instead we find limits by studying the values of functions along paths, as in the next definition.† Definition 1 Suppose that the function z = f (x, y) is defined in a circle with its center at the point (x , y ), except possibly at the point (x , y ) itself. Then the limit of f (x, y) as (x, y) approaches (x , y ) 0 0 0 0 0 0 is L and we write lim f (x, y) = L (1) (x,y)→(x0 ,y0 ) if the number f (x, y) approaches L as (x, y) approaches (x , y ) along all paths tha