tailieunhanh - Ebook Math advanced calculus: Part 2

(BQ) Part 2 book "Math advanced calculus" has contents: The derivative of a vector valued function of a vector variable, nonlinear functions, transformation of integrals, line and surface integrals, infinite series,.and other contents. | The Derivative of a Vector-Valued Function of a Vector Variable 9 Definition of the Derivative We have seen in Chapter 3 that the derivative f a of a real-valued function 2 - E 2 c E of a real variable at an accumulation point a G 2 of 2 may be characterized as follows f is differentiable at a and f à is the derivative if and only if there exists a linear function la E - E with the property that for every 0 there is a 5 e 0 such that l - - ta x - a x - a for all X G Nẳ a n 3 and f à h la h for all h G E. See Theorem . In generalizing the concept of a derivative to vector-valued functions of a vector variable from 2 s E into Em we proceed in a straightforward manner replacing the absolute value by the norm and la E - E by a linear function ỉ E - E 1. However we shall restrict our discussion to interior points of 2 rather than consider the more general case where a G 2 is merely an accumulation point of 2 . Otherwise we would not obtain a unique derivative when n 1. Definition 86. J If f 2 - Ew 2 E and if a is an interior point of 2 then is differentiable at a if and only if there exists a linear function I E - Em with the property that for every 0 there is a ô è 0 where Nẵ e a 2 such that x -f a - l x - a fi x - all for all X G Nổ j a . The function is called the derivative of at a and is denoted by a . Note that a is for given a a linear function from E into Em that is a A k a ỉ a fc for all h k G E and a A Ấ a ỉ for all h G E and any Ấ G R. 312 Definition of the Derivative 313 Example 1 Let E - Em be defined by c Z x where c e Em and where E - Em is a linear function. Then for any a e E a I x - a - x - a II x - a - x - a II 0 for all X e E . This is a generalization of the well-known result that the derivative of E - E given by x a bx is z . Example 2 Let E2 - E be defined by i 2 2 and let a a2 . We propose to show that E2 - E is the linear function that is defined by a fl 2a1co1 co2 for all u e E2. We have x - fl f a x - a 2 - a - a2 - .