By Alberto Guzman

This textual content is acceptable for a one-semester direction in what's often referred to as advert vanced calculus of a number of variables. The strategy taken the following extends straight forward effects approximately derivatives and integrals of single-variable features to capabilities in several-variable Euclidean house. The trouble-free fabric within the unmarried- and several-variable case leads evidently to major complex theorems approximately func tions of a number of variables. within the first 3 chapters, differentiability and derivatives are outlined; prop erties of derivatives reducible to the scalar, real-valued case are mentioned; and effects from the vector case, vital to the theoretical improvement of curves and surfaces, are provided. the following 3 chapters continue analogously throughout the improvement of integration thought. Integrals and integrability are de fined; houses of integrals of scalar services are mentioned; and effects approximately scalar integrals of vector features are provided. the advance of those lat ter theorems, the vector-field theorems, brings jointly a couple of effects from different chapters and emphasizes the actual functions of the theory.

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**Example text**

In Example 2, show that the approximation at the end of (a) is actually an equality. 5. Is it possible to find h(x, y) such that ~~ = yexp(x 2), ~; = x sin2 y for all x,y? 6. Assume that f and g are vector functions defined near b and continuous at b. Suppose that in every neighborhood N(b, e), there exist x and y such that g(x) = f(y). Prove that g(b) = f(b). 2 Derivatives of Scalar Functions We have seen that derivatives of vector functions can be taken component by component. Consequently, much of our investigation of such derivatives reduces to the study of derivatives of real-valued functions.

Directional Derivatives and the Gradient ........ 1. We see in Example 2 that Theorem 1 belongs to a family of similar results. 2: 1) that where a vector-variable derivative exists, there the related scalar-variable derivatives exist, but not vice versa. If / is real, then the product /,(b)(u) = [~(b) aXI ... L(b)] [UI ... aXn Un]' is a 1 x 1 matrix, but we have identified it with its only entry, the real number )UI + ... + )u n . ; of the two vectors (if, ... ;) and (UI,··· Definition. The vector grad / , un) == (if, ...

It follows that gl(t)~gl(O) has a limit, and the limit is (f" (b)(u) )(u). ) We will abbreviate (f"(b)(u)(u) by, naturally enough, j"(b)(u)2. Theorem 2. Suppose f is differentiable near b, is twice differentiable at b, and f'(b) = O. (a) If f"(b) (v)2 > 0 (or < 0) for every v (respectively, strict maximum). "# 0, then f(b) is a strict minimum 48 2. Derivatives of Scalar Functions (b) If f" (b )(v)2 > 0 for some vectors v and < 0 for others, then b is a saddle point of f. Proof. (a) Let us assume that fl/(b)(v)2 > 0 for all v =F O.