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# 15 - Contents CHAPTER 14 Multiple Integrals Double...

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CHAPTER 14 14.1 14.2 14.3 14.4 CHAPTER 15 15.1 15.2 15.3 15.4 15.5 15.6 CHAPTER 16 16.1 16.2 16.3 Contents Multiple Integrals Double Integrals Changing to Better Coordinates Triple Integrals Cylindrical and Spherical Coordinates Vector Calculus Vector Fields Line Integrals Green's Theorem Surface Integrals The Divergence Theorem Stokes' Theorem and the Curl of F Mathematics after Calculus Linear Algebra Differential Equations Discrete Mathematics Study Guide For Chapter 1 Answers to Odd-Numbered Problems Index Table of Integrals

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C H A P T E R 15 Vector Calculus Chapter 14 introduced double and triple integrals. We went from dx to jj dx dy and JIJ dx dy dz. All those integrals add up small pieces, and the limit gives area or volume or mass. What could be more natural than that? I regret to say, after the success of those multiple integrals, that something is missing. It is even more regrettable that we didn't notice it. The missing piece is nothing less than the Fundamental Theorem of Calculus. The double integral 11 dx dy equals the area. To compute it, we did not use an antiderivative of 1. At least not consciously. The method was almost trial and error, and the hard part was to find the limits of integration. This chapter goes deeper, to show how the step from a double integral to a single integral is really a new form of the Fundamental Theorem-when it is done right. Two new ideas are needed early, one pleasant and one not. You will like vector fields. You may not think so highly of line integrals. Those are ordinary single integrals like J v(x)dx, but they go along curves instead of straight lines. The nice step dx becomes the confusing step ds. Where Jdx equals the length of the interval, J ds is the length of the curve. The point is that regions are enclosed by curves, and we have to integrate along them. The Fundamental Theorem in its two-dimensional form (Green's Theorem) connects a double integral over the region to a single integral along its boundary curve. The great applications are in science and engineering, where vector fields are so natural. But there are changes in the language. Instead of an antiderivative, we speak about a potential function. Instead of the derivative, we take the "divergence" and "curl." Instead of area, we compute flux and circulation and work. Examples come first. - 1 Fields Vector 15.1 For an ordinary scalar function, the input is a number x and the output is a number f(x). For a vector field (or vector function), the input is a point (x, y) and the output is a two-dimensional vector F(x, y). There is a "field" of vectors, one at every point. 549
15 Vector Calculus In three dimensions the input point is (x, y, z) and the output vector F has three components. DEFINITION Let R be a region in the xy plane. A vectorfield F assigns to every point (x, y) in R a vector F(x, y) with two components: F(x, y) = M(x, y)i + N(x, y)j.

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