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# 16 - 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 16 Mathematics after Calculus I would like this book to do more than help you pass calculus. (I hope it does that too.) After calculus you will have choices- Which mathematics course to take next?- and these pages aim to serve as a guide. Part of the answer depends on where you are going-toward engineering or management or teaching or science or another career where mathematics plays a part. The rest of the answer depends on where the courses are going. This chapter can be a useful reference, to give a clearer idea than course titles can do: Linear Algebra Differential Equations Discrete Mathematics Advanced Calculus (with Fourier Series) Numerical Methods Statistics Pure mathematics is often divided into analysis and algebra and geometry. Those parts come together in the "mathematical way of thinking9'-a mixture of logic and ideas. It is a deep and creative subject-here we make a start. Two main courses after calculus are linear algebra and differential equations. I hope you can take both. To help you later, Sections 16.1 and 16.2 organize them by examples. First a few words to compare and contrast those two subjects. Linear algebra is about systems of equations. There are n variables to solve for. A change in one affects the others. They can be prices or velocities or currents or concentrations-outputs from any model with interconnected parts. Linear algebra makes only one assumption-the model must be linear. A change in one variable produces proportional changes in all variables. Practically every subject begins that way. (When it becomes nonlinear, we solve by a sequence of linear equations. Linear programming is nonlinear because we require x >, 0.) Elsewhere J wrote that "Linear algebra has become as basic and as applicable as calculus, and fortunately it is easier." I recommend taking it. A differential equation is continuous (from calculus), where a matrix equation is discrete (from algebra). The rate dyldt is determined by the present state y-which changes by following that rule. Section 16.2 solves y' = cy + s(t) for economics and life sciences, and y" + by' + cy = f(t) for physics and engineering. Please keep it and refer to it.
16 Mathematics after Calculus A third key direction is discrete mathematics. Matrices are a part, networks and algorithms are a bigger part. Derivatives are not a part-this is closer to algebra. It is needed in computer science. Some people have a knack for counting the ways a computer can send ten messages in parallel-and for finding the fastest way.

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