Calculus and Linear Algebra. Volume 1: Vectors in the Plane and One-Variable Calculus | Date: 15 April 2011, 02:52
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This book is the first of a two-volume text on calculus and linear algebra that is intended to provide enough material for a freshman-sophomore course. The principal objective is the integration of linear algebra and calculus.
Chapter 0, an introductory chapter, is intended for reference and review. Part of it (or all of it) can be studied in depth in accordance with the background of the students.
Chapter 1 introduces vectors in the plane; the presentation relies heavily on plane geometry.
Chapter 2 reviews and develops the idea of a function and presents the concept of limit as the first stage in developing the calculus. The least upperbound axiom is introduced at the end and it is used to prove the main theorems.
Chapter 3 is a systematic development of differential calculus, with some applications to geometry and to the sciences. The derivatives of sin x, cos x, ln x, and exp x are given with intuitive justification and are used often; rigorous proofs are deferred until Chapter 5. Accordingly, they are available for early use by students in engineering and physics. A student who has completed this chapter has a solid grounding in differential calculus. Vectors appear at several points, especially for curves in parametric form.
Chapter 4 is a similarly complete treatment of integral calculus. The introductory sections explain the ideas of definite and indefinite integrals. Then the main techniques for finding indefinite integrals are developed. Finally, the third and longest part is devoted to the definite integral, with some applications, especially to area and arc length. The definition of the integral is based on upper and lower estimates and leads to a simple proof of the main theorems for integrals of continuous functions. The Riemann integral is also defined and is shown for continuous functions to be equivalent to the definite integral. Throughout there is emphasis on computational procedures and computers.
Chapter 5 is a brief, rigorous treatment of the trigonometric, logarithmic, exponential, and related functions. This chapter can be omitted without affecting continuity, since all the main results are given elsewhere in the text.
Chapter 6 provides further applications of differential calculus—tests for maxima and minima, graphs of plane curves in rectangular and polar coordinates, Newton's method, Taylor's formula, and indeterminate forms. Much of this chapter can be studied immediately following Chapter 3 , if so desired, since integration appears only occasionally.
Chapter 7 presents applications of the definite integral to area in rectangular and polar coordinates, volumes and surface area of solids of revolution, moments of mass distributions, and centroids. Line integrals are introduced at several points. The role of integration in the physical sciences is well illustrated. There are discussions of improper integrals and the trapezoidal and Simpson's rules. Six sections are devoted to differential equations; they are included here: (1) because their development is a natural extension of
earlier theory and (2) to make them available at an early stage for students of engineering and physics.
Chapter 8 is concerned with infinite sequences and series, convergence tests, rearrangement and product theorems, power series, Taylor's formula and series, and Fourier series. Some reference is made to complex series. To a considerable extent, this chapter is independent of the others and can be studied earlier or later.
Numerous problems sets are provided throughout. Answers to selected problems appear at the end of this volume.
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