Volume 1: Vectors in the Plane and One-Variable Calculus
Wilfred Kaplan, Donald J. Lewis
Wiley/ University of Michigan Library | 2007-08-01 | ISBN-10: 1425589138 | DJvu | 712 pp | 80.5 MB
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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Volume 2: Vector Spaces, Many-Variable Calculus, and Differential Equations
Wilfred Kaplan, Donald J. Lewis
Wiley/ University of Michigan Library | 2007 | ISBN-10: 1425589340 | DJvu | 605 pp | 42.6 MB
In this second volume, the authors develop linear algebra and then apply it to geometry, many-variable calculus, and differential equations. These topics are so closely related that the subject matter is revealed here as one well-defined, tightly knit body of mathematics.
The following is a brief chapter-by-chapter summary with a few comments.
Chapter 9. Vector Spaces. Vector spaces are defined axiomatically, but it is made clear that in almost all applications verification of the axioms requires merely checking closure under the basic operations. The essential machinery of linear algebra is developed: subspaces, addition of sets, linear varieties (cosets), linear independence, basis, dimension, linear mappings, kernel, range, rank, nullity, linear transformations (mappings of a space into itself), vector spaces of linear mappings, algebras of linear transformations, inverse mapping. In many cases, geometric illustrations are given (in anticipation of Chapter 11), and many examples are chosen from the calculus.
Chapter 10. Matrices and Determinants. Matrices are introduced as linear mappings of Vn (the vector space of real n-tuples) into Vm. Hence, the general results for such mappings apply and one can, for example, at once speak of the rank of a matrix. The simple ideas of linear algebra permit one to obtain all the main results concerning simultaneous linear equations (with the exception of a few results depending on determinants). These equations are studied carefully, and are related to geometry. The operations on matrices are then developed fully—in all cases, as an application of the previously developed theory of linear mappings. Special attention is given to square matrices (linear transformations) and their inverses. Determinants are developed systematically, and their significance in geometry is pointed out. The concluding optional sections concern matrices of functions, the technique of elimination, eigenvalues, and similarity.
Chapter 11. Linear Euclidean Geometry. Here the emphasis is on 3-dimensional space, but brief optional sections indicate the generalization to n dimensions. The inner product is introduced, its properties are deduced, and R3 is defined in terms of the corresponding distance function. The cross product is also developed and is shown to be a powerful tool for studying lines and planes. Many problems of geometry are treated by linear algebra, but the emphasis is on demonstrating how effective this method is, rather than
giving a complete theory. The treatment of area and volume is related to the calculus. Linear mappings of R3 into R3 are considered; the idea of Jacobian matrix and that of its determinant appear, and their geometric meaning is stressed. Surfaces in space and cylindrical and spherical coordinates are discussed. Change of coordinates is treated rather briefly.
Chapter 12. Differential Calculus of Functions of Several Variables. Partial derivatives and differentials are developed and shown to be part of a theory emphasizing vector functions and vector operations; in particular, the gradient and the Jacobian matrix. The various chain rules are shown to be cases of one very simple rule for vector functions. Implicit and inverse functions are treated, with emphasis on the linear approximation provided by the calculus; there are corresponding applications to tangents and normals. Maxima and minima are discussed, including the case of side conditions (Lagrange multipliers); here again linear algebra is important.
Chapter 13. Integral Calculus of Functions of Several Variables. The double and triple integral are studied, with emphasis on the essential properties needed for applications. Integration in curvilinear coordinates (especially cylinder and spherical) is considered, with reference to the Jacobian formula. Numerous applications are discussed. Line integrals are studied systematically, with emphasis on Green's theorem and independence of path. The operations of divergence and curl are introduced and, through Green's theorem, shown to have geometric and physical meaning. The extensions of these ideas to space is considered briefly.
Chapter 14. Ordinary Differential Equations. This is, in effect, a short course in the subject, with emphasis on linear equations and matrix methods. The Existence Theorem is discussed (but not proved here). Practical applications are discussed, and the idea of stability is given much weight. Phase plane analysis, series methods, and numerical techniques are considered rather briefly.
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Wilfred Kaplan, Donald J. Lewis
Wiley/ University of Michigan Library | 2007-08-01 | ISBN-10: 1425589138 | DJvu | 712 pp | 80.5 MB
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.
Download
Volume 2: Vector Spaces, Many-Variable Calculus, and Differential Equations
Wilfred Kaplan, Donald J. Lewis
Wiley/ University of Michigan Library | 2007 | ISBN-10: 1425589340 | DJvu | 605 pp | 42.6 MB
In this second volume, the authors develop linear algebra and then apply it to geometry, many-variable calculus, and differential equations. These topics are so closely related that the subject matter is revealed here as one well-defined, tightly knit body of mathematics.
The following is a brief chapter-by-chapter summary with a few comments.
Chapter 9. Vector Spaces. Vector spaces are defined axiomatically, but it is made clear that in almost all applications verification of the axioms requires merely checking closure under the basic operations. The essential machinery of linear algebra is developed: subspaces, addition of sets, linear varieties (cosets), linear independence, basis, dimension, linear mappings, kernel, range, rank, nullity, linear transformations (mappings of a space into itself), vector spaces of linear mappings, algebras of linear transformations, inverse mapping. In many cases, geometric illustrations are given (in anticipation of Chapter 11), and many examples are chosen from the calculus.
Chapter 10. Matrices and Determinants. Matrices are introduced as linear mappings of Vn (the vector space of real n-tuples) into Vm. Hence, the general results for such mappings apply and one can, for example, at once speak of the rank of a matrix. The simple ideas of linear algebra permit one to obtain all the main results concerning simultaneous linear equations (with the exception of a few results depending on determinants). These equations are studied carefully, and are related to geometry. The operations on matrices are then developed fully—in all cases, as an application of the previously developed theory of linear mappings. Special attention is given to square matrices (linear transformations) and their inverses. Determinants are developed systematically, and their significance in geometry is pointed out. The concluding optional sections concern matrices of functions, the technique of elimination, eigenvalues, and similarity.
Chapter 11. Linear Euclidean Geometry. Here the emphasis is on 3-dimensional space, but brief optional sections indicate the generalization to n dimensions. The inner product is introduced, its properties are deduced, and R3 is defined in terms of the corresponding distance function. The cross product is also developed and is shown to be a powerful tool for studying lines and planes. Many problems of geometry are treated by linear algebra, but the emphasis is on demonstrating how effective this method is, rather than
giving a complete theory. The treatment of area and volume is related to the calculus. Linear mappings of R3 into R3 are considered; the idea of Jacobian matrix and that of its determinant appear, and their geometric meaning is stressed. Surfaces in space and cylindrical and spherical coordinates are discussed. Change of coordinates is treated rather briefly.
Chapter 12. Differential Calculus of Functions of Several Variables. Partial derivatives and differentials are developed and shown to be part of a theory emphasizing vector functions and vector operations; in particular, the gradient and the Jacobian matrix. The various chain rules are shown to be cases of one very simple rule for vector functions. Implicit and inverse functions are treated, with emphasis on the linear approximation provided by the calculus; there are corresponding applications to tangents and normals. Maxima and minima are discussed, including the case of side conditions (Lagrange multipliers); here again linear algebra is important.
Chapter 13. Integral Calculus of Functions of Several Variables. The double and triple integral are studied, with emphasis on the essential properties needed for applications. Integration in curvilinear coordinates (especially cylinder and spherical) is considered, with reference to the Jacobian formula. Numerous applications are discussed. Line integrals are studied systematically, with emphasis on Green's theorem and independence of path. The operations of divergence and curl are introduced and, through Green's theorem, shown to have geometric and physical meaning. The extensions of these ideas to space is considered briefly.
Chapter 14. Ordinary Differential Equations. This is, in effect, a short course in the subject, with emphasis on linear equations and matrix methods. The Existence Theorem is discussed (but not proved here). Practical applications are discussed, and the idea of stability is given much weight. Phase plane analysis, series methods, and numerical techniques are considered rather briefly.
Download
