This title is out of print.
For sophomore-level courses in Multivariable Calculus.
This text uses the language and notation of vectors and matrices to clarify issues in multivariable calculus. Accessible to anyone with a good background in single-variable calculus, it presents more linear algebra than usually found in a multivariable calculus book. Colley balances this with very clear and expansive exposition, many figures, and numerous, wide-ranging exercises. Instructors will appreciate Colley’s writing style, mathematical precision, level of rigor, and full selection of topics treated.
• Flexible presentation – Both level of rigor and technical details can be presented as the instructor prefers, making the book suitable for students with a range of backgrounds.
• Use of vector and matrix notation, particularly for differential topics – Enables a more general discussion and makes clear the analogy between concepts in single- and multivariable calculus.
• Variety of topics not usually found in a text at this level – Can be included to professor’s taste, or used as enrichment for students.
• Some emphasis on mathematical rigor, but the more technical derivations are collected at the ends of sections – Many proofs are available for reference, but positioned so as not to disrupt the flow of main ideas.
• Key results and items are set off clearly from supporting discussions.
• Large numbers of fully worked examples integrated throughout – Used both to motivate and to explicate the main ideas and techniques.
• More than 1400 exercises:
– Range from routine reinforcement of basic definitions, computations, and results, to more challenging conceptual questions.
– Includes numerous computer-based exercises, specifically noted.
• Presentation of Newton’s method for approximating solutions to systems of n equations in n unknowns.
• Discussion of numerical approximations of line integrals.
• More than 210 new and varied exercises – In a number of instances, these new problems may be used to develop supplementary topics such as:
– Matrix inverses.
– Further topics in the differential geometry of curves.
– Connections between linear algebra (eigenvalues/eigenvectors) and optimization via Lagrange multipliers.
– Improper integrals.
– Probability density functions.
– Solid angles.
• New true/false exercises at the end of each chapter (approx. 230 total).
(NOTE: Each chapter concludes with True/False Exercises and Miscellaneous Exercises.)
Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Some n-Dimensional Geometry. New Coordinate Systems.
2. Differentiation in Several Variables.
Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newton’s Method. The Chain Rule. Directional Derivatives and the Gradient.
3. Vector-Valued Functions.
Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator.
4. Maxima and Minima in Several Variables.
Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema.
5. Multiple Integration.
Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration.
6. Line Integrals.
Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields.
7. Surface Integrals and Vector Analysis.
Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations.
8. Vector Analysis in Higher Dimensions.
An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem.
Suggestions for Further Reading.
Answers to Selected Exercises.
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Susan Colley is the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College and currently Chair of the Department, having also previously served as Chair. She received S.B. and Ph.D. degrees in mathematics from the Massachusetts Institute of Technology prior to joining the faculty at Oberlin in 1983. Her research focuses on enumerative problems in algebraic geometry, particularly concerning multiple-point singularities and higher-order contact of plane curves. Professor Colley has published papers on algebraic geometry and commutative algebra, as well as articles on other mathematical subjects. She has lectured internationally on her research and has taught a wide range of subjects in undergraduate mathematics. Professor Colley is a member of several professional and honorary societies, including the American Mathematical Society, the Mathematical Association of America, Phi Beta Kappa, and Sigma Xi.
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