Vector 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.

 

Vectors: 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. 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. Vector-Valued Functions: Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator. Maxima and Minima in Several Variables: Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema. Multiple Integration: Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration. Line Integrals: Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields. Surface Integrals and Vector Analysis: Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations. Vector Analysis in Higher Dimensions: An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem.

 

For all readers interested in multivariable calculus.

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.