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.