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Vector CalculusMathematics
Multivariable CalculusMathematics

A traditional and accessible calculus text with a strong conceptual
and geometric slant that assumes a background in single variable
calculus. The text uses the language and notation of vectors and matrices
to clarify issues in multivariable calculus. It is designed to provide
a greater challenge than the multivariable material typically found
in the last four or five chapters of a threesemester calculus text.
This challenge is balanced by clear and expansive writing and an interesting
selection of material.
Uses vector and matrix notation, particularly for
differential topics, to foster a more general discussion and clarify
the analogy between concepts in single and multivariable calculus.
Presents an optional, very lucid treatment of differential
forms.
Presents a variety of topics not usually found in a text
at this level offering flexibility for students and instructors.
Incorporates more than 500 diagrams and figures that
connect analytic work to geometry and assist with visualization.
Provides some emphasis on mathematical rigor, but
presents more technical derivations at the ends of sections. Proofs
are available for reference but positioned so as to not interfere
with the main flow of ideas.
Supplies some gentle suggestions regarding problems benefiting
from or requiring a computer algebra system or visualization software.
Includes several important pedagogical features:
 Key results and items are set off clearly from supporting
discussions.
 Many fully worked examples that are integral to the text.
These examples are used both to motivate and explicate the main ideas
and techniques.
 More than 1000 exercises, from routine reinforcement
of basic definitions, computations, and results, to more challenging
conceptual questions.
Each chapter ends with a section of Miscellaneous Exercises.
Preface.
1. Vectors.
Vectors in Two and Three Dimensions. More About Vectors.
The Dot Product. The Cross Product. Equations for Planes; Distance
Problems. Some nDimensional Geometry. New Coordinate Systems.
2. Differentiation in Several Variables.
Functions of Several Variables; Graphing Surfaces. Limits.
The Derivative. Properties; Highorder Partial Derivatives. The Chain
Rule. Directional Derivatives and the Gradient.
3. VectorValued 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. Stoke's and Gauss's
Theorems. Further Vector Analysis; Maxwell's Equations. An Introduction
to Differential Forms.
Suggestions for Further Reading.
Answers to Selected Exercises.
Index.
