## Multivariable CalculusClassroom-tested and lucidly written, Multivariable Calculus gives a thorough and rigoroustreatment of differential and integral calculus of functions of several variables. Designed as ajunior-level textbook for an advanced calculus course, this book covers a variety of notions,including continuity , differentiation, multiple integrals, line and surface integrals, differentialforms, and infinite series. Numerous exercises and examples throughout the book facilitatethe student's understanding of important concepts.The level of rigor in this textbook is high; virtually every result is accompanied by a proof. Toaccommodate teachers' individual needs, the material is organized so that proofs can be deemphasizedor even omitted. Linear algebra for n-dimensional Euclidean space is developedwhen required for the calculus; for example, linear transformations are discussed for the treatmentof derivatives.Featuring a detailed discussion of differential forms and Stokes' theorem, Multivariable Calculusis an excellent textbook for junior-level advanced calculus courses and it is also usefulfor sophomores who have a strong background in single-variable calculus. A two-year calculussequence or a one-year honor calculus course is required for the most successful use of thistextbook. Students will benefit enormously from this book's systematic approach to mathematicalanalysis, which will ultimately prepare them for more advanced topics in the field. |

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### Contents

SOME PRELIMINARIES 1 The Rudiments of Set Theory | 1 |

Some Logic | 7 |

Mathematical Induction | 11 |

Inequalities and Absolute Value | 12 |

Equivalence Relations 1 5 9 12 | 15 |

EUCLIDEAN SPACES AND LINEAR TRANSFORMATIONS | 17 |

The Cartesian Plane | 19 |

Euclidean Spaces | 24 |

II | 209 |

The Proof of Proposition 2 3 | 214 |

THE INVERSE AND IMPLICIT FUNCTION THEOREMS | 218 |

The Inverse Function Theorem | 223 |

The Proof of Theorem 2 2 | 228 |

I | 230 |

II | 235 |

INTEGRATION | 240 |

The Norm in R | 28 |

Linear Transformations between Euclidean Spaces | 31 |

Matrices and Linear Transformations | 35 |

Some Operations on Linear Transformations and Matrices | 42 |

Composites of Linear Transformations and Matrix Multiplication | 47 |

Matrix Multiplication | 51 |

CONTINUOUS FUNCTIONS | 58 |

THE DERIVATIVE | 85 |

THE GEOMETRY OF EUCLIDEAN SPACES | 110 |

HIGHER ORDER DERIVATIVES AND TAYLORS | 138 |

Taylors Theorem for Functions of One Variable | 145 |

Taylors Theorem for Functions of n Variables | 154 |

COMPACT AND CONNECTED SETS | 160 |

Continuous Functions on Compact Sets | 169 |

A Characterization of Compact Sets | 172 |

Uniform Continuity | 175 |

Connected Sets | 177 |

MAXIMA AND MINIMA | 182 |

Quadratic Forms | 188 |

Criteria for Local Maxima and Minima | 194 |

I | 199 |

The Method of Lagrange Multipliers | 204 |

Properties of the Integral | 249 |

The Integral of a Function of Two Variables | 255 |

The Integral of a Function of n Variables | 263 |

Properties of the Integral | 271 |

The Proof of Theorem 6 2 | 280 |

ITERATED INTEGRALS AND THE FUBINI THEOREM | 283 |

Integrals over Nonrectangular Regions | 290 |

More Examples | 296 |

The Proof of Fubinis Theorem | 304 |

Differentiating under the Integral Sign | 306 |

The Change of Variable Formula | 309 |

LINE INTEGRALS | 318 |

SURFACE INTEGRALS | 351 |

DIFFERENTIAL FORMS | 380 |

INTEGRATION OF DIFFERENTIAL FORMS | 403 |

INFINITE SERIES | 421 |

INFINITE SERIES OF FUNCTIONS | 443 |

APPENDIX A Determinants | 477 |

APPENDIX B The Proof of the General Inverse Function Theorem | 486 |

SOLUTIONS TO SELECTED EXERCISES | 494 |

517 | |

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### Common terms and phrases

assume calculus chain rule Chapter compact sets compute continuous at v0 continuous function continuously differentiable converges uniformly coordinates critical point def,ne Determine diffeomorphism differentiable function directional derivative diverges dx2 dx2 equation Euclidean spaces example exists Figure follows formula function f give given graph Green's theorem Hint hyperplane if/is implicit function theorem induction inequality instance intersection interval iterated integral least upper bound Lemma Let f line integral line segment linear transformation maximum minimum multiplication n x n matrix nonnegative nonzero of/at open set open subset orthogonal parametrized curve parametrized surface partial derivatives partial sums partition polynomial positive definite proof of Proposition proof of Theorem properties Proposition 5.1 prove the analogue R2 defined radius of convergence real numbers region result Section sequence series converges Show statement Suppose tangent plane that/is Theorem 2.2 variables vector field vectors in R2