**AMS/MAA Textbooks**

Volume: 40;
2018;
405 pp;
Hardcover

MSC: Primary 26;

**Print ISBN: 978-1-4704-4360-3
Product Code: TEXT/40**

List Price: $79.00

AMS Member Price: $59.25

MAA Member Price: $59.25

**Electronic ISBN: 978-1-4704-4888-2
Product Code: TEXT/40.E**

List Price: $79.00

AMS Member Price: $59.25

MAA Member Price: $59.25

#### Supplemental Materials

# Calculus in 3D: Geometry, Vectors, and Multivariate Calculus

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*Zbigniew Nitecki*

MAA Press: An Imprint of the American Mathematical Society

Calculus in 3D is an accessible, well-written textbook for an honors
course in multivariable calculus for mathematically strong first- or
second-year university students. The treatment given here carefully
balances theoretical rigor, the development of student facility in the
procedures and algorithms, and inculcating intuition into underlying
geometric principles. The focus throughout is on two or three
dimensions. All of the standard multivariable material is thoroughly
covered, including vector calculus treated through both vector fields
and differential forms. There are rich collections of problems ranging
from the routine through the theoretical to deep, challenging problems
suitable for in-depth projects. Linear algebra is developed as
needed. Unusual features include a rigorous formulation of cross
products and determinants as oriented area, an in-depth treatment of
conics harking back to the classical Greek ideas, and a more extensive
than usual exploration and use of parametrized curves and surfaces.

Zbigniew Nitecki is Professor of Mathematics at Tufts University
and a leading authority on smooth dynamical systems. He is the author
of Differentiable Dynamics, MIT Press; Differential
Equations, A First Course (with M. Guterman), Saunders;
Differential Equations with Linear Algebra (with
M. Guterman), Saunders; and Calculus Deconstructed, AMS.

An instructor's manual for this title is available electronically
to those instructors who have adopted the textbook for classroom use.
Please send email to textbooks@ams.org for more
information.

#### Readership

Undergraduate students interested in honors calculus.

#### Table of Contents

# Table of Contents

## Calculus in 3D: Geometry, Vectors, and Multivariate Calculus

- Cover Cover11
- Title page i2
- Copyright ii3
- Contents iii4
- Preface v6
- Chapter 1. Coordinates and Vectors 112
- Chapter 2. Curves and Vector-Valued Functions of One Variable 6778
- Chapter 3. Differential Calculus for Real-Valued Functions of Several Variables 123134
- 3.1. Continuity and Limits 123134
- 3.2. Linear and Affine Functions 127138
- 3.3. Derivatives 132143
- 3.4. Level Curves 144155
- 3.5. Surfaces and Tangent Planes I: Graphs and Level Surfaces 158169
- 3.6. Surfaces and Tangent Planes II: Parametrized Surfaces 167178
- 3.7. Extrema 176187
- 3.8. Higher Derivatives 190201
- 3.9. Local Extrema 197208

- Chapter 4. Integral Calculus for Real-Valued Functions of Several Variables 205216
- Chapter 5. Integral Calculus for Vector Fields and Differential Forms 263274
- 5.1. Line Integrals of Vector Fields and 1-Forms 263274
- 5.2. The Fundamental Theorem for Line Integrals 272283
- 5.3. Green’s Theorem 278289
- 5.4. Green’s Theorem and 2-forms in Realstwo 289300
- 5.5. Oriented Surfaces and Flux Integrals 293304
- 5.6. Stokes’ Theorem 299310
- 5.7. 2-forms in Realsthree {} 306317
- 5.8. The Divergence Theorem 317328
- 5.9. 3-forms and the Generalized Stokes Theorem (Optional) 329340

- Appendix A. Appendix 335346
- A.1. Differentiability in the Implicit Function Theorem 335346
- A.2. Equality of Mixed Partials 336347
- A.3. The Principal Axis Theorem 339350
- A.4. Discontinuities and Integration 344355
- A.5. Linear Transformations, Matrices, and Determinants 347358
- A.6. The Inverse Mapping Theorem 353364
- A.7. Change of Coordinates: Technical Details 356367
- A.8. Surface Area: The Counterexample of Schwarz and Peano 363374
- A.9. The Poincare Lemma 367378
- A.10. Proof of Green’s Theorem 374385
- A.11. Non-Orientable Surfaces: The Möbius Band 376387
- A.12. Proof of Divergence Theorem 377388
- A.13. Answers to Selected Exercises 379390

- Bibliography 393404
- Index 397408
- Back Cover Back Cover1417