Calculus of Several Variables (Undergraduate Texts in Mathematics) |  | Author: Serge Lang
Publisher: Springer
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Sales Rank: 77424
Media: Hardcover
Edition: 3rd
Pages: 636
Number Of Items: 1
Shipping Weight (lbs): 2
Dimensions (in): 9.3 x 6.1 x 1.4
ISBN: 0387964053
Dewey Decimal Number: 515.84
EAN: 9780387964058
ASIN: 0387964053
Publication Date: February 17, 1987
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Product Description
This is a new, revised edition of this widely known text. All of the basic topics in calculus of several variables are covered, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences. The presentation is self-contained, assuming only a knowledge of basic calculus in one variable. Many completely worked-out problems have been included.
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| Customer Reviews:
 Calc III February 20, 2010
Willjam3
This book was required for my class, however I did not find it a very entertaining read. The pages are very bland and it is hard to sit through. Looking up an example for reference is fine, but unless you need it for class, I'm sure there are better Calc books out there
 A solid introduction to multi-variable calculus for the well-prepared student. August 23, 2008
N. F. Taussig (Bronx, NY)
4 out of 4 found this review helpful
Serge Lang's Calculus of Several Variables provides an effective introduction to readers with a solid understanding of single variable calculus, such as that gained by working through his A First Course in Calculus (Undergraduate Texts in Mathematics). Like that text, this one clearly conveys the key concepts, places them in context, gives the reader a sense of how mathematicians think about the subject, and teaches the reader the skills needed to solve challenging problems. Methods of solution are clearly explained, then effectively demonstrated through the examples. The exercises, which are generally challenging and occasionally daunting, reinforce these concepts and skills. Answers to most of the problems, including many fully worked out solutions, are provided in a 107 page appendix, making this text suitable for self-study. While Lang's text is not as rigorous as Tom M. Apostol's Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, many results are proved, others are proved in special cases, and still others are justified through intuitive arguments.
The first part of the text, which covers vector functions, is a prerequisite for the remaining parts of the book. It begins with vectors, including the dot (scalar) product, norm, parametric lines, planes, and the cross product. From there, Lang proceeds to differentiation of vectors, functions of several variables (scalar fields), partial differentiation, the gradient, and the Chain Rule. It is an expanded version of the corresponding chapters in his A First Course in Calculus.
The second part of the text covers maxima, minima, Lagrange multipliers, and Taylor's Formula for functions of two variables. Lang goes beyond what is covered in standard texts to address problems in which the extrema may occur on the boundary. He also delves into a discussion of quadratic forms and partial differential operators.
The third and fourth parts of the book, which can be covered before the second part, address vector fields, potential functions, line integrals (which Lang calls curve integrals in what appears to have been a vain attempt to introduce more accurate terminology), double integrals, integration with respect to polar coordinates, Green's Theorem, triple integrals, integration with respect to cylindrical and spherical coordinates, the Divergence Theorem, and Stokes' Theorem. Lang does an effective job of showing you how these topics are related.
The final section of the book introduces those aspects of linear algebra which can be applied to the calculus of several variables, including matrix operations, determinants, and linear mappings. These tools are then applied to illustrate the Jacobian and Hessian matrices, differentiability of vector fields, the Inverse Mapping Theorem, the Implicit Function Theorem, determinants as area and volume, dilations, and change of variable formulas in both two and three dimensions. The exercises in this part of the text and the subsequent appendix on scalar products of functions and the computation of Fourier series are particularly challenging.
As much as I like this text, I feel compelled to warn you that there are numerous errors, including some in the answer key. As is the case in most mathematics texts, you will have to fill in the details in some of the examples. However, in the section on center of mass, Lang assigned several problems in which you are supposed to find the y-coordinate of the center of mass of a region of the plane bounded below by the function y = f(x) and above by y = g(x) without explaining that you are supposed to integrate y dy from f(x) to g(x), then integrate the result with respect to x, before dividing by the mass. I learned this from Apostol's text. My remaining caveat is that Lang sometimes makes an assumption at the beginning of a section to simplify the subsequent discussion. That limits the usefulness of this text as a reference since you cannot simply look up the statement of a theorem, without first checking for additional assumptions stated earlier in the section.
Caveats aside, a well-prepared student who works through this text will acquire a solid understanding of the key concepts, learn to solve hard problems, and obtain a solid foundation for more advanced courses such as linear algebra and real analysis. It also provides physics students with the skills required to handle problems in undergraduate courses in classical mechanics, quantum mechanics, and electricity and magnetism .
Addendum: The statement of problem 23 in section 2 of chapter 15 is incomplete. In reading Lang's Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition, I found a complete statement of the problem. In both books, it begins "Let A be a square matrix which is of the form" [an upper triangular matrix is then shown] In this book, the remainder of the statement is missing. In Introduction to Linear Algebra, it reads "The notation means that all elements below the diagonal are equal to 0, and the elements above the diagonal are arbitrary. One may express the property by saying that a_{ij} = 0 if i > j. Such a matrix is called upper triangular. If A, B are upper triangular matrices (of the same size) what can you say about the diagonal elements of AB?"
 Great, Straight forward book. December 5, 2006
Andrew R. Garcia (Alabama)
12 out of 13 found this review helpful
This book is unlike any other calculus, most of the problems are challenging, and the examples are part of the text, not a seperate entity. It also only weighs approx 3 pounds. Without this book, I would have been lost in higher level math, such as Abstract Algebra, Topology, and Analysis. Many problems, beyond the trivial with every single problem worked out in the back. It really develops mathematical thinking and proof writing. Serge Lang is a talented instructor.
A great book if used wisely April 15, 2005
B. Jacobs
36 out of 37 found this review helpful
The previous reviewer gave this book one star, and I can see how one could have a dismal experience with it, much like one could have a horrible time backpacking through a remote wilderness area without the proper skills or guidance. On the other hand, I have taught multi-variable calculus to advanced high-school students from this book many times, and the students all loved it. (I know this from their anonymous course-evaluations.) I used the book Div, Grad, Curl and All That along with this book, as a supplemental text. The two books complemented each other nicely, one taking a more "mathy" point of view, and the other more "physicsy." The previous reviewer may not realize that there are actually oodles of worked out examples; they're all in the back of the book, instead of incorporated into the text. I like this, because it doesn't interfere with the flow of the reading. (My students think this book is a fairly straightforward read.) What I especially like about this book is its emphasis on potential theory; it makes it a deeper book than most other multivariable texts on the market.
 If the book is yellow, or Serge Lang is the author, AVOID April 24, 2003
11 out of 74 found this review helpful
Instead of wasting money on a useless calculus book written by Serge Lang, I would strongly recommend a Calculus book by Howard Anton or James Stewart. Every Serge Lang book I have read lacked examples, or if there were examples, they would be too general and not helpful in learning the material you need to know. His books seem to be more based on theories which to me are not helpful for one to learn calculus. Aside from the way it is written, compared to most other calculus books, it is too small. I also have taken a course in Linear Algebra, and his book was too small and was very hard to learn from. Most other calculus textbooks are 800-1200 pages, while his book does not come close, so there are obviously many concepts and exercises lacking. When one has trouble in a course, their text is supposed to be helpful and used as a resource, and Lang's books are anything but helpful. The only thing this book will be good for is the fireplace when I am done with it.
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