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Vector Calculus 5th
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Jerrold E. Marsden/Anthony J. Tromba
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Freeman
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fifth edition
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676
ISBN
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0716749920
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Now in its fifth edition,
Vector Calculus
helps students gain an intuitive and solid understanding of this important subject. The book¡¯s careful account is a contemporary balance between theory, application, and historical development, providing it¡¯s readers with an insight into how mathematics progresses and is in turn influenced by the natural world.
1. The Geometry of Euclidean Space
1.1 Vectors in Two- and Three-Dimensional Space
1.2 The Inner Product, Length, and Distance
1.3 Matrices, Determinants, and the Cross Product
1.4 Cylindrical and Spherical Coordinates
1.5 n-Dimensional Euclidean Space
2. Differentiation Space
2.1 The Geometry of Real-Valued Functions
2.2 Limits and Continuity
2.3 Differentiation
2.4 Introduction to Paths
2.5 Properties of the Derivative
2.6 Gradients and Directional Derivatives
3. Higher-Order Derivatives: Maxima and Minima
3.1 Iterated Partial Derivatives
3.2 Taylor¡¯s Theorem
3.3 Extrema of Real-Valued Functions
3.4 Constrained Extrema and Lagrange Multipliers
3.5 The Implicit Function Theorem
4. Vector-Valued Functions
4.1 Acceleration and Newton's Second Law
4.2 Arc Length
4.3 Vector Fields
4.4 Divergence and Curl
5. Double and Triple Integrals
5.1 Introduction
5.2 The Double Integral Over a Rectangle
5.3 The Double Integral Over More General Regions
5.4 Changing the Order of Integration
5.5 The Triple Integral
6. The Change of Variables Formula and Applications of Integration
6.1 The Geometry of Maps from R2 to R2
6.2 The Change of Variables Theorem
6.3 Applications of Double and Triple
6.4 Improper Integrals
7. Integrals
7.1 The Path Integral
7.2 Line Integrals
7.3 Parametrized Surfaces
7.4 Area of a Surface
7.5 Integrals of Scalar Functions Over Surfaces
7.6 Surface Integrals of Vector Functions
7.7 Applications to Differential Geometry, Physics and Forms of Life
8. The Integral Theorems of Vector Analysis
8.1 Green's Theorem
8.2 Stokes' Theorem
8.3 Conservative Fields
8.4 Gauss' Theorem
8.5 Applications to Physics, Engineering, and Differential Equations
8.6 Differential Forms
"Having read some of the reviews written here, I feel compelled to write some brief comments on Marsden's text. If you're the kind of math student who has always learned through repetition of techniques presented by a teacher or professor in non-mathematical language (I'd say this is the majority of non-math major undergraduates), this textbook is a bad choice. Marsden's approach is relatively simple, providing ample explanations that are concise and clear for any individual well-adjusted to "reading Math"... by that, I mean a student who is fluent in mathematical notations and in comprehending proofs. For example, test yourself (here's an actual quotation from the text, taken from the discussion on limits and continuity):
Definition, Open Sets: "Let U be a subset of R^n, we call U an open set for every point x in U there exists some r > 0 such that D(x) is contained within U." [in that sense, we might imagine the open set U to be the open disk or ball of radius r and center x denoted by D(x)]
Now, this is a simple concept; but if you find yourself struggling to understand this definition in a superficial way, you might have some problems with Marsden's text. This is the kind of language that is used throughout the text--its rather bourgeois when compared to other multivariable calculus texts (another reviewer makes the comment that Marsden's text is the self-proclaimed 'aristocrat' of Calculus III texts--I find such a comment quite fitting). If you are one of those struggling individuals that nevertheless decides to use this text, here's some advice:
(1) First, read the introductory section on prerequisites and notation--put some time into understanding how to read math and believe me, it will pay off in a big way.
(2) You have to think of this text as all or nothing. If you don't use it, fine; if you do, YOU MUST rely on the text throughout the entire course, i.e. read everything (except perhaps for the historical information)... this is critical because by reading all the chapters and sections, you'll find that you'll begin to develop an ability to read the text efficiently, i.e. your math-reading skills will increase dramatically. If instead, you choose to rely on your class notes and turn to this book when you need clarification, you won't find it simply because you won't understand what you're reading.
(3) This is obvious: work all example problems as you read the text. I've found the examples to be easier than those in the exercises (other reviewers have made this observation as well), but with some help from your TA (or novel thinking on your part), you should never feel lost in attempting to solve these problems (given that you've invested the time to really understand those examples).
(4) If you have experience in Linear Algebra, you're in a much stronger position to succeed with the text. Even though Marsden introduces the topics you'll need, obviously, the experience can only help you.
(5) In difference to the common opinion, Calculus III is actually an easy course--only right up until the end (chapters 7 & 8), does the material become much more difficult. Unfortunately, in my opinion, the most poorly written chapters in the text are chapters 7 and 8. So, a word of advice: if you find yourself coasting throughout the course, be aware that the material gets harder--and be prepared for that dramatic change in difficulty. Though Marsden 'gets the job done' in explaining the material, for all students, I'd recommend thinking about a supplement text for these 2 chapters (or very good class notes; one learns chapters 7 & 8 by working through example problem after example problem).
(6) Buy (or better yet go to your college's library and find) the student solutions manual! Assign yourself problems to practice before tests; as with all quantitative and physical science courses, one learns by solving problems, bottom line.
In summary, Marsden is a good text; its simple for the trained math reader (by that, I don't mean only professors!), but very very difficult to inexperienced students. It's not light reading, and it's going to challenge you; but for those that put in the time, you'll find that the book is very logical and well-organized in its presentation of the material."
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