- Paperback: 184 pages
- Publisher: W H Freeman & Co (Sd); 4th edition (August 1996)
- Language: English
- ISBN-10: 0716724332
- ISBN-13: 978-0716724339
- Product Dimensions: 0.2 x 8.5 x 10.8 inches
- Shipping Weight: 1 pounds
- Average Customer Review: 2.5 out of 5 stars See all reviews (2 customer reviews)
- Amazon Best Sellers Rank: #2,548,121 in Books (See Top 100 in Books)
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Study Guide for Marsden and Tromba's Vector Calculus 4th Edition
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One last point; the study hints are somewhat condescending. There seems to be the assumption that the student has no intelligence. They point out the obvious and clearly there has been no real effort spent in preparing them.
At least at the graduate level, there is a lot to be said for this act of "trading speed for depth." After all, unless the student's goal is higher mathematics, the "fine points" of Linear Algebra do not need to come "fully" into play. However, if it is directed to an undergraduate crowd as the author suggests, there is something to be said for giving a rigorous treatment of the concept of "mappings" since it is ubiquitous throughout the book, and its "fine points" come into play whether the authors wants them to or not. I am not sure how a mathematically unsophisticated sophomore is supposed to intuit the full meaning of "functions and mappings." [Are they teaching this in the High schools nowadays?]
One could argue: Why not do away with mathematical proofs altogether and go straight to the mechanics? For instance: Do we really need to spend 50 pages on the differentiation of real-valued functions? So much of it is simple straightforward extrapolation from three to n-dimensions, rarely with any hidden complications? Seems to me time could be more appropriately saved here rather than by omitting the core concepts of Linear Algebra? In fact, a lot of the notation in higher mathematics is designed to do exactly that, is it not? To wit: to summarize going from 3 to n-dimensions without having to wade through a lot of repetition.
I think the reason the latter is preferable to the former is clear: The student must feel that he fully grasps the basic concepts from the ground up. Otherwise, conceptually he is suspended in mid-air for most of the ride to the more difficult issues: with lots of heavy stuff being thrown at him all at once. The question always is: what is the proper stuff to "throw at him," and what is the proper stuff to leave out?
Those who manage to survive this conceptual high-wire act without a proper grounding (without a safety net as it were) will never really know exactly how they did it: They become good manipulators of formulas, but have no real grounding in the concepts. And those who do not, are likely to take a lot longer to understand Vector Calculus, or miss its main points altogether.
An entirely different approach is that used by Professor Richard Fyneman at Cal Tech, in his award-winning lecture series: He starts with the concepts he needs and introduces the mathematics associated with the concepts at the time he needs them. This approach is no "walk in the park" either. But once a student has grasped the particular concept in question; and mastered the mathematics associated with it: in context, there is no uncertainty and no turning back. "The whole of the complex idea and its mathematics" are mastered as one integrated piece. It seems to me that for more mature students, this is a better approach. Because if something is missing, they will naturally go searching for it and will not rest until it is mastered.
With this rather heavy-handed complaint aside, the book follows the traditional path of first being "broad but not deep," until it hits the heavy weather of the "real" Vector Calculus (line integration, curls, etc.), and then it switches courses, and becomes deep but not broad.
Amazingly, this approach seems to work. However, getting through the first two chapters, which consist of 150 pages -- exactly one-third of the book -- is pure "water torture." Doesn't having one chapter that take up 150 pages tell the authors something is amiss?