By Harold M. Edwards
In a e-book written for mathematicians, lecturers of arithmetic, and hugely stimulated scholars, Harold Edwards has taken a daring and weird method of the presentation of complex calculus. He starts off with a lucid dialogue of differential varieties and quick strikes to the elemental theorems of calculus and Stokes’ theorem. the result's actual arithmetic, either in spirit and content material, and a thrilling selection for an honors or graduate path or certainly for any mathematician wanting a refreshingly casual and versatile reintroduction to the topic. For some of these power readers, the writer has made the technique paintings within the top culture of artistic mathematics.
This reasonable softcover reprint of the 1994 version offers the various set of themes from which complex calculus classes are created in attractive unifying generalization. the writer emphasizes using differential types in linear algebra, implicit differentiation in larger dimensions utilizing the calculus of differential kinds, and the strategy of Lagrange multipliers in a common yet easy-to-use formula. There are copious workouts to assist consultant the reader in checking out knowing. The chapters might be learn in nearly any order, together with starting with the ultimate bankruptcy that includes a few of the extra conventional themes of complex calculus classes. moreover, it's perfect for a path on vector research from the differential types element of view.
The specialist mathematician will locate right here a pleasant instance of mathematical literature; the coed lucky sufficient to have passed through this publication could have an organization clutch of the character of recent arithmetic and an outstanding framework to proceed to extra complex studies.
The most vital feature…is that it truly is fun—it is enjoyable to learn the routines, it's enjoyable to learn the reviews published within the margins, it really is enjoyable just to choose a random spot within the e-book and start studying. this can be the best way arithmetic can be provided, with an pleasure and liveliness that exhibit why we're attracted to the subject.
—The American Mathematical per month (First evaluate)
An inviting, strange, high-level advent to vector calculus, established solidly on differential types. awesome exposition: casual yet refined, down-to-earth yet normal, geometrically rigorous, wonderful yet severe. extraordinary assorted functions, actual and mathematical.
—The American Mathematical per 30 days (1994) in accordance with the second one Edition
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Additional resources for Advanced Calculus: A Differential Forms Approach
4 A 1-form in 3 variables has 3 components, a 2-form 3 components, and a 3-form 1 component. How many components does a 1-form in 4 variables have? A 2-form? A 3-form? A 4-form? How many components does a k-form in n variables have? ] 5 A natural way to describe a constant flow in the plane is by saying that in time t the point (x, y) moves to (x + At, y + Bt) where A, B are the x- and y-components of the constant flow. Assuming the fluid has unit density find the 1-form which describes this planar flow.
Y)- A(x. v)l < < whenever IX - XI < o, IY - YI < o. Intuitively it means that if (x. y) is near (x. y) then A (x. y) is near A(x. v). * The integral of a 3-form over any oriented domain in space which can be enclosed in a rectangular parallelopiped, and the integral of a 1-form over any oriented domain of the line which can be enclosed in an interval, are defined by the same trick. Such domains are called bounded domains (that is, they are domains which stay within certain finite bounds). In summary, the integral of a k-form over a bounded, oriented domain of kdimensional space has been defined for k = 1, 2, 3.
0 this can be made small by making lSI and IS'I both small; that is, given E, there is a mesh size such that lSI < i5 and IS'I < i5 implies IL:(a) L:(a')l < e, and therefore the integral converges. Thus the integral converges if and only if U(S)---? 0 as lSI ---? 0. This important conclusion is perhaps more comprehensible when it is formulated as follows: The number U(S) represents the 'uncertainty' of an approximating sum I:(a) to fn A dx dy based on the subdivision S. Any approximating sum based on any refinement of S differs from any approximating sum based on S by at most V(S).
Advanced Calculus: A Differential Forms Approach by Harold M. Edwards