Advanced Calculus: A Differential Forms Approach by Harold M. Edwards

By Harold M. Edwards

In a publication written for mathematicians, lecturers of arithmetic, and hugely encouraged scholars, Harold Edwards has taken a daring and weird method of the presentation of complex calculus. He starts with a lucid dialogue of differential varieties and speedy strikes to the basic theorems of calculus and Stokes’ theorem. the result's real arithmetic, either in spirit and content material, and a thrilling selection for an honors or graduate direction or certainly for any mathematician wanting a refreshingly casual and versatile reintroduction to the topic. For a lot of these capability readers, the writer has made the procedure paintings within the most sensible culture of inventive mathematics.

This cheap softcover reprint of the 1994 variation offers the various set of issues from which complex calculus classes are created in attractive unifying generalization. the writer emphasizes using differential varieties in linear algebra, implicit differentiation in better dimensions utilizing the calculus of differential varieties, and the strategy of Lagrange multipliers in a normal yet easy-to-use formula. There are copious routines to aid advisor the reader in checking out knowing. The chapters should be learn in virtually any order, together with starting with the ultimate bankruptcy that comprises a few of the extra conventional themes of complicated calculus classes. moreover, it really is perfect for a path on vector research from the differential kinds element of view.

The expert mathematician will locate right here a pleasant instance of mathematical literature; the coed lucky adequate to have undergone this booklet can have an organization snatch of the character of recent arithmetic and a high-quality framework to proceed to extra complex studies.

The most vital feature…is that it really is fun—it is enjoyable to learn the routines, it truly is enjoyable to learn the reviews revealed within the margins, it really is enjoyable just to decide a random spot within the booklet and start examining. this is often the way in which arithmetic could be awarded, with an pleasure and liveliness that exhibit why we're drawn to the subject.

—The American Mathematical per 30 days (First evaluation)

An inviting, strange, high-level creation to vector calculus, established solidly on differential types. incredible exposition: casual yet refined, down-to-earth yet normal, geometrically rigorous, wonderful yet critical. extraordinary assorted purposes, actual and mathematical.

—The American Mathematical per 30 days (1994) in accordance with the second one Edition

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Example text

Altogether the value of dy dz - 2 dx dy on the given triangle is therefore -12. The computation is best done all at once by writing dydz- 2dxdy = (4du+2dv)(-dv)- 2(du-2dv)(4 du+2dv) = -4 du dv - 4 du dv + 16 dv du = -24dudv so that the value on the triangle (0, 0), (1, 0), (0, 1) is -12. The evaluation of an arbitrary 2-form A dy dz + B dz dx + C dx dy on an arbitrary oriented triangle (xo, Yo, zo), (x1o Y1, z1), (x2, Y2, z2) can be accomplished in the same way using the computational rules for finding pullbacks.

The 3-form A dx dy dz assigns numbers to oriented solids in xyz-space-namely, A times the oriented volume. The composition of these two operations assigns numbers to oriented solids in uvw-space. This new rule assigning numbers to oriented solids in uvw-space is clearly proportional to oriented volume; that is, it is of the form B du dv dw for some number B. This geometrically defined 3-form B du dv dw is in fact identical with the pullback of A dx dy dz under the map (I) as defined algebraically above.

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. ] 6 Let (x + tA, y + tB, z + tC) describe a flow in space. As in Exercise 5, assume the fluid has unit density and find the 2-form which describes this flow. 1 Non-Constant Forms In Chapter 1-form a constant force field is described by a A dx + B dy + C dz, where A = B = C = work required for unit displacement in x-direction work required for unit displacement in y-direction work required for unit displacement in z-direction.

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