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

By Harold M. Edwards

​​​Originally released via Houghton Mifflin corporation, Boston, 1969

In a ebook written for mathematicians, academics of arithmetic, and hugely stimulated scholars, Harold Edwards has taken a daring and strange method of the presentation of complicated calculus. He starts with a lucid dialogue of differential types and speedy strikes to the elemental theorems of calculus and Stokes’ theorem. the result's real arithmetic, either in spirit and content material, and an exhilarating selection for an honors or graduate path or certainly for any mathematician wanting a refreshingly casual and versatile reintroduction to the topic. For a lot of these power readers, the writer has made the method paintings within the most sensible culture of inventive mathematics.

This reasonable softcover reprint of the 1994 version offers the various set of issues from which complicated calculus classes are created in attractive unifying generalization. the writer emphasizes using differential varieties in linear algebra, implicit differentiation in larger dimensions utilizing the calculus of differential types, and the tactic of Lagrange multipliers in a basic yet easy-to-use formula. There are copious routines to aid advisor the reader in checking out realizing. The chapters should be learn in nearly any order, together with starting with the ultimate bankruptcy that includes many of the extra conventional issues of complicated calculus classes. furthermore, it's perfect for a path on vector research from the differential types aspect of view.

The specialist mathematician will locate right here a pleasant instance of mathematical literature; the coed lucky adequate to have passed through this booklet may have a company seize of the character of contemporary arithmetic and an excellent framework to proceed to extra complicated reports.

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Extra info for Advanced Calculus: A Differential Forms Approach

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12 3 4 5 + + not. Let Un = # of shaded squares ( U = uncertain) Cn = #of squares whose vertices all have X's (C = certain) An = 4 (:2 • Cn + ~2 · ~ · Un) (A = approximation) Find A1o. Find A2o. 3 j Definition of Certain Simple Integrals 35 the integral JD dx dy which is based on the subdivision by lines x = ± 1!.. , y = ± '!. or on any refinement of this subdivision n n differs from An by at most 2n- 2 Un. What accuracy (how many decimal places) does this estimate guarantee for the approximation A 1 o "' 1r?

What is a constant 0-form? 2 Integration If the force field is not constant, then finding the amount of work required for a given displacement requires a process of integration. The essential idea is that the 1-form (1) I I (@_ I I I / I --- A(x, y, z) dx + B(x, y, z) dy + C(x, y, z) dz which describes the force field gives the approximate amount of work required for small displacements. The amount of work required for a displacement which is not small can then be described as a limit of sums of values of (1).

0, 1) agrees with the orientation PoP1P2P3 where Po is the base of the thumb, P 1 the tip of the thumb, P2 the tip of the index finger, and P3 the tip of the third finger of a right hand held in the natural position so that these points are non-coplanar. expressing xyz directly in terms of rst and then forming the pullback or by first expressing dx dy dz as a 3-form in uvw and then expressing this 3-form as a 3-form in rst. 3, by writing out the computations explicitly in terms of the 18 coefficients a, b, c, a', ...

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