By Harold M. Edwards
Originally released via Houghton Mifflin corporation, Boston, 1969
In a publication written for mathematicians, lecturers of arithmetic, and hugely inspired scholars, Harold Edwards has taken a daring and weird method of the presentation of complicated calculus. He starts off with a lucid dialogue of differential varieties and speedy 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 short of a refreshingly casual and versatile reintroduction to the topic. For these types of strength readers, the writer has made the process paintings within the most sensible culture of inventive mathematics.
This reasonable softcover reprint of the 1994 variation offers the various set of themes from which complicated calculus classes are created in attractive unifying generalization. the writer emphasizes using differential varieties in linear algebra, implicit differentiation in greater dimensions utilizing the calculus of differential types, and the strategy of Lagrange multipliers in a normal yet easy-to-use formula. There are copious routines to aid consultant the reader in checking out knowing. The chapters could be learn in nearly any order, together with starting with the ultimate bankruptcy that comprises many of the extra conventional issues of complicated calculus classes. moreover, it's excellent for a direction on vector research from the differential kinds element of view.
The expert mathematician will locate right here a pleasant instance of mathematical literature; the scholar lucky sufficient to have undergone this ebook could have an organization take hold of of the character of contemporary arithmetic and a superior framework to proceed to extra complicated reports.
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Additional resources for Advanced Calculus: A Differential Forms Approach
The justification of these formal rules of computationwhich appear quite mysterious at first-is completely pragmatic. They are simple to apply and they give very quickly the solution to the problem of evaluating 2-Jorms. For example: In order to find the value of the 2-form dy dz - 2 dx dy on the oriented triangle whose vertices are (I, 0, I), (2, 4, I), (- I, 2, 0) one can first write this triangle as the image of the oriented triangle (0, 0), (1, 0), (0, 1) in the uv-plane under the affine mapping x = y = z = I+ u- 2v 4u + 2v - v.
Of course. that dx, dy, dz are thought of as being 'infinitesimals'. The point of view taken here, however, is that dx, dy, dz are functions assigning numbers to directed line segments. What is 'infinitesimal: then. is the line segment on which they are evaluated. Instead of saying that (2) holds for small line segments. with the approximation improving for shorter line segments, it is often said simply that work required for 'infinitesimal' displacements = A(x. y, z) dx B(x. y, z) dy C(x. y, z) dz.
3 this description was made the basis of a rigorous definition for cases in which the domain of integration is a rectangle (interval, rectangular parallelopiped) by describing explicitly what is meant by a 'finely divided polygonal approximation' to a rectangle. This definition was extended to arbitrary bounded domains in k-space by the simple trick of enclosing such a domain in a k-dimensional rectangle, setting the integrand equal to zero outside the domain, and proceeding as before. 3 does not apply, however, to integrals over curves in the plane, over curves in space, or over surfaces in space.