Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
Avalon Publishing, 1971/01/22 - 160 ページ
This little book is especially concerned with those portions of ”advanced calculus” in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential.
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Alt(a an,bn B C R basis boundary bounded function calculus closed curves closed rectangle continuously differentiable coordinate system Define f definition denoted Df(a Dif(a differentiable function div F Divergence Theorem drº equation f and g f is differentiable f is integrable Figure finite number Fubini's theorem function f Hence Hint inner product interior Jordan-measurable k-dimensional manifold k-form k-tensor least upper bound Lemma Let A C R Let f linear transformation manifold-with-boundary matrix measure Michael Spivak Möbius strip ms(f n-chain open cover open interval open rectangle open set containing orientation-preserving partial derivatives partition of unity Problem Proof prove reader Show that f singular n-cube ſº Stokes subrectangle subset suffices Suppose theorem is true unique usual orientation vector field vector space volume element