Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
Avalon Publishing, 1965 - 144 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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A C Rn appear applied basis boundary bounded calculus called Chapter choose classical clearly closed rectangle collection compact computation condition consider containing continuously differentiable converse coordinate system cover curves defined definition denned denoted depend determinant differentiable element equal equation example exists expression fact Figure function function g Give given Hence holds important inner product integrable interior interval inverse length linear transformation manifold manifold-with-boundary matrix means measure ms(f n-dimensional normal notation Note obtain open cover open rectangle open set open set containing orientation partition of unity Problem Proof prove reader result satisfies side similar simple singular n-cube Stokes subrectangle subset suffices Suppose surface Theorem true unique upper usual vector field vector space volume write written