Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
CRC Press, 2018/05/04 - 162 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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appear applied basis boundary bounded Calculus on Manifolds called Chapter choose classical clearly closed rectangle collection compact computation condition consider containing continuously differentiable converse coordinate system cover curves Define f defined definition denoted depend determinant Df(a differentiable drº element equal equation example exists expression fact Figure follows function f Give given Hence Hint holds important inner product Integration Integration on Chains interior interval k-dimensional k-form Let f linear transformation manifold-with-boundary matrix means measure normal notation Note obtain open cover open rectangle open set orientation Problem Proof prove reader result satisfies Show that f side similar simple singular n-cube ſº Stokes subrectangle subset suffices Suppose Theorem true unique upper usual vector field volume write