Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced CalculusThis 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 firstyear 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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Review: Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
User Review  Brandon Meredith  GoodreadsWhat yo mamma never told you you could do with calculus! Read full review
Review: Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
User Review  GoodreadsWhat yo mamma never told you you could do with calculus! Read full review
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Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced ... Michael Spivak No preview available  1965 
Common terms and phrases
basis boundary bounded function calculus called chain rule classical theorems closed rectangle compact set consider continuously differentiable coordinate system definition denoted Df(a Dif(a Dif(x,y differentiable function div F Divergence Theorem dy A dz dz A dx equation fccube fcdimensional manifold fcform fdxl Figure finite number Fubini's theorem function g G Rn Hence Hint induced orientation inner product integrable interior intersects inverse Jordanmeasurable l)form least upper bound Lemma Let A C Rn Let g linear transformation manifold in Rn mathematics matrix measure Michael Spivak ms(f nchain nonzero open cover open interval open rectangle open set containing orientationpreserving partial derivatives partition of unity Problem prove reader Rm is differentiable satisfies singular ncube Stokes subrectangle subset suffices Suppose Theorem 22 theorem is true unique usual orientation vector field vector space volume element