Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
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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basis boundary bounded function calculus called classical theorems closed rectangle closed set compact set consider continuously differentiable coordinate system definition denned denoted Df(a Dif(a Dif(x,y differentiable function div F Divergence Theorem Djf(a dz A dx equation fc-cube fc-dimensional manifold fc-form fdx1 Figure finite number Fubini's theorem function g G Rn Hence Hint induced orientation inner product integrable interior intersects Jaxb Jordan-measurable l)-form least upper bound Lemma Let A C Rn linear transformation manifold in Rn matrix measure Michael Spivak ms(f n-chain non-zero open cover open interval open rectangle open set containing orientation-preserving partial derivatives partition of unity Problem prove reader satisfies singular n-cube sional manifold Stokes subrectangle subset suffices Suppose Theorem 2-2 theorem is true unique usual orientation V X F vector field vector space volume element