Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced CalculusAvalon Publishing, 1965 - 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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29 ページ
... Hint : Note that any two points in A can be connected by a sequence of lines each parallel to one of the axes . ( b ) Find a function f : A → R such that Daf = 0 but ƒ is not independent of the second variable . - 2-24 . Define f : R2 ...
... Hint : Note that any two points in A can be connected by a sequence of lines each parallel to one of the axes . ( b ) Find a function f : A → R such that Daf = 0 but ƒ is not independent of the second variable . - 2-24 . Define f : R2 ...
61 ページ
... Hint : Compute fcf in two different ways for a suitable set CC [ a , b ] x [ a , b ] . 3-28 . * Use Fubini's theorem to give an easy proof that D1,2f = D2,1f if these are continuous . Hint : If D1,2f ( a ) — D2,1f ( a ) > 0 , there is a ...
... Hint : Compute fcf in two different ways for a suitable set CC [ a , b ] x [ a , b ] . 3-28 . * Use Fubini's theorem to give an easy proof that D1,2f = D2,1f if these are continuous . Hint : If D1,2f ( a ) — D2,1f ( a ) > 0 , there is a ...
105 ページ
... Hint : Integrate w find X. = - 4-30 . If w is a 1 - form on R2 - 0 such that dw = w = λ do + dg dg for some function g λ dx = dg on [ 0,1 ] to 0 , prove that for some ER and g : R20 R. Hint : If CR.1 ( w ) AR dx + d ( gR ) , CR.1 * ( w ) ...
... Hint : Integrate w find X. = - 4-30 . If w is a 1 - form on R2 - 0 such that dw = w = λ do + dg dg for some function g λ dx = dg on [ 0,1 ] to 0 , prove that for some ER and g : R20 R. Hint : If CR.1 ( w ) AR dx + d ( gR ) , CR.1 * ( w ) ...
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boundary bounded function calculus called closed curves closed rectangle continuously differentiable coordinate system Define f defined by f(x,y definition denoted Df(a Dif(a differentiable function div F dx¹ equation f(a¹ f(x¹ Figure finite number Fubini's theorem function f ƒ and g ƒ is differentiable ƒ is integrable Hint inner product interior Jordan-measurable k-dimensional manifold k-form k-tensor least upper bound Let A CR Let f lim h→0 linear transformation matrix Michael Spivak Möbius strip ms(f n-chain notation open cover open rectangle open set open set containing orientation-preserving partial derivatives partition of unity Problem Proof prove Show that ƒ singular n-cube Stokes subrectangle subset Suppose Theorem 2-2 unique usual orientation V₁ vector field vector space volume element Ʌ dx Ʌ dxi Σ Σ