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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15 ページ
... equation certainly makes no sense in the general case of a function f : R " → Rm , but can be reformulated in a way that does . If : RR is the linear transformation defined by λ ( h ) = f ' ( a ) h , then equation ( 1 ) is equivalent ...
... equation certainly makes no sense in the general case of a function f : R " → Rm , but can be reformulated in a way that does . If : RR is the linear transformation defined by λ ( h ) = f ' ( a ) h , then equation ( 1 ) is equivalent ...
45 ページ
... equation is often written simply af ax = ди af au af av ди дх + av ax Note that ƒ means something different on the two sides of the equation ! The notation df / dx , always a little too tempting , has inspired many ( usually meaningless ) ...
... equation is often written simply af ax = ди af au af av ди дх + av ax Note that ƒ means something different on the two sides of the equation ! The notation df / dx , always a little too tempting , has inspired many ( usually meaningless ) ...
122 ページ
... equations ( 1 ) . In particular , if g : A → R , we must solve n + 1 equations Dif ( a ) = Djg ( a ) , g ( a ) = 0 , in n + 1 unknowns a1 , . . . , a " , λ , which is often very simple if we leave the equation g ( a ) = 0 for last ...
... equations ( 1 ) . In particular , if g : A → R , we must solve n + 1 equations Dif ( a ) = Djg ( a ) , g ( a ) = 0 , in n + 1 unknowns a1 , . . . , a " , λ , which is often very simple if we leave the equation g ( a ) = 0 for last ...
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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