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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... dz Ʌ dx + n3 dx Ʌ dy . Moreover , on M we have ( 2 ) ( 3 ) ( 4 ) n1 dA = dy Ʌ dz . n2 dA = dz Ʌ dx . n3 dA = dx Ʌdy . Proof . Equation ( 1 ) is equivalent to the equation dA ( v , w ) = det \ n ( x ) . Є This is seen by expanding the ...
... dz Ʌ dx + n3 dx Ʌ dy . Moreover , on M we have ( 2 ) ( 3 ) ( 4 ) n1 dA = dy Ʌ dz . n2 dA = dz Ʌ dx . n3 dA = dx Ʌdy . Proof . Equation ( 1 ) is equivalent to the equation dA ( v , w ) = det \ n ( x ) . Є This is seen by expanding the ...
135 ページ
... dz Ʌ dx + F3 dx dy . Then do div F dV . According to Theorem 5-6 , on M we have = n1 dA n2 dA = dy Ʌdz , = dz Ʌ dx , n3 dA = dx Ʌdy . Therefore on aM we have ( F , n ) dA = = FindA + F2n2 dA + F3n3 dA F1 dy Ʌdz + F2 dz Ʌ dx + F3 dx dy ...
... dz Ʌ dx + F3 dx dy . Then do div F dV . According to Theorem 5-6 , on M we have = n1 dA n2 dA = dy Ʌdz , = dz Ʌ dx , n3 dA = dx Ʌdy . Therefore on aM we have ( F , n ) dA = = FindA + F2n2 dA + F3n3 dA F1 dy Ʌdz + F2 dz Ʌ dx + F3 dx dy ...
136 ページ
... dx + ẞ dy + y dz n1 - = γ θα ay ax ав θα d.S. ƒƒ [ ~ ' ( ~ 7 - 32 ) + n3 ( 3a - 31 ) + n3 ( 323 ) ] ds M მყ - ax อน F2 Proof . Define w on w = M by w F1 dx + F2 dy + F3 dz . Since VF has components D2F3 — DзF2 , DзF1 — D1F3 , D1F2 ...
... dx + ẞ dy + y dz n1 - = γ θα ay ax ав θα d.S. ƒƒ [ ~ ' ( ~ 7 - 32 ) + n3 ( 3a - 31 ) + n3 ( 323 ) ] ds M მყ - ax อน F2 Proof . Define w on w = M by w F1 dx + F2 dy + F3 dz . Since VF has components D2F3 — DзF2 , DзF1 — D1F3 , D1F2 ...
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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