An Introduction to Differentiable Manifolds and Riemannian GeometryAcademic Press, 1986/04/21 - 429 ページ An Introduction to Differentiable Manifolds and Riemannian Geometry |
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... Theorem (Mean Value Theorem) Let g be a differentiable function on an open set U c R"; let a e U and suppose that U is starlike with respect to a. Then given xe U there exists () e R, 0 < 0 < 1, such that wo-oo-j (::), -2) Proof Set f(t) ...
... Theorem (Mean Value Theorem) Let g be a differentiable function on an open set U c R"; let a e U and suppose that U is starlike with respect to a. Then given xe U there exists () e R, 0 < 0 < 1, such that wo-oo-j (::), -2) Proof Set f(t) ...
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... Theorem Let a e U be an open subset of R" which is starlike with respect to a, and let F: U → R" be differentiable on U with |&fo/öx| < K, 1 < i < m, 1 < j < n, 26 II F U NCT! O N S OF SEVER AL VARIA B LES A N D M A PP N G S.
... Theorem Let a e U be an open subset of R" which is starlike with respect to a, and let F: U → R" be differentiable on U with |&fo/öx| < K, 1 < i < m, 1 < j < n, 26 II F U NCT! O N S OF SEVER AL VARIA B LES A N D M A PP N G S.
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... Let - 1 { *(x) = —. dt, i = 1,..., n; g'( ) |. |. a + tsix - a) these are C*-functions and satisfy the two conditions. | Proof of Theorem 4.1 Using these lemmas we may complete the proof of Theorem 4.1. Suppose D is any derivation on C ...
... Let - 1 { *(x) = —. dt, i = 1,..., n; g'( ) |. |. a + tsix - a) these are C*-functions and satisfy the two conditions. | Proof of Theorem 4.1 Using these lemmas we may complete the proof of Theorem 4.1. Suppose D is any derivation on C ...
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... theorem” and contrasts strongly the behavior of C* and C* functions on R". (There exist stronger versions of this theorem as we shall see later.) (5.1) Theorem Let F c R" be a closed set and K c R" compact, Fo K = 2). Then there is a C ...
... theorem” and contrasts strongly the behavior of C* and C* functions on R". (There exist stronger versions of this theorem as we shall see later.) (5.1) Theorem Let F c R" be a closed set and K c R" compact, Fo K = 2). Then there is a C ...
41 ページ
... Let pe U, an open subset of R", and let X, e T,(R") be a vector at p. Show that X, may be extended to a C* vector field X on U. 8. In Theorem 5.1, assume only that K is closed (not necessarily compact). Does the theorem still hold? 6 ...
... Let pe U, an open subset of R", and let X, e T,(R") be a vector at p. Show that X, may be extended to a C* vector field X on U. 8. In Theorem 5.1, assume only that K is closed (not necessarily compact). Does the theorem still hold? 6 ...
目次
1 | |
20 | |
51 | |
Chapter IV Vector Fields on a Manifold | 106 |
Chapter V Tensors and Tensor Fields on Manifolds | 176 |
Chapter VI Integration on Manifolds | 229 |
Chapter VII Differentiation on Riemannian Manifolds | 297 |
Chapter VIII Curvature | 365 |
References | 417 |
Index | 423 |
多く使われている語句
algebra basis bi-invariant C*-vector field compact completes the proof components connected coordinate frames coordinate neighborhood Corollary corresponding countable covariant tensor covering curve p(t defined definition denote derivative diffeomorphism differentiable manifold dimension domain of integration element equations equivalent Euclidean space example Exercise exists fact finite fixed point formula functions geodesic geometry given Gl(n hence homeomorphism homotopy identity imbedding inner product integral curve isometry isomorphism Lemma Let F Lie group G linear map mapping F matrix notation obtain one-parameter subgroup one-to-one open set open subset oriented orthogonal orthonormal parameter plane properly discontinuously properties prove rank real numbers regular submanifold Remark Riemannian manifold Riemannian metric Section Show structure submanifold subspace suppose surface symmetric tangent space tangent vector tensor field Theorem Let topology uniquely determined vector field vector space zero