An Introduction to Differentiable Manifolds and Riemannian GeometryAcademic Press, 1986/04/21 - 429 ページ An Introduction to Differentiable Manifolds and Riemannian Geometry |
この書籍内から
検索結果1-5 / 89
25 ページ
... Let f(0) = 0, f(t) = exp(-1/t”) for t + 0; fis C* on R. Is it C" on R2) 7. Prove (1.4), that is, prove that g o f is differentiable at t = to and that its derivative is given by (1.4). 8. Prove Corollary 1.7. } 5. Sometimes it is ...
... Let f(0) = 0, f(t) = exp(-1/t”) for t + 0; fis C* on R. Is it C" on R2) 7. Prove (1.4), that is, prove that g o f is differentiable at t = to and that its derivative is given by (1.4). 8. Prove Corollary 1.7. } 5. Sometimes it is ...
26 ページ
... F is continuous if and only if its coordinate functions are. We shall say that F is differentiable, of class C", C ... 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 ...
... F is continuous if and only if its coordinate functions are. We shall say that F is differentiable, of class C", C ... 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 ...
27 ページ
... F: U → V c R" and G: V → RP so that H = Go F is defined on U, which it maps into R*. We may write the coordinate functions of H using those of F and G: h'(x) = g o F(x) = g(f'(x),..., f"(x)), i = 1,..., p. (2.3) Theorem (Chain rule) Let F ...
... F: U → V c R" and G: V → RP so that H = Go F is defined on U, which it maps into R*. We may write the coordinate functions of H using those of F and G: h'(x) = g o F(x) = g(f'(x),..., f"(x)), i = 1,..., p. (2.3) Theorem (Chain rule) Let F ...
28 ページ
... F and G are of class C" (or smooth) on U and V, respectively, then H = Go F is of class C" (or smooth) on U. Proof ... Let U be an open subset of R" and F: U → R", m = n, be a C* mapping. Suppose that F is injective (one-to-one into) ...
... F and G are of class C" (or smooth) on U and V, respectively, then H = Go F is of class C" (or smooth) on U. Proof ... Let U be an open subset of R" and F: U → R", m = n, be a C* mapping. Suppose that F is injective (one-to-one into) ...
34 ページ
... f = Yiffor every fe C*(a) which implies X, = Y,. Indeed we have noted the ... Let D be an arbitrary element of Ø(a). Then D is zero on any function fe C ... Let f(x', ..., x") be defined and C* on some open set U. If a e U, then there is ...
... f = Yiffor every fe C*(a) which implies X, = Y,. Indeed we have noted the ... Let D be an arbitrary element of Ø(a). Then D is zero on any function fe C ... Let f(x', ..., x") be defined and C* on some open set U. If a e U, then there is ...
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