Tag Archives: manifold

Smooth/Differentiable Manifold

Smooth Manifold A smooth manifold is a pair , where is a topological manifold and is a smooth structure on . Topological Manifold A topological -manifold is a topological space such that: 1)  is Hausdorff: For every distinct pair of … Continue reading

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RP^n Projective n-space

Define an equivalence relation on by writing if and only if . The quotient space is called projective -space. (This is one of the ways that we defined the projective plane .) The canonical projection is just . Define , … Continue reading

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Equivalence of C^infinity atlases

Equivalence of atlases is an equivalence relation. Each atlas on is equivalent to a unique maximal atlas on . Proof: Reflexive: If is a atlas, then is also a atlas. Symmetry: Let and be two atlases such that is also … Continue reading

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Tangent Space is Vector Space

Prove that the operation of linear combination, as in Definition 2.2.7, makes into an -dimensional vector space over . The zero vector is the infinitesimal curve represented by the constant . If , then where , defined for all sufficiently … Continue reading

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Multivariable Derivative and Partial Derivatives

If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique. Proof: Let , then becomes since . By choosing (all zeroes except in th position), … Continue reading

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