| 11. | Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces.
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| 12. | These points generate a tangent space of definite dimension " at " each point.
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| 13. | Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces.
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| 14. | Tangent spaces to an n-dimensional smooth manifold are n-dimensional linear spaces.
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| 15. | Further study of tangent measures and tangent spaces leads to the notion of a varifold.
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| 16. | The vectors in this tangent space are different from the vectors of the vector manifold.
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| 17. | The notion of approximate tangent spaces is very closely related to that of rectifiable sets.
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| 18. | Loosely speaking, rectifiable sets are precisely those for which approximate tangent spaces exist almost everywhere.
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| 19. | The tangent space at each event is a vector space of the same dimension as spacetime,.
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| 20. | For each, L ( v, q ) is a convex function of the tangent space.
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