"tangent space" meaning in All languages combined

{
  "forms": [
    {
      "form": "tangent spaces",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "tangent space (plural tangent spaces)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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          "kind": "other",
          "name": "English entries with incorrect language header",
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          "langcode": "en",
          "name": "Topology",
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      ],
      "examples": [
        {
          "ref": "2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124:",
          "text": "This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use left translations to define similar norms at the tangent spaces of Hₙ at all other points.",
          "type": "quote"
        },
        {
          "ref": "2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:",
          "text": "In a Lie group G, because left translation by an element g #x5C;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance.\nMoreover, one can give the tangent space TₑG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.",
          "type": "quote"
        },
        {
          "ref": "2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7:",
          "text": "More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!",
          "type": "quote"
        }
      ],
      "glosses": [
        "An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x."
      ],
      "holonyms": [
        {
          "word": "tangent bundle"
        }
      ],
      "id": "en-tangent_space-en-noun-GcXoCAJm",
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        [
          "vector space",
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        [
          "set",
          "set"
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        [
          "vector",
          "vector"
        ],
        [
          "tangent",
          "tangent"
        ]
      ],
      "qualifier": "differential topology",
      "raw_glosses": [
        "(topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x."
      ],
      "related": [
        {
          "word": "cotangent space"
        }
      ],
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      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "vector space tangent to a manifold",
          "tags": [
            "masculine"
          ],
          "word": "spazio tangente"
        },
        {
          "code": "ja",
          "lang": "Japanese",
          "sense": "vector space tangent to a manifold",
          "word": "接空間"
        }
      ]
    }
  ],
  "word": "tangent space"
}

{
  "forms": [
    {
      "form": "tangent spaces",
      "tags": [
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      "expansion": "tangent space (plural tangent spaces)",
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  "holonyms": [
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      "word": "tangent bundle"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "cotangent space"
    }
  ],
  "senses": [
    {
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        {
          "ref": "2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124:",
          "text": "This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use left translations to define similar norms at the tangent spaces of Hₙ at all other points.",
          "type": "quote"
        },
        {
          "ref": "2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:",
          "text": "In a Lie group G, because left translation by an element g #x5C;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance.\nMoreover, one can give the tangent space TₑG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.",
          "type": "quote"
        },
        {
          "ref": "2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7:",
          "text": "More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!",
          "type": "quote"
        }
      ],
      "glosses": [
        "An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x."
      ],
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        [
          "set",
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        [
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          "vector"
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        [
          "tangent",
          "tangent"
        ]
      ],
      "qualifier": "differential topology",
      "raw_glosses": [
        "(topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x."
      ],
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        "mathematics",
        "sciences",
        "topology"
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  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "vector space tangent to a manifold",
      "tags": [
        "masculine"
      ],
      "word": "spazio tangente"
    },
    {
      "code": "ja",
      "lang": "Japanese",
      "sense": "vector space tangent to a manifold",
      "word": "接空間"
    }
  ],
  "word": "tangent space"
}