"differential structure" meaning in English

{
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    {
      "form": "differential structures",
      "tags": [
        "plural"
      ]
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      "expansion": "differential structure (plural differential structures)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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      "examples": [
        {
          "ref": "2002, R.W. Carroll, Calculus Revisited, Springer, pages 12–37",
          "text": "First let (M,#x5C;tau) be a topological space. The sheaf #x5C;mathfrak#x7B;G#x7D; of real continuous functions on (M,#x5C;tau) is said to be a differential structure on M if for any open set U#x5C;in#x5C;tau, any functions f#x5F;i#x5C;in#x5C;mathfrak#x7B;G#x7D;(U), and any w#x5C;inC#x5C;infty(#x5C;mathbb#x7B;R#x7D;ⁿ), the superposition w#x5C;circ(f#x5F;1,#x5C;dotsf#x5F;n)#x5C;in#x5C;mathfrak#x7B;G#x7D;(U).",
          "type": "quotation"
        },
        {
          "ref": "2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29",
          "text": "It is important to emphasise that, among the various choices for #x5C;lambdaⁱ#x5F;#x7B;jk#x7D; and A#x7B;i#x5C;a#x7D;#x5F;#x7B;#x5C;j#x5C;b#x7D;, some are intrinsic to the differential structure of the manifold #x5C;mathcal#x7B;M#x7D;. In other words, among all the operators of D-differentiation, some arise from the differential structure of #x5C;mathcal#x7B;M#x7D;.[…]On the other hand, there exist operators of D-differentiation that do not follow from the differential structure of #x5C;mathcal#x7B;M#x7D;.",
          "type": "quotation"
        },
        {
          "ref": "2010, Vladimir Igorevich Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, page 369",
          "text": "This chapter is concerned with differentiable measures on general measurable spaces and on measurable spaces equipped with certain differential structures enabling us to consider differentiations along vector fields.",
          "type": "quotation"
        },
        {
          "ref": "2015, Stephen Bruce Sontz, Principal Bundles: The Classical Case, Springer, page 12",
          "text": "Given a Hausdorff topological space M with differential structures #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;2 (these being maximal smooth atlases), we say that #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;1 are equivalent if there is a diffeomorphism #x5C;phi#x3A;(M,#x5C;mathcal#x7B;A#x7D;#x5F;1)#x5C;rightarrow(M,#x5C;mathcal#x7B;A#x7D;#x5F;2) from M with the first differential structure to M with the second differential structure. Note that #x5C;phi need not be the identity function.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it."
      ],
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        [
          "differentiation",
          "differentiation"
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          "function",
          "function"
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      "raw_glosses": [
        "(topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it."
      ],
      "related": [
        {
          "word": "differentiable manifold"
        },
        {
          "word": "differential manifold"
        },
        {
          "word": "smooth manifold"
        }
      ],
      "synonyms": [
        {
          "word": "differentiable structure"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
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      "wikipedia": [
        "differential structure"
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    }
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  "word": "differential structure"
}

{
  "forms": [
    {
      "form": "differential structures",
      "tags": [
        "plural"
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      "expansion": "differential structure (plural differential structures)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "differentiable manifold"
    },
    {
      "word": "differential manifold"
    },
    {
      "word": "smooth manifold"
    }
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        {
          "ref": "2002, R.W. Carroll, Calculus Revisited, Springer, pages 12–37",
          "text": "First let (M,#x5C;tau) be a topological space. The sheaf #x5C;mathfrak#x7B;G#x7D; of real continuous functions on (M,#x5C;tau) is said to be a differential structure on M if for any open set U#x5C;in#x5C;tau, any functions f#x5F;i#x5C;in#x5C;mathfrak#x7B;G#x7D;(U), and any w#x5C;inC#x5C;infty(#x5C;mathbb#x7B;R#x7D;ⁿ), the superposition w#x5C;circ(f#x5F;1,#x5C;dotsf#x5F;n)#x5C;in#x5C;mathfrak#x7B;G#x7D;(U).",
          "type": "quotation"
        },
        {
          "ref": "2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29",
          "text": "It is important to emphasise that, among the various choices for #x5C;lambdaⁱ#x5F;#x7B;jk#x7D; and A#x7B;i#x5C;a#x7D;#x5F;#x7B;#x5C;j#x5C;b#x7D;, some are intrinsic to the differential structure of the manifold #x5C;mathcal#x7B;M#x7D;. In other words, among all the operators of D-differentiation, some arise from the differential structure of #x5C;mathcal#x7B;M#x7D;.[…]On the other hand, there exist operators of D-differentiation that do not follow from the differential structure of #x5C;mathcal#x7B;M#x7D;.",
          "type": "quotation"
        },
        {
          "ref": "2010, Vladimir Igorevich Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, page 369",
          "text": "This chapter is concerned with differentiable measures on general measurable spaces and on measurable spaces equipped with certain differential structures enabling us to consider differentiations along vector fields.",
          "type": "quotation"
        },
        {
          "ref": "2015, Stephen Bruce Sontz, Principal Bundles: The Classical Case, Springer, page 12",
          "text": "Given a Hausdorff topological space M with differential structures #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;2 (these being maximal smooth atlases), we say that #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;1 are equivalent if there is a diffeomorphism #x5C;phi#x3A;(M,#x5C;mathcal#x7B;A#x7D;#x5F;1)#x5C;rightarrow(M,#x5C;mathcal#x7B;A#x7D;#x5F;2) from M with the first differential structure to M with the second differential structure. Note that #x5C;phi need not be the identity function.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it."
      ],
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        [
          "differentiation",
          "differentiation"
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          "function",
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      "raw_glosses": [
        "(topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "differential structure"
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  "synonyms": [
    {
      "word": "differentiable structure"
    }
  ],
  "word": "differential structure"
}