. "\uBBF8\uBD84\uAE30\uD558\uD559\uC5D0\uC11C \uB9AC \uBBF8\uBD84(Lie\u5FAE\u5206, \uC601\uC5B4: Lie derivative)\uC740 \uB9E4\uB044\uB7EC\uC6B4 \uB2E4\uC591\uCCB4 \uC704\uC5D0\uC11C \uC544\uD540 \uC811\uC18D \uC5C6\uC774 \uC815\uC758\uB420 \uC218 \uC788\uB294, \uD150\uC11C\uC7A5\uC758 \uBBF8\uBD84 \uC5F0\uC0B0\uC774\uB2E4. \uAE30\uD638\uB294 ."@ko . . . . . "Derivada de Lie"@es . . . "1117685273"^^ . . . . . . . . . . . . . . "\u041F\u043E\u0445\u0456\u0434\u043D\u0430 \u041B\u0456"@uk . . . . . . "Em matem\u00E1tica, uma derivada de Lie \u00E9 uma na \u00E1lgebra de fun\u00E7\u00F5es diferenci\u00E1veis sobre uma variedade diferenci\u00E1vel , cuja defini\u00E7\u00E3o pode estender-se \u00E0 \u00E1lgebra tensorial da variedade. Obtem-se ent\u00E3o o que em topologia diferencial se denomina deriva\u00E7\u00E3o tensorial: uma aplica\u00E7\u00E3o -linear sobre o conjunto de tensores de tipo (r,s), que preserva o tipo tensorial e satisfaz a regra do produto de Leibniz e que comuta com as Contra\u00E7\u00E3o de tensor|contra\u00E7\u00F5es]]. Para definir a derivada de Lie sobre o conjunto de tensores de tipo (r,s) basta definir-se sua a\u00E7\u00E3o sobre fun\u00E7\u00F5es e sobre campos de vetores: Assim, se X \u00E9 um campo diferenci\u00E1vel de vetores, se define a derivada de Lie em rela\u00E7\u00E3o a X como a \u00FAnica deriva\u00E7\u00E3o tensorial tal que: \n* para toda fun\u00E7\u00E3o diferenci\u00E1vel f. \n* para todo campo diferenci\u00E1vel Y, onde [.,.] \u00E9 o . A derivada assim definida satisfar\u00E1 automaticamente as propriedades citadas de uma deriva\u00E7\u00E3o tensorial: \n* a regra do produto comutar\u00E1 com as contra\u00E7\u00F5es. O espa\u00E7o vetorial de todas as derivadas de Lie em M forma por sua vez uma \u00E1lgebra de Lie infinita dimensional em rela\u00E7\u00E3o ao colchete de Lie."@pt . . . "\u5728\u5FAE\u5206\u5E7E\u4F55\u4E2D\uFF0C\u674E\u5BFC\u6570\uFF08Lie derivative\uFF09\u662F\u4E00\u500B\u4EE5\u7D22\u752B\u65AF\u00B7\u674E\u547D\u540D\u7684\u7B97\u5B50\uFF0C\u4F5C\u7528\u5728\u6D41\u5F62\u4E0A\u7684\u5F35\u91CF\u5834\uFF0C\u5411\u91CF\u5834\u6216\u51FD\u6570\uFF0C\u5C07\u8A72\u5F35\u91CF\u6CBF\u8457\u67D0\u500B\u5411\u91CF\u5834\u7684\u6D41\u505A\u65B9\u5411\u5C0E\u6578\u3002\u56E0\u70BA\u8A72\u4F5C\u7528\u5728\u5EA7\u6A19\u8B8A\u63DB\u4E0B\u4FDD\u6301\u4E0D\u8B8A\uFF0C\u56E0\u6B64\uFF0C\u8A72\u674E\u5C0E\u6578\u5728\u4E00\u822C\u7684\u6D41\u5F62\u4E0A\u90FD\u662F\u5B9A\u7FA9\u826F\u597D\u7684\u3002 \u6240\u6709\u674E\u5BFC\u6570\u7EC4\u6210\u7684\u5411\u91CF\u7A7A\u95F4\u5BF9\u5E94\u4E8E\u5982\u4E0B\u7684\u674E\u62EC\u53F7\u6784\u6210\u4E00\u4E2A\u65E0\u9650\u7EF4\u674E\u4EE3\u6570\u3002 \u674E\u5BFC\u6570\u7528\u5411\u91CF\u573A\u8868\u793A\uFF0C\u8FD9\u4E9B\u5411\u91CF\u573A\u53EF\u770B\u4F5CM\u4E0A\u7684\u6D41\uFF08flow, \u4E5F\u5C31\u662F\u65F6\u53D8\u5FAE\u5206\u540C\u80DA\uFF09\u7684\u3002\u4ECE\u53E6\u4E00\u89D2\u5EA6\u770B\uFF0CM\u4E0A\u7684\u5FAE\u5206\u540C\u80DA\u7EC4\u6210\u7684\u7FA4\uFF0C\u6709\u5176\u5BF9\u5E94\u7684\u674E\u5BFC\u6570\u7684\u674E\u4EE3\u6570\u7ED3\u6784\uFF0C\u5728\u67D0\u79CD\u610F\u4E49\u4E0A\u548C\u674E\u7FA4\u7406\u8BBA\u76F4\u63A5\u76F8\u5173\u3002"@zh . . "Lie derivative"@en . . . . . "\uB9AC \uBBF8\uBD84"@ko . . . . . "En matem\u00E1tica, una derivada de Lie es una derivaci\u00F3n en el \u00E1lgebra de funciones diferenciables sobre una variedad diferenciable , cuya definici\u00F3n puede extenderse al \u00E1lgebra tensorial de la variedad. Obtenemos entonces lo que en topolog\u00EDa diferencial se denomina derivaci\u00F3n tensorial:una aplicaci\u00F3n -lineal sobre el conjunto de tensores de tipo (r,s), que preserva el tipo tensorial y satisface la regla del producto de Leibniz y que conmuta con las contracciones. \n* para toda funci\u00F3n diferenciable f. \n* para todo campo diferenciable Y. Donde [,] es el corchete de Lie."@es . . . . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u30EA\u30FC\u5FAE\u5206\uFF08\u30EA\u30FC\u3073\u3076\u3093\u3001\u82F1: Lie derivative\uFF09\u306F\u3001\u591A\u69D8\u4F53 M \u4E0A\u306E\u30C6\u30F3\u30BD\u30EB\u5834\u5168\u4F53\u306E\u6210\u3059\u591A\u5143\u74B0\u4E0A\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u5FAE\u5206\uFF08\u5C0E\u5206\u3068\u3082\uFF09\u306E\u4E00\u7A2E\u3067\u3042\u308B\u3002\u30BD\u30D5\u30B9\u30FB\u30EA\u30FC\u306B\u3061\u306A\u3093\u3067\u540D\u3065\u3051\u3089\u308C\u305F\u3002M \u4E0A\u306E\u30EA\u30FC\u5FAE\u5206\u5168\u4F53\u306E\u6210\u3059\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306F\u6B21\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u30EA\u30FC\u62EC\u5F27\u7A4D \u306B\u3064\u3044\u3066\u7121\u9650\u6B21\u5143\u306E\u30EA\u30FC\u74B0\u3092\u6210\u3059\u3002\u30EA\u30FC\u5FAE\u5206\u306F M \u4E0A\u306E\u6D41\u308C\uFF08flow; \u30D5\u30ED\u30FC\u3001 \u306A\u5FAE\u5206\u540C\u76F8\u5199\u50CF\uFF09\u306E\u3068\u3057\u3066\u30D9\u30AF\u30C8\u30EB\u5834\u306B\u3088\u3063\u3066\u8868\u3055\u308C\u308B\u3002\u3082\u3046\u5C11\u3057\u5225\u306A\u8A00\u3044\u65B9\u3092\u3059\u308C\u3070\u3001\u30EA\u30FC\u7FA4\u8AD6\u306E\u65B9\u6CD5\u306E\u76F4\u63A5\u306E\u985E\u4F3C\u7269\u3067\u306F\u3042\u308B\u304C\u3001M \u4E0A\u306E\u5FAE\u5206\u540C\u76F8\u5199\u50CF\u5168\u4F53\u306E\u6210\u3059\u7FA4\u306F\u4ED8\u968F\u3059\u308B\u30EA\u30FC\u74B0\u69CB\u9020\uFF08\u3082\u3061\u308D\u3093\u305D\u308C\u306F\u30EA\u30FC\u5FAE\u5206\u5168\u4F53\u306E\u306A\u3059\u30EA\u30FC\u74B0\u306E\u3053\u3068\u3060\u304C\uFF09\u3092\u6301\u3064\u3068\u3044\u3046\u3053\u3068\u304C\u3067\u304D\u308B\u3002"@ja . . "\uBBF8\uBD84\uAE30\uD558\uD559\uC5D0\uC11C \uB9AC \uBBF8\uBD84(Lie\u5FAE\u5206, \uC601\uC5B4: Lie derivative)\uC740 \uB9E4\uB044\uB7EC\uC6B4 \uB2E4\uC591\uCCB4 \uC704\uC5D0\uC11C \uC544\uD540 \uC811\uC18D \uC5C6\uC774 \uC815\uC758\uB420 \uC218 \uC788\uB294, \uD150\uC11C\uC7A5\uC758 \uBBF8\uBD84 \uC5F0\uC0B0\uC774\uB2E4. \uAE30\uD638\uB294 ."@ko . . . . "In differential geometry, the Lie derivative (/li\u02D0/ LEE), named after Sophus Lie by W\u0142adys\u0142aw \u015Alebodzi\u0144ski, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted . The differential operator is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word \"Lie\" is dropped, and one simply speaks of the derivative of a function. The Lie derivative of a vector field Y with respect to another vector field X is known as the \"Lie bracket\" of X and Y, and is often denoted [X,Y] instead of . The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity valid for any vector fields X and Y and any tensor field T. Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory. Generalisations exist for spinor fields, fibre bundles with connection and vector-valued differential forms."@en . . "p/l058560"@en . . . . . . "In matematica, la derivata di Lie, cos\u00EC chiamata in onore di Sophus Lie da parte di , calcola la variazione di un campo vettoriale, pi\u00F9 in generale di un campo tensoriale, lungo il flusso di un altro campo vettoriale. L'idea base della derivata di Lie \u00E8 quella di confrontare due tensori, uno l'evoluto dell'altro, lungo una stessa curva che \u00E8 soluzione di un opportuno campo vettoriale e facendo il limite per lo spostamento infinitesimale. Tale derivata \u00E8 strettamente correlata con l'idea che sottende la derivata di una sezione lungo una curva."@it . . . . . "La d\u00E9riv\u00E9e de Lie est une op\u00E9ration de diff\u00E9rentiation naturelle sur les champs de tenseurs, en particulier lesformes diff\u00E9rentielles, g\u00E9n\u00E9ralisant la d\u00E9rivation directionnelle d'une fonction surun ouvert de ou plus g\u00E9n\u00E9ralement surune vari\u00E9t\u00E9 diff\u00E9rentielle. On note ici M une vari\u00E9t\u00E9 diff\u00E9rentielle de dimension n, \u03A9M l'espace des formes diff\u00E9rentielles sur M et X un champ de vecteurs sur M."@fr . . . . . . . . . . "In matematica, la derivata di Lie, cos\u00EC chiamata in onore di Sophus Lie da parte di , calcola la variazione di un campo vettoriale, pi\u00F9 in generale di un campo tensoriale, lungo il flusso di un altro campo vettoriale. L'idea base della derivata di Lie \u00E8 quella di confrontare due tensori, uno l'evoluto dell'altro, lungo una stessa curva che \u00E8 soluzione di un opportuno campo vettoriale e facendo il limite per lo spostamento infinitesimale. Tale derivata \u00E8 strettamente correlata con l'idea che sottende la derivata di una sezione lungo una curva."@it . . . "Derivada de Lie"@pt . "D\u00E9riv\u00E9e de Lie"@fr . . . . . . . "En matem\u00E1tica, una derivada de Lie es una derivaci\u00F3n en el \u00E1lgebra de funciones diferenciables sobre una variedad diferenciable , cuya definici\u00F3n puede extenderse al \u00E1lgebra tensorial de la variedad. Obtenemos entonces lo que en topolog\u00EDa diferencial se denomina derivaci\u00F3n tensorial:una aplicaci\u00F3n -lineal sobre el conjunto de tensores de tipo (r,s), que preserva el tipo tensorial y satisface la regla del producto de Leibniz y que conmuta con las contracciones. Para definir la derivada de Lie sobre el conjunto de tensores de tipo (r,s) bastar\u00E1 con definir su acci\u00F3n sobre funciones y sobre campos de vectores:As\u00ED, si X es un campo diferenciable de vectores, se define la derivada de Lie con respecto a X como la \u00FAnica derivaci\u00F3n tensorial tal que:\u200B \n* para toda funci\u00F3n diferenciable f. \n* para todo campo diferenciable Y. Donde [,] es el corchete de Lie. La derivada as\u00ED definida satisfar\u00E1 autom\u00E1ticamente las propiedades citadas de una derivaci\u00F3n tensorial: \n* la regla del producto \n* conmutar\u00E1 con las contracciones. El espacio vectorial de todas las derivadas de Lie en M forma a su vez un \u00E1lgebra de Lie infinito dimensional con respecto al corchete de Lie. Aunque menos habitual, tambi\u00E9n se denota a la derivada de Lie de respecto de un campo como . Esta notaci\u00F3n, en ocasiones m\u00E1s limpia que la anterior pues evita sub\u00EDndices, proviene del profesor ."@es . "\u041F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F \u041B\u0438"@ru . "Lie-Ableitung"@de . "\u041F\u043E\u0445\u0456\u0434\u043D\u0430 \u041B\u0456 \u0442\u0435\u043D\u0437\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u044F \u0437\u0430 \u043D\u0430\u043F\u0440\u044F\u043C\u043A\u043E\u043C \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u044F \u2014 \u0433\u043E\u043B\u043E\u0432\u043D\u0430 \u043B\u0456\u043D\u0456\u0439\u043D\u0430 \u0447\u0430\u0441\u0442\u0438\u043D\u0430 \u043F\u0440\u0438\u0440\u043E\u0441\u0442\u0443 \u0442\u0435\u043D\u0437\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u044F \u043F\u0440\u0438 \u0439\u043E\u0433\u043E \u043F\u0435\u0440\u0435\u0442\u0432\u043E\u0440\u0435\u043D\u043D\u0456, \u044F\u043A\u0435 \u0456\u043D\u0434\u0443\u043A\u043E\u0432\u0430\u043D\u0435 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E\u044E \u043E\u0434\u043D\u043E\u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0438\u0447\u043D\u043E\u044E \u0433\u0440\u0443\u043F\u043E\u044E \u0434\u0438\u0444\u0435\u043E\u043C\u043E\u0440\u0444\u0456\u0437\u043C\u0456\u0432 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443, \u0449\u043E \u043F\u043E\u0440\u043E\u0434\u0436\u0435\u043D\u0430 \u043F\u043E\u043B\u0435\u043C . \u0417\u0430\u0437\u0432\u0438\u0447\u0430\u0439 \u043F\u043E\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F ."@uk . . . . . "In de differentiaalmeetkunde, een deelgebied van de wiskunde, evalueert de lie-afgeleide, door de Poolse wiskundige naar Sophus Lie vernoemd, de verandering van een vectorveld of meer in het algemeen een langs de van een ander vectorveld. Deze verandering is co\u00F6rdinaatinvariant en om die reden kan de lie-afgeleide worden gedefinieerd op elke differentieerbare vari\u00EBteit."@nl . . . . . "294437"^^ . . . . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u30EA\u30FC\u5FAE\u5206\uFF08\u30EA\u30FC\u3073\u3076\u3093\u3001\u82F1: Lie derivative\uFF09\u306F\u3001\u591A\u69D8\u4F53 M \u4E0A\u306E\u30C6\u30F3\u30BD\u30EB\u5834\u5168\u4F53\u306E\u6210\u3059\u591A\u5143\u74B0\u4E0A\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u5FAE\u5206\uFF08\u5C0E\u5206\u3068\u3082\uFF09\u306E\u4E00\u7A2E\u3067\u3042\u308B\u3002\u30BD\u30D5\u30B9\u30FB\u30EA\u30FC\u306B\u3061\u306A\u3093\u3067\u540D\u3065\u3051\u3089\u308C\u305F\u3002M \u4E0A\u306E\u30EA\u30FC\u5FAE\u5206\u5168\u4F53\u306E\u6210\u3059\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306F\u6B21\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u30EA\u30FC\u62EC\u5F27\u7A4D \u306B\u3064\u3044\u3066\u7121\u9650\u6B21\u5143\u306E\u30EA\u30FC\u74B0\u3092\u6210\u3059\u3002\u30EA\u30FC\u5FAE\u5206\u306F M \u4E0A\u306E\u6D41\u308C\uFF08flow; \u30D5\u30ED\u30FC\u3001 \u306A\u5FAE\u5206\u540C\u76F8\u5199\u50CF\uFF09\u306E\u3068\u3057\u3066\u30D9\u30AF\u30C8\u30EB\u5834\u306B\u3088\u3063\u3066\u8868\u3055\u308C\u308B\u3002\u3082\u3046\u5C11\u3057\u5225\u306A\u8A00\u3044\u65B9\u3092\u3059\u308C\u3070\u3001\u30EA\u30FC\u7FA4\u8AD6\u306E\u65B9\u6CD5\u306E\u76F4\u63A5\u306E\u985E\u4F3C\u7269\u3067\u306F\u3042\u308B\u304C\u3001M \u4E0A\u306E\u5FAE\u5206\u540C\u76F8\u5199\u50CF\u5168\u4F53\u306E\u6210\u3059\u7FA4\u306F\u4ED8\u968F\u3059\u308B\u30EA\u30FC\u74B0\u69CB\u9020\uFF08\u3082\u3061\u308D\u3093\u305D\u308C\u306F\u30EA\u30FC\u5FAE\u5206\u5168\u4F53\u306E\u306A\u3059\u30EA\u30FC\u74B0\u306E\u3053\u3068\u3060\u304C\uFF09\u3092\u6301\u3064\u3068\u3044\u3046\u3053\u3068\u304C\u3067\u304D\u308B\u3002"@ja . "In der Analysis bezeichnet die Lie-Ableitung (nach Sophus Lie) die Ableitung eines Vektorfeldes oder allgemeiner eines Tensorfeldes entlang eines Vektorfeldes.Auf dem Raum der Vektorfelder wird durch die Lie-Ableitung eine Lie-Klammer definiert, die Jacobi-Lie-Klammer genannt wird. Der Raum der Vektorfelder wird durch diese Operation zu einer Lie-Algebra. In der Allgemeinen Relativit\u00E4tstheorie und in der geometrischen Formulierung der Hamiltonschen Mechanik wird die Lie-Ableitung verwendet, um Symmetrien aufzudecken, diese zur L\u00F6sung von Problemen auszunutzen und beispielsweise Konstanten der Bewegung zu finden."@de . "Derivata di Lie"@it . . . . "In der Analysis bezeichnet die Lie-Ableitung (nach Sophus Lie) die Ableitung eines Vektorfeldes oder allgemeiner eines Tensorfeldes entlang eines Vektorfeldes.Auf dem Raum der Vektorfelder wird durch die Lie-Ableitung eine Lie-Klammer definiert, die Jacobi-Lie-Klammer genannt wird. Der Raum der Vektorfelder wird durch diese Operation zu einer Lie-Algebra."@de . 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"\u5728\u5FAE\u5206\u5E7E\u4F55\u4E2D\uFF0C\u674E\u5BFC\u6570\uFF08Lie derivative\uFF09\u662F\u4E00\u500B\u4EE5\u7D22\u752B\u65AF\u00B7\u674E\u547D\u540D\u7684\u7B97\u5B50\uFF0C\u4F5C\u7528\u5728\u6D41\u5F62\u4E0A\u7684\u5F35\u91CF\u5834\uFF0C\u5411\u91CF\u5834\u6216\u51FD\u6570\uFF0C\u5C07\u8A72\u5F35\u91CF\u6CBF\u8457\u67D0\u500B\u5411\u91CF\u5834\u7684\u6D41\u505A\u65B9\u5411\u5C0E\u6578\u3002\u56E0\u70BA\u8A72\u4F5C\u7528\u5728\u5EA7\u6A19\u8B8A\u63DB\u4E0B\u4FDD\u6301\u4E0D\u8B8A\uFF0C\u56E0\u6B64\uFF0C\u8A72\u674E\u5C0E\u6578\u5728\u4E00\u822C\u7684\u6D41\u5F62\u4E0A\u90FD\u662F\u5B9A\u7FA9\u826F\u597D\u7684\u3002 \u6240\u6709\u674E\u5BFC\u6570\u7EC4\u6210\u7684\u5411\u91CF\u7A7A\u95F4\u5BF9\u5E94\u4E8E\u5982\u4E0B\u7684\u674E\u62EC\u53F7\u6784\u6210\u4E00\u4E2A\u65E0\u9650\u7EF4\u674E\u4EE3\u6570\u3002 \u674E\u5BFC\u6570\u7528\u5411\u91CF\u573A\u8868\u793A\uFF0C\u8FD9\u4E9B\u5411\u91CF\u573A\u53EF\u770B\u4F5CM\u4E0A\u7684\u6D41\uFF08flow, \u4E5F\u5C31\u662F\u65F6\u53D8\u5FAE\u5206\u540C\u80DA\uFF09\u7684\u3002\u4ECE\u53E6\u4E00\u89D2\u5EA6\u770B\uFF0CM\u4E0A\u7684\u5FAE\u5206\u540C\u80DA\u7EC4\u6210\u7684\u7FA4\uFF0C\u6709\u5176\u5BF9\u5E94\u7684\u674E\u5BFC\u6570\u7684\u674E\u4EE3\u6570\u7ED3\u6784\uFF0C\u5728\u67D0\u79CD\u610F\u4E49\u4E0A\u548C\u674E\u7FA4\u7406\u8BBA\u76F4\u63A5\u76F8\u5173\u3002"@zh . . . . . "La d\u00E9riv\u00E9e de Lie est une op\u00E9ration de diff\u00E9rentiation naturelle sur les champs de tenseurs, en particulier lesformes diff\u00E9rentielles, g\u00E9n\u00E9ralisant la d\u00E9rivation directionnelle d'une fonction surun ouvert de ou plus g\u00E9n\u00E9ralement surune vari\u00E9t\u00E9 diff\u00E9rentielle. On note ici M une vari\u00E9t\u00E9 diff\u00E9rentielle de dimension n, \u03A9M l'espace des formes diff\u00E9rentielles sur M et X un champ de vecteurs sur M."@fr . . "\u0627\u0634\u062A\u0642\u0627\u0642 \u0644\u064A"@ar . . . . . . . . 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"In de differentiaalmeetkunde, een deelgebied van de wiskunde, evalueert de lie-afgeleide, door de Poolse wiskundige naar Sophus Lie vernoemd, de verandering van een vectorveld of meer in het algemeen een langs de van een ander vectorveld. Deze verandering is co\u00F6rdinaatinvariant en om die reden kan de lie-afgeleide worden gedefinieerd op elke differentieerbare vari\u00EBteit."@nl . "Em matem\u00E1tica, uma derivada de Lie \u00E9 uma na \u00E1lgebra de fun\u00E7\u00F5es diferenci\u00E1veis sobre uma variedade diferenci\u00E1vel , cuja defini\u00E7\u00E3o pode estender-se \u00E0 \u00E1lgebra tensorial da variedade. Obtem-se ent\u00E3o o que em topologia diferencial se denomina deriva\u00E7\u00E3o tensorial: uma aplica\u00E7\u00E3o -linear sobre o conjunto de tensores de tipo (r,s), que preserva o tipo tensorial e satisfaz a regra do produto de Leibniz e que comuta com as Contra\u00E7\u00E3o de tensor|contra\u00E7\u00F5es]]. Para definir a derivada de Lie sobre o conjunto de tensores de tipo (r,s) basta definir-se sua a\u00E7\u00E3o sobre fun\u00E7\u00F5es e sobre campos de vetores:"@pt . "In differential geometry, the Lie derivative (/li\u02D0/ LEE), named after Sophus Lie by W\u0142adys\u0142aw \u015Alebodzi\u0144ski, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. valid for any vector fields X and Y and any tensor field T."@en . . "\u0627\u0634\u062A\u0642\u0627\u0642 \u0644\u064A \u0647\u0648 \u0646\u0648\u0639 \u0645\u0646 \u0627\u0644\u0627\u0634\u062A\u0642\u0627\u0642\u0627\u062A \u0644\u0644\u062F\u0627\u0644\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u064A\u0629. \u0633\u0645\u064A\u062A \u0628\u0627\u0633\u0645 \u0648\u0627\u0636\u0639\u0647\u0627 \u0633\u0648\u0641\u0648\u0633 \u0644\u064A. \u064A\u0642\u064A\u0645 \u0647\u0630\u0627 \u0627\u0644\u0627\u0634\u062A\u0642\u0627\u0642 \u0627\u0644\u062D\u0642\u0644 \u0627\u0644\u0627\u062A\u062C\u0627\u0647\u064A \u0639\u0644\u0649 \u0637\u0648\u0644 \u062A\u062F\u0641\u0642 \u062D\u0642\u0644 \u0627\u062A\u062C\u0627\u0647\u064A \u0622\u062E\u0631. \u0625\u0646 \u0627\u0634\u062A\u0642\u0627\u0642 \u0644\u064A \u0647\u0648 \u0627\u0634\u062A\u0642\u0627\u0642 \u0639\u0644\u0649 \u062C\u0628\u0631 \u0644\u0645\u062A\u0639\u062F\u062F \u0634\u0639\u0628 M. \u0627\u0644\u062D\u0642\u0644 \u0627\u0644\u0627\u062A\u062C\u0627\u0647\u064A \u0644\u0645\u0634\u062A\u0642 \u0644\u064A \u0639\u0644\u0649 M \u064A\u0634\u0643\u0644 \u062C\u0628\u0631 \u0644\u064A \u0644\u0627\u0646\u0647\u0627\u0626\u064A \u0627\u0644\u0628\u0639\u062F \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0645\u0639\u0631\u0641\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062A\u0627\u0644\u064A:"@ar . . . . . "36130"^^ . . . . . . "\u674E\u5BFC\u6570"@zh . 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"Lie derivative"@en . "Lie-afgeleide"@nl . . . . . . . . . . . "\u30EA\u30FC\u5FAE\u5206"@ja . . . . . . . . . . . . .