"\u5728\u6570\u5B66\u4E2D\uFF0C\u5F31\u5FAE\u5206\uFF08Weak Derivative\uFF09\u662F\u4E00\u4E2A\u51FD\u6570\u7684\u5FAE\u5206\uFF08\u5F3A\u5FAE\u5206\uFF09\u6982\u5FF5\u7684\u63A8\u5E7F\uFF0C\u5B83\u53EF\u4EE5\u4F5C\u7528\u4E8E\u90A3\u4E9B\u52D2\u8D1D\u683C\u53EF\u79EF\uFF08Lebesgue Integrable\uFF09\u7684\u51FD\u6570\uFF0C\u800C\u4E0D\u5FC5\u9884\u8BBE\u51FD\u6570\u7684\u53EF\u5FAE\u6027\uFF08\u4E8B\u5B9E\u4E0A\u5927\u90E8\u5206\u53EF\u4EE5\u5F31\u5FAE\u5206\u7684\u51FD\u6570\u5E76\u4E0D\u53EF\u5FAE\uFF09\u3002\u4E00\u4E2A\u5178\u578B\u7684\u52D2\u8D1D\u683C\u53EF\u79EF\u51FD\u6570\u7684\u7A7A\u95F4\u662F\u3002\u5728\u5206\u5E03\u4E2D\uFF0C\u53EF\u4EE5\u5B9A\u4E49\u4E00\u4E2A\u66F4\u4E00\u822C\u7684\u5FAE\u5206\u6982\u5FF5\u3002"@zh . "Eine schwache Ableitung ist in der Funktionalanalysis, einem Teilgebiet der Mathematik, eine Erweiterung des Begriffs der gew\u00F6hnlichen (klassischen) Ableitung. Er erm\u00F6glicht es, Funktionen eine Ableitung zuzuordnen, die nicht (stark bzw. im klassischen Sinne) differenzierbar sind. Schwache Ableitungen spielen eine gro\u00DFe Rolle in der Theorie der partiellen Differentialgleichungen. R\u00E4ume schwach differenzierbarer Funktionen sind die Sobolev-R\u00E4ume. Ein noch allgemeinerer Begriff der Ableitung ist die Distributionenableitung."@de . "\uD574\uC11D\uD559\uC5D0\uC11C \uC57D\uB3C4\uD568\uC218(\u5F31\u5C0E\u51FD\u6578, \uC601\uC5B4: weak derivative)\uB294 \uC77C\uBC18\uC801\uC778 \uB3C4\uD568\uC218\uC758 \uAC1C\uB150\uC758 \uC77C\uBC18\uD654\uC774\uB2E4. \uC774\uB97C \uD1B5\uD558\uC5EC \uACE0\uC804\uC801\uC73C\uB85C \uB3C4\uD568\uC218\uB97C \uCDE8\uD560 \uC218 \uC5C6\uB294 \uD568\uC218\uB4E4\uC758 \uB3C4\uD568\uC218\uB97C \uCDE8\uD560 \uC218 \uC788\uB2E4."@ko . . "In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space . The method of integration by parts holds that for differentiable functions and we have A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions \u03C6 vanishing at the boundary points."@en . "S\u0142aba pochodna \u2013 rozszerzenie poj\u0119cia pochodnej na funkcje lokalnie ca\u0142kowalne. Poj\u0119cie s\u0142abej pochodnej ma szerokie zastosowania w teorii r\u00F3wna\u0144 r\u00F3\u017Cniczkowych cz\u0105stkowych."@pl . . . . . . . "In matematica, la derivata debole \u00E8 una generalizzazione del concetto di derivata di una funzione a funzioni non necessariamente differenziabili, ma solamente integrabili, ovvero funzioni che appartengono allo spazio L1. La definizione di derivata debole origina le soluzioni deboli in spazi di Sobolev di problemi differenziali alle derivate parziali, frequenti in diversi settori dell'analisi, in particolare dell'analisi funzionale."@it . . "\u00AB\u0421\u043B\u0430\u0431\u043A\u0430 \u043F\u043E\u0445\u0456\u0434\u043D\u0430\u00BB (\u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0446\u0456) \u2014 \u0443\u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0435\u043D\u0435 \u043F\u043E\u043D\u044F\u0442\u0442\u044F \u043F\u043E\u0445\u0456\u0434\u043D\u043E\u0457 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 (\u00AB\u0441\u0438\u043B\u044C\u043D\u0430 \u043F\u043E\u0445\u0456\u0434\u043D\u0430\u00BB) \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0439, \u0456\u043D\u0442\u0435\u0433\u0440\u043E\u0432\u043D\u0438\u0445 \u0437\u0430 \u041B\u0435\u0431\u0435\u0433\u043E\u043C (\u0442\u043E\u0431\u0442\u043E \u0437 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 ), \u0430\u043B\u0435 \u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043E\u0432\u043D\u0438\u0445."@uk . . . . "\u0421\u043B\u0430\u0431\u0430\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F"@ru . "Schwache Ableitung"@de . . "Derivada d\u00E9bil"@es . . "S\u0142aba pochodna"@pl . . . . . "Derivata debole"@it . "En math\u00E9matiques, une fonction \u00E0 d\u00E9riv\u00E9e faible est une g\u00E9n\u00E9ralisation du concept de la d\u00E9riv\u00E9e d'une fonction (d\u00E9riv\u00E9e forte) pour les fonctions non suppos\u00E9es diff\u00E9rentiables, mais seulement int\u00E9grables, c'est-\u00E0-dire dans l'espace Lp : L1([a , b])."@fr . "\u5728\u6570\u5B66\u4E2D\uFF0C\u5F31\u5FAE\u5206\uFF08Weak Derivative\uFF09\u662F\u4E00\u4E2A\u51FD\u6570\u7684\u5FAE\u5206\uFF08\u5F3A\u5FAE\u5206\uFF09\u6982\u5FF5\u7684\u63A8\u5E7F\uFF0C\u5B83\u53EF\u4EE5\u4F5C\u7528\u4E8E\u90A3\u4E9B\u52D2\u8D1D\u683C\u53EF\u79EF\uFF08Lebesgue Integrable\uFF09\u7684\u51FD\u6570\uFF0C\u800C\u4E0D\u5FC5\u9884\u8BBE\u51FD\u6570\u7684\u53EF\u5FAE\u6027\uFF08\u4E8B\u5B9E\u4E0A\u5927\u90E8\u5206\u53EF\u4EE5\u5F31\u5FAE\u5206\u7684\u51FD\u6570\u5E76\u4E0D\u53EF\u5FAE\uFF09\u3002\u4E00\u4E2A\u5178\u578B\u7684\u52D2\u8D1D\u683C\u53EF\u79EF\u51FD\u6570\u7684\u7A7A\u95F4\u662F\u3002\u5728\u5206\u5E03\u4E2D\uFF0C\u53EF\u4EE5\u5B9A\u4E49\u4E00\u4E2A\u66F4\u4E00\u822C\u7684\u5FAE\u5206\u6982\u5FF5\u3002"@zh . . . . . "\u5F31\u5FAE\u5206"@zh . . . . . . . "1411087"^^ . . "6606"^^ . . . "\u6570\u5B66\u306E\u5206\u91CE\u306B\u304A\u3051\u308B\u5F31\u5FAE\u5206\uFF08\u3058\u3083\u304F\u3073\u3076\u3093\u3001\u82F1: weak derivative\uFF09\u3068\u306F\u3001\u901A\u5E38\u306E\u610F\u5473\u3067\u306E\u95A2\u6570\u306E\u5FAE\u5206\uFF08\u5F37\u5FAE\u5206\uFF09\u306E\u6982\u5FF5\u3092\u3001\u5FAE\u5206\u53EF\u80FD\u3068\u306F\u9650\u3089\u306A\u3044\u304C\u7A4D\u5206\u53EF\u80FD\u3067\u3042\u308B\u95A2\u6570\uFF08\u30EB\u30D9\u30FC\u30B0\u7A7A\u9593\u306B\u5C5E\u3059\u308B\u95A2\u6570\uFF09\u306B\u5BFE\u3057\u3066\u4E00\u822C\u5316\u3057\u305F\u3082\u306E\u3067\u3042\u308B\u3002\u3088\u308A\u4E00\u822C\u7684\u306A\u5B9A\u7FA9\u306B\u3064\u3044\u3066\u306F\u3001\u5206\u5E03\uFF08distribution\uFF09\u3092\u53C2\u7167\u3055\u308C\u305F\u3044\u3002"@ja . . . . . "\u0421\u043B\u0430\u0431\u043A\u0430 \u043F\u043E\u0445\u0456\u0434\u043D\u0430"@uk . . . . "\u00AB\u0421\u043B\u0430\u0431\u0430\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F\u00BB (\u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435) \u2014 \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u0438\u0435 \u043F\u043E\u043D\u044F\u0442\u0438\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 (\u00AB\u0441\u0438\u043B\u044C\u043D\u0430\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F\u00BB) \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u0439, \u0438\u043D\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u0435\u043C\u044B\u0445 \u043F\u043E \u041B\u0435\u0431\u0435\u0433\u0443 (\u0442\u043E \u0435\u0441\u0442\u044C \u0438\u0437 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 ), \u043D\u043E \u043D\u0435 \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0445\u0441\u044F \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0440\u0443\u0435\u043C\u044B\u043C\u0438."@ru . "\u6570\u5B66\u306E\u5206\u91CE\u306B\u304A\u3051\u308B\u5F31\u5FAE\u5206\uFF08\u3058\u3083\u304F\u3073\u3076\u3093\u3001\u82F1: weak derivative\uFF09\u3068\u306F\u3001\u901A\u5E38\u306E\u610F\u5473\u3067\u306E\u95A2\u6570\u306E\u5FAE\u5206\uFF08\u5F37\u5FAE\u5206\uFF09\u306E\u6982\u5FF5\u3092\u3001\u5FAE\u5206\u53EF\u80FD\u3068\u306F\u9650\u3089\u306A\u3044\u304C\u7A4D\u5206\u53EF\u80FD\u3067\u3042\u308B\u95A2\u6570\uFF08\u30EB\u30D9\u30FC\u30B0\u7A7A\u9593\u306B\u5C5E\u3059\u308B\u95A2\u6570\uFF09\u306B\u5BFE\u3057\u3066\u4E00\u822C\u5316\u3057\u305F\u3082\u306E\u3067\u3042\u308B\u3002\u3088\u308A\u4E00\u822C\u7684\u306A\u5B9A\u7FA9\u306B\u3064\u3044\u3066\u306F\u3001\u5206\u5E03\uFF08distribution\uFF09\u3092\u53C2\u7167\u3055\u308C\u305F\u3044\u3002"@ja . . . . "En matem\u00E1ticas, la derivada d\u00E9bil es una generalizaci\u00F3n del concepto de derivada de una funci\u00F3n que se asume como no diferenciable, pero s\u00ED integrable, es decir, residen en un espacio Lp . V\u00E9ase distribuci\u00F3n para una definici\u00F3n a\u00FAn m\u00E1s generalizada."@es . . . "In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space . The method of integration by parts holds that for differentiable functions and we have A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions \u03C6 vanishing at the boundary points."@en . "\u5F31\u5FAE\u5206"@ja . . . . . . . . "Eine schwache Ableitung ist in der Funktionalanalysis, einem Teilgebiet der Mathematik, eine Erweiterung des Begriffs der gew\u00F6hnlichen (klassischen) Ableitung. Er erm\u00F6glicht es, Funktionen eine Ableitung zuzuordnen, die nicht (stark bzw. im klassischen Sinne) differenzierbar sind. Schwache Ableitungen spielen eine gro\u00DFe Rolle in der Theorie der partiellen Differentialgleichungen. R\u00E4ume schwach differenzierbarer Funktionen sind die Sobolev-R\u00E4ume. Ein noch allgemeinerer Begriff der Ableitung ist die Distributionenableitung."@de . . "Weak derivative"@en . "En matem\u00E0tiques, una derivada feble \u00E9s una generalitzaci\u00F3 del concepte de derivada d'una funci\u00F3 (derivada forta) per a funcions no derivables, sin\u00F3 nom\u00E9s integrables, \u00E9s a dir que pertanyen a l'Espai de Lebesgue . Vegeu per a una definici\u00F3 fins i tot m\u00E9s general."@ca . "En matem\u00E1ticas, la derivada d\u00E9bil es una generalizaci\u00F3n del concepto de derivada de una funci\u00F3n que se asume como no diferenciable, pero s\u00ED integrable, es decir, residen en un espacio Lp . V\u00E9ase distribuci\u00F3n para una definici\u00F3n a\u00FAn m\u00E1s generalizada."@es . . . . "\u00AB\u0421\u043B\u0430\u0431\u0430\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F\u00BB (\u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435) \u2014 \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u0438\u0435 \u043F\u043E\u043D\u044F\u0442\u0438\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 (\u00AB\u0441\u0438\u043B\u044C\u043D\u0430\u044F \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u0430\u044F\u00BB) \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u0439, \u0438\u043D\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u0435\u043C\u044B\u0445 \u043F\u043E \u041B\u0435\u0431\u0435\u0433\u0443 (\u0442\u043E \u0435\u0441\u0442\u044C \u0438\u0437 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 ), \u043D\u043E \u043D\u0435 \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0445\u0441\u044F \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0440\u0443\u0435\u043C\u044B\u043C\u0438."@ru . "\u00AB\u0421\u043B\u0430\u0431\u043A\u0430 \u043F\u043E\u0445\u0456\u0434\u043D\u0430\u00BB (\u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0446\u0456) \u2014 \u0443\u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0435\u043D\u0435 \u043F\u043E\u043D\u044F\u0442\u0442\u044F \u043F\u043E\u0445\u0456\u0434\u043D\u043E\u0457 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 (\u00AB\u0441\u0438\u043B\u044C\u043D\u0430 \u043F\u043E\u0445\u0456\u0434\u043D\u0430\u00BB) \u0434\u043B\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0439, \u0456\u043D\u0442\u0435\u0433\u0440\u043E\u0432\u043D\u0438\u0445 \u0437\u0430 \u041B\u0435\u0431\u0435\u0433\u043E\u043C (\u0442\u043E\u0431\u0442\u043E \u0437 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 ), \u0430\u043B\u0435 \u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043E\u0432\u043D\u0438\u0445."@uk . . . "En matem\u00E0tiques, una derivada feble \u00E9s una generalitzaci\u00F3 del concepte de derivada d'una funci\u00F3 (derivada forta) per a funcions no derivables, sin\u00F3 nom\u00E9s integrables, \u00E9s a dir que pertanyen a l'Espai de Lebesgue . Vegeu per a una definici\u00F3 fins i tot m\u00E9s general."@ca . . . "S\u0142aba pochodna \u2013 rozszerzenie poj\u0119cia pochodnej na funkcje lokalnie ca\u0142kowalne. Poj\u0119cie s\u0142abej pochodnej ma szerokie zastosowania w teorii r\u00F3wna\u0144 r\u00F3\u017Cniczkowych cz\u0105stkowych."@pl . . . . . "\uC57D\uB3C4\uD568\uC218"@ko . . . . . . "Derivada feble"@ca . . . "\uD574\uC11D\uD559\uC5D0\uC11C \uC57D\uB3C4\uD568\uC218(\u5F31\u5C0E\u51FD\u6578, \uC601\uC5B4: weak derivative)\uB294 \uC77C\uBC18\uC801\uC778 \uB3C4\uD568\uC218\uC758 \uAC1C\uB150\uC758 \uC77C\uBC18\uD654\uC774\uB2E4. \uC774\uB97C \uD1B5\uD558\uC5EC \uACE0\uC804\uC801\uC73C\uB85C \uB3C4\uD568\uC218\uB97C \uCDE8\uD560 \uC218 \uC5C6\uB294 \uD568\uC218\uB4E4\uC758 \uB3C4\uD568\uC218\uB97C \uCDE8\uD560 \uC218 \uC788\uB2E4."@ko . . . "Fonction \u00E0 d\u00E9riv\u00E9e faible"@fr . . . . . . . "1084938050"^^ . . . . . . . . "En math\u00E9matiques, une fonction \u00E0 d\u00E9riv\u00E9e faible est une g\u00E9n\u00E9ralisation du concept de la d\u00E9riv\u00E9e d'une fonction (d\u00E9riv\u00E9e forte) pour les fonctions non suppos\u00E9es diff\u00E9rentiables, mais seulement int\u00E9grables, c'est-\u00E0-dire dans l'espace Lp : L1([a , b])."@fr . . . . . . "In matematica, la derivata debole \u00E8 una generalizzazione del concetto di derivata di una funzione a funzioni non necessariamente differenziabili, ma solamente integrabili, ovvero funzioni che appartengono allo spazio L1. La definizione di derivata debole origina le soluzioni deboli in spazi di Sobolev di problemi differenziali alle derivate parziali, frequenti in diversi settori dell'analisi, in particolare dell'analisi funzionale."@it .