. . "\uC218\uD559\uC5D0\uC11C \uB77C\uD50C\uB77C\uC2A4 \uC5F0\uC0B0\uC790(Laplace\u6F14\u7B97\u5B50, \uC601\uC5B4: Laplace operator) \uB610\uB294 \uB77C\uD50C\uB77C\uC2DC\uC548(\uC601\uC5B4: Laplacian)\uC740 2\uCC28 \uBBF8\uBD84 \uC5F0\uC0B0\uC790\uC758 \uC77C\uC885\uC73C\uB85C, \uAE30\uC6B8\uAE30\uC758 \uBC1C\uC0B0\uC774\uB2E4. \uAE30\uD638\uB294 \u0394(\uADF8\uB9AC\uC2A4 \uB300\uBB38\uC790 \uB378\uD0C0) \uB610\uB294 \u22072\uC774\uB2E4."@ko . "\u6570\u5B66\u306B\u304A\u3051\u308B\u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20\uFF08\u30E9\u30D7\u30E9\u30B9\u3055\u3088\u3046\u305D\u3001\u82F1: Laplace operator\uFF09\u3042\u308B\u3044\u306F\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\uFF08\u82F1: Laplacian)\u306F\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593\u4E0A\u306E\u51FD\u6570\u306E\u52FE\u914D\u306E\u767A\u6563\u3068\u3057\u3066\u4E0E\u3048\u3089\u308C\u308B\u5FAE\u5206\u4F5C\u7528\u7D20\u3067\u3042\u308B\u3002\u8A18\u53F7\u3067\u306F \u2207\u00B7\u2207, \u22072, \u3042\u308B\u3044\u306F \u2206 \u3067\u8868\u3055\u308C\u308B\u306E\u304C\u666E\u901A\u3067\u3042\u308B\u3002\u51FD\u6570 f \u306E\u70B9 p \u306B\u304A\u3051\u308B\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3 \u2206f(p) \u306F\uFF08\u6B21\u5143\u306B\u4F9D\u5B58\u3059\u308B\u5B9A\u6570\u306E\u9055\u3044\u3092\u9664\u3044\u3066\uFF09\u70B9 p \u3092\u4E2D\u5FC3\u3068\u3059\u308B\u7403\u9762\u3092\u534A\u5F84\u304C\u5897\u5927\u3059\u308B\u3088\u3046\u306B\u52D5\u304B\u3059\u3068\u304D\u306E f(p) \u304B\u3089\u5F97\u3089\u308C\u308B\u5E73\u5747\u5024\u306B\u306A\u3063\u3066\u3044\u308B\u3002\u76F4\u4EA4\u5EA7\u6A19\u7CFB\u306B\u304A\u3044\u3066\u306F\u3001\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\u306F\u5404\u72EC\u7ACB\u5909\u6570\u306B\u95A2\u3059\u308B\u51FD\u6570\u306E\u4E8C\u968E\uFF08\u975E\u6DF7\u5408\uFF09\u504F\u5C0E\u51FD\u6570\u306E\u548C\u3068\u3057\u3066\u4E0E\u3048\u3089\u308C\u3001\u307E\u305F\u307B\u304B\u306B\u5186\u7B52\u5EA7\u6A19\u7CFB\u3084\u7403\u5EA7\u6A19\u7CFB\u306A\u3069\u306E\u5EA7\u6A19\u7CFB\u306B\u304A\u3044\u3066\u3082\u6709\u7528\u306A\u8868\u793A\u3092\u6301\u3064\u3002 \u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20\u306E\u540D\u79F0\u306F\u3001\u5929\u4F53\u529B\u5B66\u306E\u7814\u7A76\u306B\u540C\u4F5C\u7528\u7D20\u3092\u6700\u521D\u306B\u7528\u3044\u305F\u30D5\u30E9\u30F3\u30B9\u4EBA\u6570\u5B66\u8005\u306E\u30D4\u30A8\u30FC\u30EB\uFF1D\u30B7\u30E2\u30F3\u30FB\u30C9\u30FB\u30E9\u30D7\u30E9\u30B9 (1749\u20131827) \u306B\u56E0\u3093\u3067\u3044\u308B\u3002\u540C\u4F5C\u7528\u7D20\u306F\u4E0E\u3048\u3089\u308C\u305F\u91CD\u529B\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u306B\u9069\u7528\u3059\u308B\u3068\u8CEA\u91CF\u5BC6\u5EA6\u306E\u5B9A\u6570\u500D\u3092\u4E0E\u3048\u308B\u3002\u73FE\u5728\u3067\u306F\u30E9\u30D7\u30E9\u30B9\u65B9\u7A0B\u5F0F\u3068\u547C\u3070\u308C\u308B\u65B9\u7A0B\u5F0F \u2206f = 0 \u306E\u89E3\u306F\u8ABF\u548C\u51FD\u6570\u3068\u547C\u3070\u308C\u3001\u81EA\u7531\u7A7A\u9593\u306B\u304A\u3044\u3066\u53EF\u80FD\u306A\u91CD\u529B\u5834\u3092\u8868\u73FE\u3059\u308B\u3082\u306E\u3067\u3042\u308B\u3002 \u5FAE\u5206\u65B9\u7A0B\u5F0F\u306B\u304A\u3044\u3066\u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20\u306F\u96FB\u6C17\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u3001\u91CD\u529B\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u3001\u71B1\u3084\u6D41\u4F53\u306E\u62E1\u6563\u65B9\u7A0B\u5F0F\u3001\u6CE2\u306E\u4F1D\u642C\u3001\u91CF\u5B50\u529B\u5B66\u3068\u3044\u3063\u305F\u3001\u591A\u304F\u306E\u7269\u7406\u73FE\u8C61\u3092\u8A18\u8FF0\u3059\u308B\u306E\u306B\u73FE\u308C\u308B\u3002\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\u306F\u3001\u51FD\u6570\u306E\u52FE\u914D\u30D5\u30ED\u30FC\u306E\u6D41\u675F\u5BC6\u5EA6\u3092\u8868\u3059\u3002"@ja . . . . . . . . . . . . . . . . . . . . "\u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633"@ar . . . . . . . . "27546"^^ . . "\u041E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 (\u043B\u0430\u043F\u043B\u0430\u0441\u0438\u0430\u0301\u043D, \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u0434\u0435\u043B\u044C\u0442\u0430) \u2014 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440, \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435 \u0433\u043B\u0430\u0434\u043A\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0438\u0439 \u0438 \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u043C\u044B\u0439 \u0441\u0438\u043C\u0432\u043E\u043B\u043E\u043C . \u0424\u0443\u043D\u043A\u0446\u0438\u0438 \u043E\u043D \u0441\u0442\u0430\u0432\u0438\u0442 \u0432 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0438\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u044E \u0432 n-\u043C\u0435\u0440\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435. \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0434\u043B\u044F \u0432\u0435\u043A\u0442\u043E\u0440\u0430 : \u041B\u0430\u043F\u043B\u0430\u0441\u0438\u0430\u043D \u0432\u0435\u043A\u0442\u043E\u0440\u0430 - \u0442\u043E\u0436\u0435 \u0432\u0435\u043A\u0442\u043E\u0440."@ru . . . . "\u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0623\u0648 \u0644\u0627\u0628\u0644\u0627\u0633\u064A\u0627\u0646 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Laplace operator \u0623\u0648 Laplacian)\u200F \u0648\u0631\u0645\u0632\u0647 \u0623\u0648 \u0625\u062D\u062F\u0649 \u0627\u0644\u0645\u0624\u062B\u0631\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0648\u0647\u0648 \u0645\u0646 \u0627\u0644\u0645\u0624\u062B\u0631\u0627\u062A \u0627\u0644\u0645\u0647\u0645\u0629 \u0641\u064A \u0645\u062C\u0627\u0644 \u062D\u0633\u0627\u0628 \u0627\u0644\u0645\u062A\u062C\u0647\u0627\u062A \u0648\u0643\u0630\u0644\u0643 \u062D\u0633\u0627\u0628 \u0627\u0644\u062A\u0641\u0627\u0636\u0644 \u0648\u0627\u0644\u062A\u0643\u0627\u0645\u0644 \u0645\u062A\u0639\u062F\u062F \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0648\u0633\u0645\u0649 \u0627\u0644\u0645\u0624\u062B\u0631 \u0628\u0647\u0630\u0627 \u0627\u0644\u0627\u0633\u0645 \u0639\u0631\u0641\u0627\u0646\u0627\u064B \u0644\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u064A \u0627\u0644\u0641\u0631\u0646\u0633\u064A \u0628\u064A\u064A\u0631 \u0644\u0627\u0628\u0644\u0627\u0633. \u064A\u0638\u0647\u0631 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0641\u064A \u0645\u0639\u0627\u062F\u0644\u0627\u062A \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u062A\u0635\u0641 \u0627\u0644\u0639\u062F\u064A\u062F \u0645\u0646 \u0627\u0644\u0638\u0648\u0627\u0647\u0631 \u0627\u0644\u0641\u064A\u0632\u064A\u0627\u0626\u064A\u0629\u060C \u0645\u062B\u0644 \u0627\u0644\u0643\u0645\u0648\u0646\u0627\u062A \u0627\u0644\u0643\u0647\u0631\u0628\u0627\u0626\u064A\u0629 \u0648\u0627\u0644\u062C\u0627\u0630\u0628\u064A\u0629\u060C \u0648\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631 \u0644\u0644\u062D\u0631\u0627\u0631\u0629 \u0648\u062A\u062F\u0641\u0642 \u0627\u0644\u0645\u0648\u0627\u0626\u0639\u060C \u0648\u0627\u0646\u062A\u0634\u0627\u0631 \u0627\u0644\u0645\u0648\u062C\u0629\u060C \u0648\u0645\u064A\u0643\u0627\u0646\u064A\u0643\u0627 \u0627\u0644\u0643\u0645\u061B \u0643\u0645\u0627 \u0623\u0646 \u0647\u0630\u0627 \u0627\u0644\u0645\u0624\u062B\u0631 \u064A\u064F\u0633\u062A\u062E\u062F\u0645 \u0623\u064A\u0636\u064B\u0627 \u0641\u064A \u0645\u064A\u062F\u0627\u0646 \u0641\u064A\u0632\u064A\u0627\u0621 \u0627\u0644\u0623\u0631\u0636. \u064A\u0645\u062B\u0644 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0643\u062B\u0627\u0641\u0629 \u0627\u0644\u062A\u062F\u0641\u0642 \u0644\u062A\u062F\u0641\u0642 \u0627\u0644\u062A\u062F\u0631\u062C \u0644\u0644\u062F\u0627\u0644\u0629."@ar . . . . . "L'op\u00E9rateur laplacien, ou simplement le laplacien, est l'op\u00E9rateur diff\u00E9rentiel d\u00E9fini par l'application de l'op\u00E9rateur gradient suivie de l'application de l'op\u00E9rateur divergence : Intuitivement, il combine et relie la description statique d'un champ (d\u00E9crit par son gradient) aux effets dynamiques (la divergence) de ce champ dans l'espace et le temps. C'est l'exemple le plus simple et le plus r\u00E9pandu d'op\u00E9rateur elliptique. Il appara\u00EEt dans la formulation math\u00E9matique de nombreuses disciplines th\u00E9oriques, comme la g\u00E9ophysique, l'\u00E9lectrostatique, la thermodynamique, la m\u00E9canique classique et quantique. On le retrouve syst\u00E9matiquement dans les expressions de l'\u00E9quation de Laplace, de l'\u00E9quation de Poisson, de l'\u00E9quation de la chaleur et l'\u00E9quation d'onde. L'op\u00E9rateur laplacien appliqu\u00E9 deux fois est appel\u00E9 bilaplacien."@fr . "L'op\u00E9rateur laplacien, ou simplement le laplacien, est l'op\u00E9rateur diff\u00E9rentiel d\u00E9fini par l'application de l'op\u00E9rateur gradient suivie de l'application de l'op\u00E9rateur divergence : Intuitivement, il combine et relie la description statique d'un champ (d\u00E9crit par son gradient) aux effets dynamiques (la divergence) de ce champ dans l'espace et le temps. C'est l'exemple le plus simple et le plus r\u00E9pandu d'op\u00E9rateur elliptique. L'op\u00E9rateur laplacien appliqu\u00E9 deux fois est appel\u00E9 bilaplacien."@fr . "\u62C9\u666E\u62C9\u65AF\u7B97\u5B50"@zh . . . "Laplacian"@en . . . "Laplace operator"@en . . . . . . . "Operador laplaci\u00E0"@ca . . . "Laplaceoperatorn eller Laplaces operator \u00E4r inom vektoranalysen en differentialoperator. Den har f\u00E5tt sitt namn efter Pierre Simon de Laplace. Laplaceoperatorn \u00E4r lika med summan av alla andra ordningens partiella derivator av en beroende variabel. Laplaceoperatorn \u00E4r en elliptisk operator med m\u00E5nga till\u00E4mpningar inom fysiken och matematiken. F\u00F6r ett skal\u00E4rf\u00E4lt \u03C6 kan Laplaceoperatorn uttryckas div(grad \u03C6), eller likv\u00E4rdigt med hj\u00E4lp av nabla-symbolen i kvadrat, \u22072: Samt f\u00F6r vektorf\u00E4lt : \u22072 kan \u00E4ven skrivas som \u2206. Operatorn f\u00F6rekommer, till exempel, i Laplaces ekvation."@sv . . "Op\u00E9rateur laplacien"@fr . . . . . . . . . "En c\u00E1lculo vectorial, el operador laplaciano o laplaciano es un operador diferencial el\u00EDptico de segundo orden, denotado como \u0394, relacionado con ciertos problemas de minimizaci\u00F3n de ciertas magnitudes sobre un cierto dominio. El operador tiene ese nombre en reconocimiento a Pierre-Simon Laplace que estudi\u00F3 soluciones de ecuaciones diferenciales en derivadas parciales en las que aparec\u00EDa dicho operador."@es . . "\u5728\u6578\u5B78\u4EE5\u53CA\u7269\u7406\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u6216\u662F\u62C9\u666E\u62C9\u65AF\u7B97\u7B26\uFF08\u82F1\u8A9E\uFF1ALaplace operator, Laplacian\uFF09\u662F\u7531\u6B27\u51E0\u91CC\u5F97\u7A7A\u95F4\u4E2D\u7684\u4E00\u500B\u51FD\u6570\u7684\u68AF\u5EA6\u7684\u6563\u5EA6\u7ED9\u51FA\u7684\u5FAE\u5206\u7B97\u5B50\uFF0C\u901A\u5E38\u5BEB\u6210 \u3001 \u6216 \u3002 \u9019\u540D\u5B57\u662F\u70BA\u4E86\u7D00\u5FF5\u6CD5\u56FD\u6570\u5B66\u5BB6\u76AE\u8036-\u897F\u8499\u00B7\u62C9\u666E\u62C9\u65AF\uFF081749\u20131827\uFF09\u800C\u547D\u540D\u7684\u3002\u4ED6\u5728\u7814\u7A76\u5929\u4F53\u529B\u5B66\u5728\u6578\u5B78\u4E2D\u9996\u6B21\u5E94\u7528\u7B97\u5B50\uFF0C\u5F53\u5B83\u88AB\u65BD\u52A0\u5230\u4E00\u4E2A\u7ED9\u5B9A\u7684\u91CD\u529B\u4F4D\uFF08Gravitational potential\uFF09\u7684\u65F6\u5019\uFF0C\u5176\u4E2D\u6240\u8FF0\u7B97\u5B50\u7ED9\u51FA\u7684\u8D28\u91CF\u5BC6\u5EA6\u7684\u5E38\u6570\u500D\u3002\u7D93\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u904B\u7B97\u70BA\u96F6 \u7684\u51FD\u6578\u7A31\u70BA\u8C03\u548C\u51FD\u6570\uFF0C\u73B0\u5728\u79F0\u4E3A\u62C9\u666E\u62C9\u65AF\u65B9\u7A0B\uFF0C\u548C\u4EE3\u8868\u4E86\u5728\u81EA\u7531\u7A7A\u95F4\u4E2D\u7684\u53EF\u80FD\u7684\u91CD\u529B\u573A\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u6709\u8A31\u591A\u7528\u9014\uFF0C\u6B64\u5916\u4E5F\u662F\u692D\u5706\u7B97\u5B50\u4E2D\u7684\u4E00\u500B\u91CD\u8981\u4F8B\u5B50\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u51FA\u73B0\u63CF\u8FF0\u8BB8\u591A\u7269\u7406\u73B0\u8C61\u7684\u5FAE\u5206\u65B9\u7A0B\u88E1\u3002\u4F8B\u5982\uFF0C\u5E38\u7528\u65BC\u6CE2\u65B9\u7A0B\u7684\u6578\u5B78\u6A21\u578B\u3001\u71B1\u50B3\u5C0E\u65B9\u7A0B\u3001\u6D41\u4F53\u529B\u5B66\u4EE5\u53CA\u4EA5\u59C6\u970D\u8332\u65B9\u7A0B\u3002\u5728\u975C\u96FB\u5B78\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u65B9\u7A0B\u548C\u6CCA\u677E\u65B9\u7A0B\u7684\u61C9\u7528\u96A8\u8655\u53EF\u898B\u3002\u5728\u91CF\u5B50\u529B\u5B78\u4E2D\uFF0C\u5176\u4EE3\u8868\u859B\u4E01\u683C\u65B9\u7A0B\u4E2D\u7684\u52D5\u80FD\u9805\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u662F\u6700\u7B80\u5355\u7684\u692D\u5706\u7B97\u5B50\uFF0C\u5E76\u4E14\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u662F\u970D\u5947\u7406\u8AD6\u7684\u6838\u5FC3\uFF0C\u4E26\u4E14\u662F\u5FB7\u62C9\u59C6\u4E0A\u540C\u8ABF\u7684\u7D50\u679C\u3002\u5728\u56FE\u50CF\u5904\u7406\u548C\u8BA1\u7B97\u673A\u89C6\u89C9\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u5DF2\u7ECF\u88AB\u7528\u4E8E\u8BF8\u5982\u548C\u8FB9\u7F18\u68C0\u6D4B\u7B49\u7684\u5404\u79CD\u4EFB\u52A1\u3002"@zh . . . . . . . "En c\u00E1lculo vectorial, el operador laplaciano o laplaciano es un operador diferencial el\u00EDptico de segundo orden, denotado como \u0394, relacionado con ciertos problemas de minimizaci\u00F3n de ciertas magnitudes sobre un cierto dominio. El operador tiene ese nombre en reconocimiento a Pierre-Simon Laplace que estudi\u00F3 soluciones de ecuaciones diferenciales en derivadas parciales en las que aparec\u00EDa dicho operador. Expresado en coordenadas cartesianas es igual a la suma de todas las segundas derivadas parciales no mixtas dependientes de una variable. Se hace uso del s\u00EDmbolo delta (\u0394) o nabla cuadrado para representarlo. Si , son un campo escalar y un campo vectorial respectivamente, el laplaciano de ambos puede escribirse en t\u00E9rminos del operador nabla como:"@es . . . . . . . . . . . . . . . . . "In matematica e fisica, in particolare nel calcolo differenziale vettoriale, l'operatore di Laplace o laplaciano, il cui nome \u00E8 dovuto a Pierre Simon Laplace, \u00E8 un operatore differenziale del secondo ordine definito come la divergenza del gradiente di una funzione in uno spazio euclideo, ed \u00E8 solitamente rappresentato dai simboli , , o ."@it . . . . . . . . . . . . . "Em matem\u00E1tica e f\u00EDsica, o Laplaciano ou Operador de Laplace (ou ainda operador de Laplace-Beltrami), denotado por ou , sendo o operador nabla, \u00E9 um operador diferencial de segunda ordem. O Laplaciano, nome dado em homenagem a Pierre-Simon Laplace, aparece naturalmente em diversas equa\u00E7\u00F5es diferenciais parciais que modelam problemas f\u00EDsicos, tais como potencial el\u00E9trico e gravitacional, propaga\u00E7\u00E3o de ondas, condu\u00E7\u00E3o de calor e fluidos, e tamb\u00E9m fazendo parte das equa\u00E7\u00F5es de Poisson para eletrost\u00E1tica e da equa\u00E7\u00E3o de Schr\u00F6dinger independente do tempo."@pt . . . "p/l057510"@en . . . . . "Operator Laplace\u2019a, laplasjan \u2013 operator r\u00F3\u017Cniczkowy drugiego rz\u0119du, wprowadzony przez Pierre\u2019a Simona de Laplace\u2019a. W uk\u0142adzie kartezja\u0144skim 3-wymiarowym ma posta\u0107: Operator ten uog\u00F3lnia si\u0119 na przestrzenie euklidesowe -wymiarowe z dowolnymi uk\u0142adami wsp\u00F3\u0142rz\u0119dnych krzywoliniowych (w tym ze wsp\u00F3\u0142rz\u0119dnymi kartezja\u0144skimi) oraz na dowolne przestrzenie riemannowskie i pseudoriemannowskie."@pl . . . "\uC218\uD559\uC5D0\uC11C \uB77C\uD50C\uB77C\uC2A4 \uC5F0\uC0B0\uC790(Laplace\u6F14\u7B97\u5B50, \uC601\uC5B4: Laplace operator) \uB610\uB294 \uB77C\uD50C\uB77C\uC2DC\uC548(\uC601\uC5B4: Laplacian)\uC740 2\uCC28 \uBBF8\uBD84 \uC5F0\uC0B0\uC790\uC758 \uC77C\uC885\uC73C\uB85C, \uAE30\uC6B8\uAE30\uC758 \uBC1C\uC0B0\uC774\uB2E4. \uAE30\uD638\uB294 \u0394(\uADF8\uB9AC\uC2A4 \uB300\uBB38\uC790 \uB378\uD0C0) \uB610\uB294 \u22072\uC774\uB2E4."@ko . . . . . . . "Der Laplace-Operator ist ein mathematischer Operator, der zuerst von Pierre-Simon Laplace eingef\u00FChrt wurde. Es handelt sich um einen linearen Differentialoperator innerhalb der mehrdimensionalen Analysis. Er wird meist durch das Zeichen , den Gro\u00DFbuchstaben Delta des griechischen Alphabets, notiert. Der Laplace-Operator kommt in vielen Differentialgleichungen vor, die das Verhalten physikalischer Felder beschreiben. Beispiele sind die Poisson-Gleichung der Elektrostatik, die Navier-Stokes-Gleichungen f\u00FCr Str\u00F6mungen von Fl\u00FCssigkeiten oder Gasen und die Diffusionsgleichung f\u00FCr die W\u00E4rmeleitung."@de . . . . . . . . "Der Laplace-Operator ist ein mathematischer Operator, der zuerst von Pierre-Simon Laplace eingef\u00FChrt wurde. Es handelt sich um einen linearen Differentialoperator innerhalb der mehrdimensionalen Analysis. Er wird meist durch das Zeichen , den Gro\u00DFbuchstaben Delta des griechischen Alphabets, notiert. Der Laplace-Operator kommt in vielen Differentialgleichungen vor, die das Verhalten physikalischer Felder beschreiben. Beispiele sind die Poisson-Gleichung der Elektrostatik, die Navier-Stokes-Gleichungen f\u00FCr Str\u00F6mungen von Fl\u00FCssigkeiten oder Gasen und die Diffusionsgleichung f\u00FCr die W\u00E4rmeleitung."@de . "Laplaca operatoro"@eo . . . . . . . "\u041E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 \u2014 \u0434\u0456\u044F \u043D\u0430\u0434 \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u0438\u043C \u0430\u0431\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u043C \u043F\u043E\u043B\u0435\u043C, \u0449\u043E \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F, \u044F\u043A \u0441\u0443\u043C\u0430 \u0434\u0440\u0443\u0433\u0438\u0445 \u0447\u0430\u0441\u0442\u043A\u043E\u0432\u0438\u0445 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u0445 \u043F\u043E \u043A\u043E\u0436\u043D\u0456\u0439 \u0434\u0435\u043A\u0430\u0440\u0442\u043E\u0432\u0456\u0439 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442\u0456. \u041F\u043E\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0430\u0431\u043E . \u0414\u043B\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0447\u0430\u0441\u0442\u043E \u0432\u0438\u043A\u043E\u0440\u0438\u0441\u0442\u043E\u0432\u0443\u0454\u0442\u044C\u0441\u044F \u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0456\u0439 \u0456 \u0442\u0435\u043E\u0440\u0435\u0442\u0438\u0447\u043D\u0456\u0439 \u0444\u0456\u0437\u0438\u0446\u0456. \u0421\u043F\u0440\u0430\u0432\u0435\u0434\u043B\u0438\u0432\u0435 \u0441\u043F\u0456\u0432\u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u043D\u044F: . \u041D\u0430\u0437\u0432\u0430\u043D\u0438\u0439 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430."@uk . "Laplace\u016Fv oper\u00E1tor"@cs . . . . "Laplace operator"@en . . . . . . . . "Laplace-Operator"@de . . "Laplace\u016Fv oper\u00E1tor je diferenci\u00E1ln\u00ED oper\u00E1tor definovan\u00FD jako divergence gradientu skal\u00E1rn\u00EDho, nebo obecn\u011B tenzorov\u00E9ho pole nazvan\u00FD podle Pierre-Simona Laplace. Je-li aplikov\u00E1n na skal\u00E1rn\u00ED pole, v\u00FDsledkem je skal\u00E1rn\u00ED pole, je-li aplikov\u00E1n na tenzorov\u00E9 pole, v\u00FDsledkem je tenzorov\u00E9 pole stejn\u00E9ho \u0159\u00E1du. Zna\u010D\u00ED se symbolem ."@cs . . "In matematica e fisica, in particolare nel calcolo differenziale vettoriale, l'operatore di Laplace o laplaciano, il cui nome \u00E8 dovuto a Pierre Simon Laplace, \u00E8 un operatore differenziale del secondo ordine definito come la divergenza del gradiente di una funzione in uno spazio euclideo, ed \u00E8 solitamente rappresentato dai simboli , , o . Si tratta di un operatore ellittico, che in coordinate cartesiane \u00E8 definito come la somma delle derivate parziali seconde non miste rispetto alle coordinate. L'operatore di Laplace pu\u00F2 operare da due fino ad n dimensioni e pu\u00F2 essere applicato sia a campi scalari, sia a campi vettoriali. Le funzioni di classe che annullano il laplaciano, ovvero che soddisfano l'equazione di Laplace, sono le funzioni armoniche. L'operatore di Laplace viene generalizzato a spazi non euclidei, dove si presenta anche nella forma, ad esempio, di operatore ellittico, iperbolico. In particolare, nello spaziotempo di Minkowski l'operatore di Laplace-Beltrami diventa l'operatore di d'Alembert. Il laplaciano viene impiegato, ad esempio, per modellare la propagazione ondosa ed il flusso del calore, comparendo nell'equazione di Helmholtz. Riveste un ruolo centrale anche in elettrostatica, dove \u00E8 utilizzato nell'equazione di Laplace e nell'equazione di Poisson. In meccanica quantistica rappresenta l'osservabile energia cinetica ed \u00E8 presente nell'equazione di Schr\u00F6dinger. In idraulica viene utilizzato per ricavare l'espressione della in funzione delle caratteristiche di una corrente intubata nel regime laminare. Infine, l'operatore di Laplace si trova al centro della teoria di Hodge e dei risultati della coomologia di De Rham."@it . "\u041E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 (\u043B\u0430\u043F\u043B\u0430\u0441\u0438\u0430\u0301\u043D, \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u0434\u0435\u043B\u044C\u0442\u0430) \u2014 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440, \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435 \u0433\u043B\u0430\u0434\u043A\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0438\u0439 \u0438 \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u043C\u044B\u0439 \u0441\u0438\u043C\u0432\u043E\u043B\u043E\u043C . \u0424\u0443\u043D\u043A\u0446\u0438\u0438 \u043E\u043D \u0441\u0442\u0430\u0432\u0438\u0442 \u0432 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0438\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u044E \u0432 n-\u043C\u0435\u0440\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435. \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u044D\u043A\u0432\u0438\u0432\u0430\u043B\u0435\u043D\u0442\u0435\u043D \u043F\u043E\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C\u0443 \u0432\u0437\u044F\u0442\u0438\u044E \u043E\u043F\u0435\u0440\u0430\u0446\u0438\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043D\u0442\u0430 \u0438 \u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043D\u0446\u0438\u0438: , \u0442\u0430\u043A\u0438\u043C \u043E\u0431\u0440\u0430\u0437\u043E\u043C, \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0435 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0432 \u0442\u043E\u0447\u043A\u0435 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0438\u0441\u0442\u043E\u043B\u043A\u043E\u0432\u0430\u043D\u043E \u043A\u0430\u043A \u043F\u043B\u043E\u0442\u043D\u043E\u0441\u0442\u044C \u0438\u0441\u0442\u043E\u0447\u043D\u0438\u043A\u043E\u0432 (\u0441\u0442\u043E\u043A\u043E\u0432) \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u044F \u0432 \u044D\u0442\u043E\u0439 \u0442\u043E\u0447\u043A\u0435. \u0412 \u0434\u0435\u043A\u0430\u0440\u0442\u043E\u0432\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0435 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0447\u0430\u0441\u0442\u043E \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u0442\u0441\u044F \u0441\u043B\u0435\u0434\u0443\u044E\u0449\u0438\u043C \u043E\u0431\u0440\u0430\u0437\u043E\u043C , \u0442\u043E \u0435\u0441\u0442\u044C \u0432 \u0432\u0438\u0434\u0435 \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0430 \u043D\u0430\u0431\u043B\u0430 \u043D\u0430 \u0441\u0435\u0431\u044F. \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0441\u0438\u043C\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u043D. \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0434\u043B\u044F \u0432\u0435\u043A\u0442\u043E\u0440\u0430 : \u041B\u0430\u043F\u043B\u0430\u0441\u0438\u0430\u043D \u0432\u0435\u043A\u0442\u043E\u0440\u0430 - \u0442\u043E\u0436\u0435 \u0432\u0435\u043A\u0442\u043E\u0440."@ru . . "\u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0623\u0648 \u0644\u0627\u0628\u0644\u0627\u0633\u064A\u0627\u0646 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Laplace operator \u0623\u0648 Laplacian)\u200F \u0648\u0631\u0645\u0632\u0647 \u0623\u0648 \u0625\u062D\u062F\u0649 \u0627\u0644\u0645\u0624\u062B\u0631\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0648\u0647\u0648 \u0645\u0646 \u0627\u0644\u0645\u0624\u062B\u0631\u0627\u062A \u0627\u0644\u0645\u0647\u0645\u0629 \u0641\u064A \u0645\u062C\u0627\u0644 \u062D\u0633\u0627\u0628 \u0627\u0644\u0645\u062A\u062C\u0647\u0627\u062A \u0648\u0643\u0630\u0644\u0643 \u062D\u0633\u0627\u0628 \u0627\u0644\u062A\u0641\u0627\u0636\u0644 \u0648\u0627\u0644\u062A\u0643\u0627\u0645\u0644 \u0645\u062A\u0639\u062F\u062F \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0648\u0633\u0645\u0649 \u0627\u0644\u0645\u0624\u062B\u0631 \u0628\u0647\u0630\u0627 \u0627\u0644\u0627\u0633\u0645 \u0639\u0631\u0641\u0627\u0646\u0627\u064B \u0644\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u064A \u0627\u0644\u0641\u0631\u0646\u0633\u064A \u0628\u064A\u064A\u0631 \u0644\u0627\u0628\u0644\u0627\u0633. \u064A\u0638\u0647\u0631 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0641\u064A \u0645\u0639\u0627\u062F\u0644\u0627\u062A \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u062A\u0635\u0641 \u0627\u0644\u0639\u062F\u064A\u062F \u0645\u0646 \u0627\u0644\u0638\u0648\u0627\u0647\u0631 \u0627\u0644\u0641\u064A\u0632\u064A\u0627\u0626\u064A\u0629\u060C \u0645\u062B\u0644 \u0627\u0644\u0643\u0645\u0648\u0646\u0627\u062A \u0627\u0644\u0643\u0647\u0631\u0628\u0627\u0626\u064A\u0629 \u0648\u0627\u0644\u062C\u0627\u0630\u0628\u064A\u0629\u060C \u0648\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631 \u0644\u0644\u062D\u0631\u0627\u0631\u0629 \u0648\u062A\u062F\u0641\u0642 \u0627\u0644\u0645\u0648\u0627\u0626\u0639\u060C \u0648\u0627\u0646\u062A\u0634\u0627\u0631 \u0627\u0644\u0645\u0648\u062C\u0629\u060C \u0648\u0645\u064A\u0643\u0627\u0646\u064A\u0643\u0627 \u0627\u0644\u0643\u0645\u061B \u0643\u0645\u0627 \u0623\u0646 \u0647\u0630\u0627 \u0627\u0644\u0645\u0624\u062B\u0631 \u064A\u064F\u0633\u062A\u062E\u062F\u0645 \u0623\u064A\u0636\u064B\u0627 \u0641\u064A \u0645\u064A\u062F\u0627\u0646 \u0641\u064A\u0632\u064A\u0627\u0621 \u0627\u0644\u0623\u0631\u0636. \u064A\u0645\u062B\u0644 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 \u0643\u062B\u0627\u0641\u0629 \u0627\u0644\u062A\u062F\u0641\u0642 \u0644\u062A\u062F\u0641\u0642 \u0627\u0644\u062A\u062F\u0631\u062C \u0644\u0644\u062F\u0627\u0644\u0629."@ar . "Operatore di Laplace"@it . . "Operator Laplace\u2019a"@pl . "En matematiko, laplaca operatoro a\u016D operatoro de Laplace, skribata kiel , a\u016D , estas diferenciala operatoro de dua ordo en la n-dimensia e\u016Dklida spaco Rn. \u011Ci estas difinita kiel la diver\u011Denco de la gradiento (matematiko) , kie f estas dufoje diferencialebla reelo-valora funkcio Ekvivalente, la laplaca operatoro de f estas la sumo de \u0109iuj nemiksitaj duaj partaj deriva\u0135oj la\u016D la karteziaj koordinatoj xi: tiel, en du-dimensia kazo kie x kaj y estas la karteziaj koordinatoj de la xy-ebeno, laplaca operatoro estas"@eo . . . . . "In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian \u0394f\u200A(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f\u200A(p). The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749\u20131827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation \u0394f = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum. The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow, the wave equation describes wave propagation, and the Schr\u00F6dinger equation in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology."@en . . "En matematiko, laplaca operatoro a\u016D operatoro de Laplace, skribata kiel , a\u016D , estas diferenciala operatoro de dua ordo en la n-dimensia e\u016Dklida spaco Rn. \u011Ci estas difinita kiel la diver\u011Denco de la gradiento (matematiko) , kie f estas dufoje diferencialebla reelo-valora funkcio Ekvivalente, la laplaca operatoro de f estas la sumo de \u0109iuj nemiksitaj duaj partaj deriva\u0135oj la\u016D la karteziaj koordinatoj xi: tiel, en du-dimensia kazo kie x kaj y estas la karteziaj koordinatoj de la xy-ebeno, laplaca operatoro estas tiel, en tri-dimensia kazo kie x, y, z estas la karteziaj koordinatoj, laplaca operatoro estas"@eo . . . . . . . . . . . . . . . . . . . . . . "Laplacian"@en . . . . . . . . . . . . . . . . . . . . . . . . "\u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20"@ja . . . "In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian \u0394f\u200A(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f\u200A(p)."@en . "\u5728\u6578\u5B78\u4EE5\u53CA\u7269\u7406\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u6216\u662F\u62C9\u666E\u62C9\u65AF\u7B97\u7B26\uFF08\u82F1\u8A9E\uFF1ALaplace operator, Laplacian\uFF09\u662F\u7531\u6B27\u51E0\u91CC\u5F97\u7A7A\u95F4\u4E2D\u7684\u4E00\u500B\u51FD\u6570\u7684\u68AF\u5EA6\u7684\u6563\u5EA6\u7ED9\u51FA\u7684\u5FAE\u5206\u7B97\u5B50\uFF0C\u901A\u5E38\u5BEB\u6210 \u3001 \u6216 \u3002 \u9019\u540D\u5B57\u662F\u70BA\u4E86\u7D00\u5FF5\u6CD5\u56FD\u6570\u5B66\u5BB6\u76AE\u8036-\u897F\u8499\u00B7\u62C9\u666E\u62C9\u65AF\uFF081749\u20131827\uFF09\u800C\u547D\u540D\u7684\u3002\u4ED6\u5728\u7814\u7A76\u5929\u4F53\u529B\u5B66\u5728\u6578\u5B78\u4E2D\u9996\u6B21\u5E94\u7528\u7B97\u5B50\uFF0C\u5F53\u5B83\u88AB\u65BD\u52A0\u5230\u4E00\u4E2A\u7ED9\u5B9A\u7684\u91CD\u529B\u4F4D\uFF08Gravitational potential\uFF09\u7684\u65F6\u5019\uFF0C\u5176\u4E2D\u6240\u8FF0\u7B97\u5B50\u7ED9\u51FA\u7684\u8D28\u91CF\u5BC6\u5EA6\u7684\u5E38\u6570\u500D\u3002\u7D93\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u904B\u7B97\u70BA\u96F6 \u7684\u51FD\u6578\u7A31\u70BA\u8C03\u548C\u51FD\u6570\uFF0C\u73B0\u5728\u79F0\u4E3A\u62C9\u666E\u62C9\u65AF\u65B9\u7A0B\uFF0C\u548C\u4EE3\u8868\u4E86\u5728\u81EA\u7531\u7A7A\u95F4\u4E2D\u7684\u53EF\u80FD\u7684\u91CD\u529B\u573A\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u6709\u8A31\u591A\u7528\u9014\uFF0C\u6B64\u5916\u4E5F\u662F\u692D\u5706\u7B97\u5B50\u4E2D\u7684\u4E00\u500B\u91CD\u8981\u4F8B\u5B50\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u51FA\u73B0\u63CF\u8FF0\u8BB8\u591A\u7269\u7406\u73B0\u8C61\u7684\u5FAE\u5206\u65B9\u7A0B\u88E1\u3002\u4F8B\u5982\uFF0C\u5E38\u7528\u65BC\u6CE2\u65B9\u7A0B\u7684\u6578\u5B78\u6A21\u578B\u3001\u71B1\u50B3\u5C0E\u65B9\u7A0B\u3001\u6D41\u4F53\u529B\u5B66\u4EE5\u53CA\u4EA5\u59C6\u970D\u8332\u65B9\u7A0B\u3002\u5728\u975C\u96FB\u5B78\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u65B9\u7A0B\u548C\u6CCA\u677E\u65B9\u7A0B\u7684\u61C9\u7528\u96A8\u8655\u53EF\u898B\u3002\u5728\u91CF\u5B50\u529B\u5B78\u4E2D\uFF0C\u5176\u4EE3\u8868\u859B\u4E01\u683C\u65B9\u7A0B\u4E2D\u7684\u52D5\u80FD\u9805\u3002 \u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u662F\u6700\u7B80\u5355\u7684\u692D\u5706\u7B97\u5B50\uFF0C\u5E76\u4E14\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u662F\u970D\u5947\u7406\u8AD6\u7684\u6838\u5FC3\uFF0C\u4E26\u4E14\u662F\u5FB7\u62C9\u59C6\u4E0A\u540C\u8ABF\u7684\u7D50\u679C\u3002\u5728\u56FE\u50CF\u5904\u7406\u548C\u8BA1\u7B97\u673A\u89C6\u89C9\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\u5DF2\u7ECF\u88AB\u7528\u4E8E\u8BF8\u5982\u548C\u8FB9\u7F18\u68C0\u6D4B\u7B49\u7684\u5404\u79CD\u4EFB\u52A1\u3002"@zh . . . . . . . "Laplaceoperatorn eller Laplaces operator \u00E4r inom vektoranalysen en differentialoperator. Den har f\u00E5tt sitt namn efter Pierre Simon de Laplace. Laplaceoperatorn \u00E4r lika med summan av alla andra ordningens partiella derivator av en beroende variabel. Laplaceoperatorn \u00E4r en elliptisk operator med m\u00E5nga till\u00E4mpningar inom fysiken och matematiken. F\u00F6r ett skal\u00E4rf\u00E4lt \u03C6 kan Laplaceoperatorn uttryckas div(grad \u03C6), eller likv\u00E4rdigt med hj\u00E4lp av nabla-symbolen i kvadrat, \u22072: Samt f\u00F6r vektorf\u00E4lt : \u22072 kan \u00E4ven skrivas som \u2206. Operatorn f\u00F6rekommer, till exempel, i Laplaces ekvation."@sv . . . "174706"^^ . "En c\u00E0lcul vectorial, l'operador laplaci\u00E0 \u00E9s un operador diferencial el\u00B7l\u00EDptic de segon ordre, denotat com \u0394, relacionat amb certs problemes de minimitzaci\u00F3 de determinades magnituds sobre un cert domini. L'operador t\u00E9 aquest nom en reconeixement de Pierre-Simon Laplace, que va estudiar solucions d'equacions diferencials en derivades parcials en qu\u00E8 apareixia aquest operador. Expressat en coordenades cartesianes, \u00E9s igual a la suma de totes les segones derivades parcials no mixtes dependents d'una variable. Correspon a div (grad \u03C6), d'on l'\u00FAs del s\u00EDmbol delta (\u0394) o nabla quadrat per a representar-lo. Si , s\u00F3n un camp escalar i un camp vectorial respectivament, el laplaci\u00E0 de tots dos es pot escriure en termes de l'operador nabla com:"@ca . . . . . . . . . . . . "Laplaciano"@pt . . . . . "Em matem\u00E1tica e f\u00EDsica, o Laplaciano ou Operador de Laplace (ou ainda operador de Laplace-Beltrami), denotado por ou , sendo o operador nabla, \u00E9 um operador diferencial de segunda ordem. O Laplaciano, nome dado em homenagem a Pierre-Simon Laplace, aparece naturalmente em diversas equa\u00E7\u00F5es diferenciais parciais que modelam problemas f\u00EDsicos, tais como potencial el\u00E9trico e gravitacional, propaga\u00E7\u00E3o de ondas, condu\u00E7\u00E3o de calor e fluidos, e tamb\u00E9m fazendo parte das equa\u00E7\u00F5es de Poisson para eletrost\u00E1tica e da equa\u00E7\u00E3o de Schr\u00F6dinger independente do tempo."@pt . . . . "1124133580"^^ . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430"@ru . . . . . "En c\u00E0lcul vectorial, l'operador laplaci\u00E0 \u00E9s un operador diferencial el\u00B7l\u00EDptic de segon ordre, denotat com \u0394, relacionat amb certs problemes de minimitzaci\u00F3 de determinades magnituds sobre un cert domini. L'operador t\u00E9 aquest nom en reconeixement de Pierre-Simon Laplace, que va estudiar solucions d'equacions diferencials en derivades parcials en qu\u00E8 apareixia aquest operador."@ca . . . "Operador laplaciano"@es . . . . . . . . . "\u041E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 \u2014 \u0434\u0456\u044F \u043D\u0430\u0434 \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u0438\u043C \u0430\u0431\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u043C \u043F\u043E\u043B\u0435\u043C, \u0449\u043E \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F, \u044F\u043A \u0441\u0443\u043C\u0430 \u0434\u0440\u0443\u0433\u0438\u0445 \u0447\u0430\u0441\u0442\u043A\u043E\u0432\u0438\u0445 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u0445 \u043F\u043E \u043A\u043E\u0436\u043D\u0456\u0439 \u0434\u0435\u043A\u0430\u0440\u0442\u043E\u0432\u0456\u0439 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442\u0456. \u041F\u043E\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0430\u0431\u043E . \u0414\u043B\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 \u0447\u0430\u0441\u0442\u043E \u0432\u0438\u043A\u043E\u0440\u0438\u0441\u0442\u043E\u0432\u0443\u0454\u0442\u044C\u0441\u044F \u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0456\u0439 \u0456 \u0442\u0435\u043E\u0440\u0435\u0442\u0438\u0447\u043D\u0456\u0439 \u0444\u0456\u0437\u0438\u0446\u0456. \u0421\u043F\u0440\u0430\u0432\u0435\u0434\u043B\u0438\u0432\u0435 \u0441\u043F\u0456\u0432\u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u043D\u044F: . \u041D\u0430\u0437\u0432\u0430\u043D\u0438\u0439 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430."@uk . . "Operator Laplace\u2019a, laplasjan \u2013 operator r\u00F3\u017Cniczkowy drugiego rz\u0119du, wprowadzony przez Pierre\u2019a Simona de Laplace\u2019a. W uk\u0142adzie kartezja\u0144skim 3-wymiarowym ma posta\u0107: Operator ten uog\u00F3lnia si\u0119 na przestrzenie euklidesowe -wymiarowe z dowolnymi uk\u0142adami wsp\u00F3\u0142rz\u0119dnych krzywoliniowych (w tym ze wsp\u00F3\u0142rz\u0119dnymi kartezja\u0144skimi) oraz na dowolne przestrzenie riemannowskie i pseudoriemannowskie."@pl . . "De laplace-operator, ook wel laplaciaan genoemd, is een differentiaaloperator genoemd naar de Franse wiskundige Pierre-Simon Laplace en aangeduid door het symbool \u2206. In de natuurkunde vindt de operator toepassing bij de beschrijving van voortplanting van golven (golfvergelijking), bij warmtetransport en in de elektrostatica in de laplacevergelijking. In de kwantummechanica stelt de laplace-operator de kinetische energie voor in de schr\u00F6dingervergelijking. De functies waarvoor de laplaciaan gelijk is aan nul, worden in de wiskunde harmonische functies genoemd. . Alternatief kan men schrijven:"@nl . . . . . . "\uB77C\uD50C\uB77C\uC2A4 \uC5F0\uC0B0\uC790"@ko . . "Laplace\u016Fv oper\u00E1tor je diferenci\u00E1ln\u00ED oper\u00E1tor definovan\u00FD jako divergence gradientu skal\u00E1rn\u00EDho, nebo obecn\u011B tenzorov\u00E9ho pole nazvan\u00FD podle Pierre-Simona Laplace. Je-li aplikov\u00E1n na skal\u00E1rn\u00ED pole, v\u00FDsledkem je skal\u00E1rn\u00ED pole, je-li aplikov\u00E1n na tenzorov\u00E9 pole, v\u00FDsledkem je tenzorov\u00E9 pole stejn\u00E9ho \u0159\u00E1du. Zna\u010D\u00ED se symbolem ."@cs . . "Laplace-operator"@nl . . . . . . . . . . . . . . . . "Laplaceoperatorn"@sv . . . . . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430"@uk . "\u6570\u5B66\u306B\u304A\u3051\u308B\u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20\uFF08\u30E9\u30D7\u30E9\u30B9\u3055\u3088\u3046\u305D\u3001\u82F1: Laplace operator\uFF09\u3042\u308B\u3044\u306F\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\uFF08\u82F1: Laplacian)\u306F\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593\u4E0A\u306E\u51FD\u6570\u306E\u52FE\u914D\u306E\u767A\u6563\u3068\u3057\u3066\u4E0E\u3048\u3089\u308C\u308B\u5FAE\u5206\u4F5C\u7528\u7D20\u3067\u3042\u308B\u3002\u8A18\u53F7\u3067\u306F \u2207\u00B7\u2207, \u22072, \u3042\u308B\u3044\u306F \u2206 \u3067\u8868\u3055\u308C\u308B\u306E\u304C\u666E\u901A\u3067\u3042\u308B\u3002\u51FD\u6570 f \u306E\u70B9 p \u306B\u304A\u3051\u308B\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3 \u2206f(p) \u306F\uFF08\u6B21\u5143\u306B\u4F9D\u5B58\u3059\u308B\u5B9A\u6570\u306E\u9055\u3044\u3092\u9664\u3044\u3066\uFF09\u70B9 p \u3092\u4E2D\u5FC3\u3068\u3059\u308B\u7403\u9762\u3092\u534A\u5F84\u304C\u5897\u5927\u3059\u308B\u3088\u3046\u306B\u52D5\u304B\u3059\u3068\u304D\u306E f(p) \u304B\u3089\u5F97\u3089\u308C\u308B\u5E73\u5747\u5024\u306B\u306A\u3063\u3066\u3044\u308B\u3002\u76F4\u4EA4\u5EA7\u6A19\u7CFB\u306B\u304A\u3044\u3066\u306F\u3001\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\u306F\u5404\u72EC\u7ACB\u5909\u6570\u306B\u95A2\u3059\u308B\u51FD\u6570\u306E\u4E8C\u968E\uFF08\u975E\u6DF7\u5408\uFF09\u504F\u5C0E\u51FD\u6570\u306E\u548C\u3068\u3057\u3066\u4E0E\u3048\u3089\u308C\u3001\u307E\u305F\u307B\u304B\u306B\u5186\u7B52\u5EA7\u6A19\u7CFB\u3084\u7403\u5EA7\u6A19\u7CFB\u306A\u3069\u306E\u5EA7\u6A19\u7CFB\u306B\u304A\u3044\u3066\u3082\u6709\u7528\u306A\u8868\u793A\u3092\u6301\u3064\u3002 \u30E9\u30D7\u30E9\u30B9\u4F5C\u7528\u7D20\u306E\u540D\u79F0\u306F\u3001\u5929\u4F53\u529B\u5B66\u306E\u7814\u7A76\u306B\u540C\u4F5C\u7528\u7D20\u3092\u6700\u521D\u306B\u7528\u3044\u305F\u30D5\u30E9\u30F3\u30B9\u4EBA\u6570\u5B66\u8005\u306E\u30D4\u30A8\u30FC\u30EB\uFF1D\u30B7\u30E2\u30F3\u30FB\u30C9\u30FB\u30E9\u30D7\u30E9\u30B9 (1749\u20131827) \u306B\u56E0\u3093\u3067\u3044\u308B\u3002\u540C\u4F5C\u7528\u7D20\u306F\u4E0E\u3048\u3089\u308C\u305F\u91CD\u529B\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u306B\u9069\u7528\u3059\u308B\u3068\u8CEA\u91CF\u5BC6\u5EA6\u306E\u5B9A\u6570\u500D\u3092\u4E0E\u3048\u308B\u3002\u73FE\u5728\u3067\u306F\u30E9\u30D7\u30E9\u30B9\u65B9\u7A0B\u5F0F\u3068\u547C\u3070\u308C\u308B\u65B9\u7A0B\u5F0F \u2206f = 0 \u306E\u89E3\u306F\u8ABF\u548C\u51FD\u6570\u3068\u547C\u3070\u308C\u3001\u81EA\u7531\u7A7A\u9593\u306B\u304A\u3044\u3066\u53EF\u80FD\u306A\u91CD\u529B\u5834\u3092\u8868\u73FE\u3059\u308B\u3082\u306E\u3067\u3042\u308B\u3002"@ja . "De laplace-operator, ook wel laplaciaan genoemd, is een differentiaaloperator genoemd naar de Franse wiskundige Pierre-Simon Laplace en aangeduid door het symbool \u2206. In de natuurkunde vindt de operator toepassing bij de beschrijving van voortplanting van golven (golfvergelijking), bij warmtetransport en in de elektrostatica in de laplacevergelijking. In de kwantummechanica stelt de laplace-operator de kinetische energie voor in de schr\u00F6dingervergelijking. De functies waarvoor de laplaciaan gelijk is aan nul, worden in de wiskunde harmonische functies genoemd. Voor een scalaire functie op een -dimensionale Euclidische ruimte is de laplace-operator gedefinieerd door: Hierin staat voor de tweede parti\u00EBle afgeleide naar de variabele . Als operator schrijft men daarom wel: . Alternatief kan men schrijven: Ook kan de laplace-operator (in rechthoekige co\u00F6rdinaten) uitgedrukt worden in de operator nabla (\u2207):"@nl . . . . . . . . .