[{"data":1,"prerenderedAt":4216},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fhesse-matrix":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fhesse-matrix":4203},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":368,"checks":4138,"conditions":4163,"conditionsHtml":4164,"deprecated":4165,"description":4132,"extension":4166,"figure":4165,"figureCaption":4165,"figureHtml":4165,"formulaId":4167,"latex":4168,"latexHtml":4169,"level":4170,"meta":4171,"navigation":4172,"path":4173,"redirectFrom":4174,"related":4175,"seo":4180,"stem":4181,"subject":581,"summary":4182,"tags":4183,"variables":4189,"__hash__":4202},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fhesse-matrix.md","Hesse-Matrix",[550,551,552,553,554],"Hessematrix","Hesse Matrix berechnen","Matrix der zweiten Ableitungen","Hessian","Hesse-Matrix positiv definit",{"type":556,"value":557,"toc":4131},"minimark",[558,563,1039,1079,1082,1799,1902,1906,2430,3011,3571,3811,3815],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,570,571,617,618,675,676,965,966,1038],"p",{},"Die ",[568,569,548],"strong",{}," sammelt alle zweiten partiellen Ableitungen einer Funktion. Bei ",[572,573,576,598],"span",{"className":574},[575],"katex",[572,577,580],{"className":578},[579],"katex-mathml",[581,582,584],"math",{"xmlns":583},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[585,586,587,594],"semantics",{},[588,589,590],"mrow",{},[591,592,593],"mi",{},"n",[595,596,593],"annotation",{"encoding":597},"application\u002Fx-tex",[572,599,603],{"className":600,"ariaHidden":602},[601],"katex-html","true",[572,604,607,612],{"className":605},[606],"base",[572,608],{"className":609,"style":611},[610],"strut","height:0.4306em;",[572,613,593],{"className":614},[615,616],"mord","mathnormal"," Variablen ist sie die ",[572,619,621,641],{"className":620},[575],[572,622,624],{"className":623},[579],[581,625,626],{"xmlns":583},[585,627,628,638],{},[588,629,630,632,636],{},[591,631,593],{},[633,634,635],"mo",{},"×",[591,637,593],{},[595,639,640],{"encoding":597},"n \\times n",[572,642,644,666],{"className":643,"ariaHidden":602},[601],[572,645,647,651,654,659,663],{"className":646},[606],[572,648],{"className":649,"style":650},[610],"height:0.6667em;vertical-align:-0.0833em;",[572,652,593],{"className":653},[615,616],[572,655],{"className":656,"style":658},[657],"mspace","margin-right:0.2222em;",[572,660,635],{"className":661},[662],"mbin",[572,664],{"className":665,"style":658},[657],[572,667,669,672],{"className":668},[606],[572,670],{"className":671,"style":611},[610],[572,673,593],{"className":674},[615,616],"-Matrix mit den Einträgen ",[572,677,679,736],{"className":678},[575],[572,680,682],{"className":681},[579],[581,683,684],{"xmlns":583},[585,685,686,733],{},[588,687,688],{},[689,690,691,707],"mfrac",{},[588,692,693,704],{},[694,695,696,700],"msup",{},[591,697,699],{"mathvariant":698},"normal","∂",[701,702,703],"mn",{},"2",[591,705,706],{},"f",[588,708,709,711,720,724,726],{},[591,710,699],{"mathvariant":698},[712,713,714,717],"msub",{},[591,715,716],{},"x",[591,718,719],{},"i",[721,722,723],"mtext",{}," ",[591,725,699],{"mathvariant":698},[712,727,728,730],{},[591,729,716],{},[591,731,732],{},"j",[595,734,735],{"encoding":597},"\\frac{\\partial^2 f}{\\partial x_i\\, \\partial x_j}",[572,737,739],{"className":738,"ariaHidden":602},[601],[572,740,742,746],{"className":741},[606],[572,743],{"className":744,"style":745},[610],"height:1.6123em;vertical-align:-0.5423em;",[572,747,749,754,961],{"className":748},[615],[572,750],{"className":751},[752,753],"mopen","nulldelimiter",[572,755,757],{"className":756},[689],[572,758,762,952],{"className":759},[760,761],"vlist-t","vlist-t2",[572,763,766,949],{"className":764},[765],"vlist-r",[572,767,771,891,902],{"className":768,"style":770},[769],"vlist","height:1.07em;",[572,772,774,779],{"style":773},"top:-2.655em;",[572,775],{"className":776,"style":778},[777],"pstrut","height:3em;",[572,780,786],{"className":781},[782,783,784,785],"sizing","reset-size6","size3","mtight",[572,787,789,793,842,846,849],{"className":788},[615,785],[572,790,699],{"className":791,"style":792},[615,785],"margin-right:0.0556em;",[572,794,796,799],{"className":795},[615,785],[572,797,716],{"className":798},[615,616,785],[572,800,803],{"className":801},[802],"msupsub",[572,804,806,833],{"className":805},[760,761],[572,807,809,828],{"className":808},[765],[572,810,813],{"className":811,"style":812},[769],"height:0.3281em;",[572,814,816,820],{"style":815},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[572,817],{"className":818,"style":819},[777],"height:2.5em;",[572,821,825],{"className":822},[782,823,824,785],"reset-size3","size1",[572,826,719],{"className":827},[615,616,785],[572,829,832],{"className":830},[831],"vlist-s","​",[572,834,836],{"className":835},[765],[572,837,840],{"className":838,"style":839},[769],"height:0.143em;",[572,841],{},[572,843],{"className":844,"style":845},[657,785],"margin-right:0.1952em;",[572,847,699],{"className":848,"style":792},[615,785],[572,850,852,855],{"className":851},[615,785],[572,853,716],{"className":854},[615,616,785],[572,856,858],{"className":857},[802],[572,859,861,882],{"className":860},[760,761],[572,862,864,879],{"className":863},[765],[572,865,867],{"className":866,"style":812},[769],[572,868,869,872],{"style":815},[572,870],{"className":871,"style":819},[777],[572,873,875],{"className":874},[782,823,824,785],[572,876,732],{"className":877,"style":878},[615,616,785],"margin-right:0.0572em;",[572,880,832],{"className":881},[831],[572,883,885],{"className":884},[765],[572,886,889],{"className":887,"style":888},[769],"height:0.2819em;",[572,890],{},[572,892,894,897],{"style":893},"top:-3.23em;",[572,895],{"className":896,"style":778},[777],[572,898],{"className":899,"style":901},[900],"frac-line","border-bottom-width:0.04em;",[572,903,905,908],{"style":904},"top:-3.4461em;",[572,906],{"className":907,"style":778},[777],[572,909,911],{"className":910},[782,783,784,785],[572,912,914,945],{"className":913},[615,785],[572,915,917,920],{"className":916},[615,785],[572,918,699],{"className":919,"style":792},[615,785],[572,921,923],{"className":922},[802],[572,924,926],{"className":925},[760],[572,927,929],{"className":928},[765],[572,930,933],{"className":931,"style":932},[769],"height:0.8913em;",[572,934,936,939],{"style":935},"top:-2.931em;margin-right:0.0714em;",[572,937],{"className":938,"style":819},[777],[572,940,942],{"className":941},[782,823,824,785],[572,943,703],{"className":944},[615,785],[572,946,706],{"className":947,"style":948},[615,616,785],"margin-right:0.1076em;",[572,950,832],{"className":951},[831],[572,953,955],{"className":954},[765],[572,956,959],{"className":957,"style":958},[769],"height:0.5423em;",[572,960],{},[572,962],{"className":963},[964,753],"mclose",". Sie spielt für Funktionen mehrerer Variablen die Rolle der zweiten Ableitung ",[572,967,969,992],{"className":968},[575],[572,970,972],{"className":971},[579],[581,973,974],{"xmlns":583},[585,975,976,989],{},[588,977,978],{},[694,979,980,982],{},[591,981,706],{},[588,983,984,987],{},[633,985,986],{"mathvariant":698},"′",[633,988,986],{"mathvariant":698},[595,990,991],{"encoding":597},"f''",[572,993,995],{"className":994,"ariaHidden":602},[601],[572,996,998,1002],{"className":997},[606],[572,999],{"className":1000,"style":1001},[610],"height:0.9463em;vertical-align:-0.1944em;",[572,1003,1005,1008],{"className":1004},[615],[572,1006,706],{"className":1007,"style":948},[615,616],[572,1009,1011],{"className":1010},[802],[572,1012,1014],{"className":1013},[760],[572,1015,1017],{"className":1016},[765],[572,1018,1021],{"className":1019,"style":1020},[769],"height:0.7519em;",[572,1022,1024,1028],{"style":1023},"top:-3.063em;margin-right:0.05em;",[572,1025],{"className":1026,"style":1027},[777],"height:2.7em;",[572,1029,1031],{"className":1030},[782,783,784,785],[572,1032,1034],{"className":1033},[615,785],[572,1035,1037],{"className":1036},[615,785],"′′",".",[564,1040,1041,1042,1071,1072,1075,1076,1038],{},"Ist ",[572,1043,1045,1058],{"className":1044},[575],[572,1046,1048],{"className":1047},[579],[581,1049,1050],{"xmlns":583},[585,1051,1052,1056],{},[588,1053,1054],{},[591,1055,706],{},[595,1057,706],{"encoding":597},[572,1059,1061],{"className":1060,"ariaHidden":602},[601],[572,1062,1064,1068],{"className":1063},[606],[572,1065],{"className":1066,"style":1067},[610],"height:0.8889em;vertical-align:-0.1944em;",[572,1069,706],{"className":1070,"style":948},[615,616]," zweimal stetig differenzierbar, sind die gemischten Ableitungen nach dem ",[568,1073,1074],{},"Satz von Schwarz"," gleich, und die Hesse-Matrix ist ",[568,1077,1078],{},"symmetrisch",[564,1080,1081],{},"Sie taucht in der Taylor-Näherung zweiter Ordnung auf:",[572,1083,1086],{"className":1084},[1085],"katex-display",[572,1087,1089,1222],{"className":1088},[575],[572,1090,1092],{"className":1091},[579],[581,1093,1095],{"xmlns":583,"display":1094},"block",[585,1096,1097,1219],{},[588,1098,1099,1101,1105,1114,1117,1124,1127,1130,1132,1134,1140,1142,1144,1147,1149,1151,1157,1159,1162,1168,1170,1181,1183,1194,1201,1203,1209,1211,1213],{},[591,1100,706],{},[633,1102,1104],{"stretchy":1103},"false","(",[1106,1107,1108,1111],"mover",{"accent":602},[591,1109,1110],{},"a",[633,1112,1113],{},"⃗",[633,1115,1116],{},"+",[1106,1118,1119,1122],{"accent":602},[591,1120,1121],{},"h",[633,1123,1113],{},[633,1125,1126],{"stretchy":1103},")",[633,1128,1129],{},"≈",[591,1131,706],{},[633,1133,1104],{"stretchy":1103},[1106,1135,1136,1138],{"accent":602},[591,1137,1110],{},[633,1139,1113],{},[633,1141,1126],{"stretchy":1103},[633,1143,1116],{},[591,1145,1146],{"mathvariant":698},"∇",[591,1148,706],{},[633,1150,1104],{"stretchy":1103},[1106,1152,1153,1155],{"accent":602},[591,1154,1110],{},[633,1156,1113],{},[633,1158,1126],{"stretchy":1103},[633,1160,1161],{},"⋅",[1106,1163,1164,1166],{"accent":602},[591,1165,1121],{},[633,1167,1113],{},[633,1169,1116],{},[1171,1172,1174],"mstyle",{"scriptlevel":1173,"displaystyle":1103},"0",[689,1175,1176,1179],{},[701,1177,1178],{},"1",[701,1180,703],{},[721,1182,723],{},[694,1184,1185,1191],{},[1106,1186,1187,1189],{"accent":602},[591,1188,1121],{},[633,1190,1113],{},[591,1192,1193],{},"T",[712,1195,1196,1199],{},[591,1197,1198],{},"H",[591,1200,706],{},[633,1202,1104],{"stretchy":1103},[1106,1204,1205,1207],{"accent":602},[591,1206,1110],{},[633,1208,1113],{},[633,1210,1126],{"stretchy":1103},[721,1212,723],{},[1106,1214,1215,1217],{"accent":602},[591,1216,1121],{},[633,1218,1113],{},[595,1220,1221],{"encoding":597},"f(\\vec a + \\vec h) \\approx f(\\vec a) + \\nabla f(\\vec a) \\cdot \\vec h + \\tfrac{1}{2}\\, \\vec h^{T} H_f(\\vec a)\\, \\vec h",[572,1223,1225,1298,1356,1415,1477,1528],{"className":1224,"ariaHidden":602},[601],[572,1226,1228,1232,1235,1238,1289,1292,1295],{"className":1227},[606],[572,1229],{"className":1230,"style":1231},[610],"height:1em;vertical-align:-0.25em;",[572,1233,706],{"className":1234,"style":948},[615,616],[572,1236,1104],{"className":1237},[752],[572,1239,1242],{"className":1240},[615,1241],"accent",[572,1243,1245],{"className":1244},[760],[572,1246,1248],{"className":1247},[765],[572,1249,1252,1261],{"className":1250,"style":1251},[769],"height:0.714em;",[572,1253,1255,1258],{"style":1254},"top:-3em;",[572,1256],{"className":1257,"style":778},[777],[572,1259,1110],{"className":1260},[615,616],[572,1262,1263,1266],{"style":1254},[572,1264],{"className":1265,"style":778},[777],[572,1267,1271],{"className":1268,"style":1270},[1269],"accent-body","left:-0.2355em;",[572,1272,1276],{"className":1273,"style":1275},[1274],"overlay","height:0.714em;width:0.471em;",[1277,1278,1285],"svg",{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[1286,1287],"path",{"d":1288},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[572,1290],{"className":1291,"style":658},[657],[572,1293,1116],{"className":1294},[662],[572,1296],{"className":1297,"style":658},[657],[572,1299,1301,1305,1342,1345,1349,1353],{"className":1300},[606],[572,1302],{"className":1303,"style":1304},[610],"height:1.2274em;vertical-align:-0.25em;",[572,1306,1308],{"className":1307},[615,1241],[572,1309,1311],{"className":1310},[760],[572,1312,1314],{"className":1313},[765],[572,1315,1318,1326],{"className":1316,"style":1317},[769],"height:0.9774em;",[572,1319,1320,1323],{"style":1254},[572,1321],{"className":1322,"style":778},[777],[572,1324,1121],{"className":1325},[615,616],[572,1327,1329,1332],{"style":1328},"top:-3.2634em;",[572,1330],{"className":1331,"style":778},[777],[572,1333,1335],{"className":1334,"style":1270},[1269],[572,1336,1338],{"className":1337,"style":1275},[1274],[1277,1339,1340],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1341],{"d":1288},[572,1343,1126],{"className":1344},[964],[572,1346],{"className":1347,"style":1348},[657],"margin-right:0.2778em;",[572,1350,1129],{"className":1351},[1352],"mrel",[572,1354],{"className":1355,"style":1348},[657],[572,1357,1359,1362,1365,1368,1403,1406,1409,1412],{"className":1358},[606],[572,1360],{"className":1361,"style":1231},[610],[572,1363,706],{"className":1364,"style":948},[615,616],[572,1366,1104],{"className":1367},[752],[572,1369,1371],{"className":1370},[615,1241],[572,1372,1374],{"className":1373},[760],[572,1375,1377],{"className":1376},[765],[572,1378,1380,1388],{"className":1379,"style":1251},[769],[572,1381,1382,1385],{"style":1254},[572,1383],{"className":1384,"style":778},[777],[572,1386,1110],{"className":1387},[615,616],[572,1389,1390,1393],{"style":1254},[572,1391],{"className":1392,"style":778},[777],[572,1394,1396],{"className":1395,"style":1270},[1269],[572,1397,1399],{"className":1398,"style":1275},[1274],[1277,1400,1401],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1402],{"d":1288},[572,1404,1126],{"className":1405},[964],[572,1407],{"className":1408,"style":658},[657],[572,1410,1116],{"className":1411},[662],[572,1413],{"className":1414,"style":658},[657],[572,1416,1418,1421,1424,1427,1430,1465,1468,1471,1474],{"className":1417},[606],[572,1419],{"className":1420,"style":1231},[610],[572,1422,1146],{"className":1423},[615],[572,1425,706],{"className":1426,"style":948},[615,616],[572,1428,1104],{"className":1429},[752],[572,1431,1433],{"className":1432},[615,1241],[572,1434,1436],{"className":1435},[760],[572,1437,1439],{"className":1438},[765],[572,1440,1442,1450],{"className":1441,"style":1251},[769],[572,1443,1444,1447],{"style":1254},[572,1445],{"className":1446,"style":778},[777],[572,1448,1110],{"className":1449},[615,616],[572,1451,1452,1455],{"style":1254},[572,1453],{"className":1454,"style":778},[777],[572,1456,1458],{"className":1457,"style":1270},[1269],[572,1459,1461],{"className":1460,"style":1275},[1274],[1277,1462,1463],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1464],{"d":1288},[572,1466,1126],{"className":1467},[964],[572,1469],{"className":1470,"style":658},[657],[572,1472,1161],{"className":1473},[662],[572,1475],{"className":1476,"style":658},[657],[572,1478,1480,1484,1519,1522,1525],{"className":1479},[606],[572,1481],{"className":1482,"style":1483},[610],"height:1.0608em;vertical-align:-0.0833em;",[572,1485,1487],{"className":1486},[615,1241],[572,1488,1490],{"className":1489},[760],[572,1491,1493],{"className":1492},[765],[572,1494,1496,1504],{"className":1495,"style":1317},[769],[572,1497,1498,1501],{"style":1254},[572,1499],{"className":1500,"style":778},[777],[572,1502,1121],{"className":1503},[615,616],[572,1505,1506,1509],{"style":1328},[572,1507],{"className":1508,"style":778},[777],[572,1510,1512],{"className":1511,"style":1270},[1269],[572,1513,1515],{"className":1514,"style":1275},[1274],[1277,1516,1517],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1518],{"d":1288},[572,1520],{"className":1521,"style":658},[657],[572,1523,1116],{"className":1524},[662],[572,1526],{"className":1527,"style":658},[657],[572,1529,1531,1535,1606,1610,1676,1720,1723,1758,1761,1764],{"className":1530},[606],[572,1532],{"className":1533,"style":1534},[610],"height:1.3224em;vertical-align:-0.345em;",[572,1536,1538,1541,1603],{"className":1537},[615],[572,1539],{"className":1540},[752,753],[572,1542,1544],{"className":1543},[689],[572,1545,1547,1594],{"className":1546},[760,761],[572,1548,1550,1591],{"className":1549},[765],[572,1551,1554,1568,1576],{"className":1552,"style":1553},[769],"height:0.8451em;",[572,1555,1556,1559],{"style":773},[572,1557],{"className":1558,"style":778},[777],[572,1560,1562],{"className":1561},[782,783,784,785],[572,1563,1565],{"className":1564},[615,785],[572,1566,703],{"className":1567},[615,785],[572,1569,1570,1573],{"style":893},[572,1571],{"className":1572,"style":778},[777],[572,1574],{"className":1575,"style":901},[900],[572,1577,1579,1582],{"style":1578},"top:-3.394em;",[572,1580],{"className":1581,"style":778},[777],[572,1583,1585],{"className":1584},[782,783,784,785],[572,1586,1588],{"className":1587},[615,785],[572,1589,1178],{"className":1590},[615,785],[572,1592,832],{"className":1593},[831],[572,1595,1597],{"className":1596},[765],[572,1598,1601],{"className":1599,"style":1600},[769],"height:0.345em;",[572,1602],{},[572,1604],{"className":1605},[964,753],[572,1607],{"className":1608,"style":1609},[657],"margin-right:0.1667em;",[572,1611,1613,1648],{"className":1612},[615],[572,1614,1616],{"className":1615},[615,1241],[572,1617,1619],{"className":1618},[760],[572,1620,1622],{"className":1621},[765],[572,1623,1625,1633],{"className":1624,"style":1317},[769],[572,1626,1627,1630],{"style":1254},[572,1628],{"className":1629,"style":778},[777],[572,1631,1121],{"className":1632},[615,616],[572,1634,1635,1638],{"style":1328},[572,1636],{"className":1637,"style":778},[777],[572,1639,1641],{"className":1640,"style":1270},[1269],[572,1642,1644],{"className":1643,"style":1275},[1274],[1277,1645,1646],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1647],{"d":1288},[572,1649,1651],{"className":1650},[802],[572,1652,1654],{"className":1653},[760],[572,1655,1657],{"className":1656},[765],[572,1658,1660],{"className":1659,"style":932},[769],[572,1661,1663,1666],{"style":1662},"top:-3.113em;margin-right:0.05em;",[572,1664],{"className":1665,"style":1027},[777],[572,1667,1669],{"className":1668},[782,783,784,785],[572,1670,1672],{"className":1671},[615,785],[572,1673,1193],{"className":1674,"style":1675},[615,616,785],"margin-right:0.1389em;",[572,1677,1679,1683],{"className":1678},[615],[572,1680,1198],{"className":1681,"style":1682},[615,616],"margin-right:0.0813em;",[572,1684,1686],{"className":1685},[802],[572,1687,1689,1711],{"className":1688},[760,761],[572,1690,1692,1708],{"className":1691},[765],[572,1693,1696],{"className":1694,"style":1695},[769],"height:0.3361em;",[572,1697,1699,1702],{"style":1698},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[572,1700],{"className":1701,"style":1027},[777],[572,1703,1705],{"className":1704},[782,783,784,785],[572,1706,706],{"className":1707,"style":948},[615,616,785],[572,1709,832],{"className":1710},[831],[572,1712,1714],{"className":1713},[765],[572,1715,1718],{"className":1716,"style":1717},[769],"height:0.2861em;",[572,1719],{},[572,1721,1104],{"className":1722},[752],[572,1724,1726],{"className":1725},[615,1241],[572,1727,1729],{"className":1728},[760],[572,1730,1732],{"className":1731},[765],[572,1733,1735,1743],{"className":1734,"style":1251},[769],[572,1736,1737,1740],{"style":1254},[572,1738],{"className":1739,"style":778},[777],[572,1741,1110],{"className":1742},[615,616],[572,1744,1745,1748],{"style":1254},[572,1746],{"className":1747,"style":778},[777],[572,1749,1751],{"className":1750,"style":1270},[1269],[572,1752,1754],{"className":1753,"style":1275},[1274],[1277,1755,1756],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1757],{"d":1288},[572,1759,1126],{"className":1760},[964],[572,1762],{"className":1763,"style":1609},[657],[572,1765,1767],{"className":1766},[615,1241],[572,1768,1770],{"className":1769},[760],[572,1771,1773],{"className":1772},[765],[572,1774,1776,1784],{"className":1775,"style":1317},[769],[572,1777,1778,1781],{"style":1254},[572,1779],{"className":1780,"style":778},[777],[572,1782,1121],{"className":1783},[615,616],[572,1785,1786,1789],{"style":1328},[572,1787],{"className":1788,"style":778},[777],[572,1790,1792],{"className":1791,"style":1270},[1269],[572,1793,1795],{"className":1794,"style":1275},[1274],[1277,1796,1797],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1798],{"d":1288},[564,1800,1801,1802,1897,1898,1901],{},"Daraus folgt ihre Hauptanwendung: In einem kritischen Punkt (",[572,1803,1805,1830],{"className":1804},[575],[572,1806,1808],{"className":1807},[579],[581,1809,1810],{"xmlns":583},[585,1811,1812,1827],{},[588,1813,1814,1816,1818,1821],{},[591,1815,1146],{"mathvariant":698},[591,1817,706],{},[633,1819,1820],{},"=",[1106,1822,1823,1825],{"accent":602},[701,1824,1173],{},[633,1826,1113],{},[595,1828,1829],{"encoding":597},"\\nabla f = \\vec 0",[572,1831,1833,1854],{"className":1832,"ariaHidden":602},[601],[572,1834,1836,1839,1842,1845,1848,1851],{"className":1835},[606],[572,1837],{"className":1838,"style":1067},[610],[572,1840,1146],{"className":1841},[615],[572,1843,706],{"className":1844,"style":948},[615,616],[572,1846],{"className":1847,"style":1348},[657],[572,1849,1820],{"className":1850},[1352],[572,1852],{"className":1853,"style":1348},[657],[572,1855,1857,1861],{"className":1856},[606],[572,1858],{"className":1859,"style":1860},[610],"height:0.9274em;",[572,1862,1864],{"className":1863},[615,1241],[572,1865,1867],{"className":1866},[760],[572,1868,1870],{"className":1869},[765],[572,1871,1873,1881],{"className":1872,"style":1860},[769],[572,1874,1875,1878],{"style":1254},[572,1876],{"className":1877,"style":778},[777],[572,1879,1173],{"className":1880},[615],[572,1882,1884,1887],{"style":1883},"top:-3.2134em;",[572,1885],{"className":1886,"style":778},[777],[572,1888,1890],{"className":1889,"style":1270},[1269],[572,1891,1893],{"className":1892,"style":1275},[1274],[1277,1894,1895],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,1896],{"d":1288},") entscheidet die ",[568,1899,1900],{},"Definitheit"," der Hesse-Matrix über die Art der Stelle. Positiv definit (alle Eigenwerte positiv) bedeutet lokales Minimum, negativ definit lokales Maximum, indefinit Sattelpunkt.",[559,1903,1905],{"id":1904},"beispiele","Beispiele",[564,1907,1908,1909,2112,2113,2300,2301,2429],{},"Für ",[572,1910,1912,1964],{"className":1911},[575],[572,1913,1915],{"className":1914},[579],[581,1916,1917],{"xmlns":583},[585,1918,1919,1961],{},[588,1920,1921,1923,1925,1927,1930,1933,1935,1937,1944,1946,1948,1950,1956,1959],{},[591,1922,706],{},[633,1924,1104],{"stretchy":1103},[591,1926,716],{},[633,1928,1929],{"separator":602},",",[591,1931,1932],{},"y",[633,1934,1126],{"stretchy":1103},[633,1936,1820],{},[694,1938,1939,1941],{},[591,1940,716],{},[701,1942,1943],{},"3",[633,1945,1116],{},[701,1947,703],{},[591,1949,716],{},[694,1951,1952,1954],{},[591,1953,1932],{},[701,1955,703],{},[633,1957,1958],{},"−",[591,1960,1932],{},[595,1962,1963],{"encoding":597},"f(x, y) = x^3 + 2xy^2 - y",[572,1965,1967,2005,2051,2102],{"className":1966,"ariaHidden":602},[601],[572,1968,1970,1973,1976,1979,1982,1986,1989,1993,1996,1999,2002],{"className":1969},[606],[572,1971],{"className":1972,"style":1231},[610],[572,1974,706],{"className":1975,"style":948},[615,616],[572,1977,1104],{"className":1978},[752],[572,1980,716],{"className":1981},[615,616],[572,1983,1929],{"className":1984},[1985],"mpunct",[572,1987],{"className":1988,"style":1609},[657],[572,1990,1932],{"className":1991,"style":1992},[615,616],"margin-right:0.0359em;",[572,1994,1126],{"className":1995},[964],[572,1997],{"className":1998,"style":1348},[657],[572,2000,1820],{"className":2001},[1352],[572,2003],{"className":2004,"style":1348},[657],[572,2006,2008,2012,2042,2045,2048],{"className":2007},[606],[572,2009],{"className":2010,"style":2011},[610],"height:0.8974em;vertical-align:-0.0833em;",[572,2013,2015,2018],{"className":2014},[615],[572,2016,716],{"className":2017},[615,616],[572,2019,2021],{"className":2020},[802],[572,2022,2024],{"className":2023},[760],[572,2025,2027],{"className":2026},[765],[572,2028,2031],{"className":2029,"style":2030},[769],"height:0.8141em;",[572,2032,2033,2036],{"style":1023},[572,2034],{"className":2035,"style":1027},[777],[572,2037,2039],{"className":2038},[782,783,784,785],[572,2040,1943],{"className":2041},[615,785],[572,2043],{"className":2044,"style":658},[657],[572,2046,1116],{"className":2047},[662],[572,2049],{"className":2050,"style":658},[657],[572,2052,2054,2058,2061,2064,2093,2096,2099],{"className":2053},[606],[572,2055],{"className":2056,"style":2057},[610],"height:1.0085em;vertical-align:-0.1944em;",[572,2059,703],{"className":2060},[615],[572,2062,716],{"className":2063},[615,616],[572,2065,2067,2070],{"className":2066},[615],[572,2068,1932],{"className":2069,"style":1992},[615,616],[572,2071,2073],{"className":2072},[802],[572,2074,2076],{"className":2075},[760],[572,2077,2079],{"className":2078},[765],[572,2080,2082],{"className":2081,"style":2030},[769],[572,2083,2084,2087],{"style":1023},[572,2085],{"className":2086,"style":1027},[777],[572,2088,2090],{"className":2089},[782,783,784,785],[572,2091,703],{"className":2092},[615,785],[572,2094],{"className":2095,"style":658},[657],[572,2097,1958],{"className":2098},[662],[572,2100],{"className":2101,"style":658},[657],[572,2103,2105,2109],{"className":2104},[606],[572,2106],{"className":2107,"style":2108},[610],"height:0.625em;vertical-align:-0.1944em;",[572,2110,1932],{"className":2111,"style":1992},[615,616]," ist ",[572,2114,2116,2154],{"className":2115},[575],[572,2117,2119],{"className":2118},[579],[581,2120,2121],{"xmlns":583},[585,2122,2123,2151],{},[588,2124,2125,2131,2133,2135,2141,2143,2145],{},[712,2126,2127,2129],{},[591,2128,706],{},[591,2130,716],{},[633,2132,1820],{},[701,2134,1943],{},[694,2136,2137,2139],{},[591,2138,716],{},[701,2140,703],{},[633,2142,1116],{},[701,2144,703],{},[694,2146,2147,2149],{},[591,2148,1932],{},[701,2150,703],{},[595,2152,2153],{"encoding":597},"f_x = 3x^2 + 2y^2",[572,2155,2157,2215,2262],{"className":2156,"ariaHidden":602},[601],[572,2158,2160,2163,2206,2209,2212],{"className":2159},[606],[572,2161],{"className":2162,"style":1067},[610],[572,2164,2166,2169],{"className":2165},[615],[572,2167,706],{"className":2168,"style":948},[615,616],[572,2170,2172],{"className":2171},[802],[572,2173,2175,2197],{"className":2174},[760,761],[572,2176,2178,2194],{"className":2177},[765],[572,2179,2182],{"className":2180,"style":2181},[769],"height:0.1514em;",[572,2183,2185,2188],{"style":2184},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[572,2186],{"className":2187,"style":1027},[777],[572,2189,2191],{"className":2190},[782,783,784,785],[572,2192,716],{"className":2193},[615,616,785],[572,2195,832],{"className":2196},[831],[572,2198,2200],{"className":2199},[765],[572,2201,2204],{"className":2202,"style":2203},[769],"height:0.15em;",[572,2205],{},[572,2207],{"className":2208,"style":1348},[657],[572,2210,1820],{"className":2211},[1352],[572,2213],{"className":2214,"style":1348},[657],[572,2216,2218,2221,2224,2253,2256,2259],{"className":2217},[606],[572,2219],{"className":2220,"style":2011},[610],[572,2222,1943],{"className":2223},[615],[572,2225,2227,2230],{"className":2226},[615],[572,2228,716],{"className":2229},[615,616],[572,2231,2233],{"className":2232},[802],[572,2234,2236],{"className":2235},[760],[572,2237,2239],{"className":2238},[765],[572,2240,2242],{"className":2241,"style":2030},[769],[572,2243,2244,2247],{"style":1023},[572,2245],{"className":2246,"style":1027},[777],[572,2248,2250],{"className":2249},[782,783,784,785],[572,2251,703],{"className":2252},[615,785],[572,2254],{"className":2255,"style":658},[657],[572,2257,1116],{"className":2258},[662],[572,2260],{"className":2261,"style":658},[657],[572,2263,2265,2268,2271],{"className":2264},[606],[572,2266],{"className":2267,"style":2057},[610],[572,2269,703],{"className":2270},[615],[572,2272,2274,2277],{"className":2273},[615],[572,2275,1932],{"className":2276,"style":1992},[615,616],[572,2278,2280],{"className":2279},[802],[572,2281,2283],{"className":2282},[760],[572,2284,2286],{"className":2285},[765],[572,2287,2289],{"className":2288,"style":2030},[769],[572,2290,2291,2294],{"style":1023},[572,2292],{"className":2293,"style":1027},[777],[572,2295,2297],{"className":2296},[782,783,784,785],[572,2298,703],{"className":2299},[615,785]," und ",[572,2302,2304,2335],{"className":2303},[575],[572,2305,2307],{"className":2306},[579],[581,2308,2309],{"xmlns":583},[585,2310,2311,2332],{},[588,2312,2313,2319,2321,2324,2326,2328,2330],{},[712,2314,2315,2317],{},[591,2316,706],{},[591,2318,1932],{},[633,2320,1820],{},[701,2322,2323],{},"4",[591,2325,716],{},[591,2327,1932],{},[633,2329,1958],{},[701,2331,1178],{},[595,2333,2334],{"encoding":597},"f_y = 4xy - 1",[572,2336,2338,2394,2419],{"className":2337,"ariaHidden":602},[601],[572,2339,2341,2345,2385,2388,2391],{"className":2340},[606],[572,2342],{"className":2343,"style":2344},[610],"height:0.9805em;vertical-align:-0.2861em;",[572,2346,2348,2351],{"className":2347},[615],[572,2349,706],{"className":2350,"style":948},[615,616],[572,2352,2354],{"className":2353},[802],[572,2355,2357,2377],{"className":2356},[760,761],[572,2358,2360,2374],{"className":2359},[765],[572,2361,2363],{"className":2362,"style":2181},[769],[572,2364,2365,2368],{"style":2184},[572,2366],{"className":2367,"style":1027},[777],[572,2369,2371],{"className":2370},[782,783,784,785],[572,2372,1932],{"className":2373,"style":1992},[615,616,785],[572,2375,832],{"className":2376},[831],[572,2378,2380],{"className":2379},[765],[572,2381,2383],{"className":2382,"style":1717},[769],[572,2384],{},[572,2386],{"className":2387,"style":1348},[657],[572,2389,1820],{"className":2390},[1352],[572,2392],{"className":2393,"style":1348},[657],[572,2395,2397,2401,2404,2407,2410,2413,2416],{"className":2396},[606],[572,2398],{"className":2399,"style":2400},[610],"height:0.8389em;vertical-align:-0.1944em;",[572,2402,2323],{"className":2403},[615],[572,2405,716],{"className":2406},[615,616],[572,2408,1932],{"className":2409,"style":1992},[615,616],[572,2411],{"className":2412,"style":658},[657],[572,2414,1958],{"className":2415},[662],[572,2417],{"className":2418,"style":658},[657],[572,2420,2422,2426],{"className":2421},[606],[572,2423],{"className":2424,"style":2425},[610],"height:0.6444em;",[572,2427,1178],{"className":2428},[615],". Daraus:",[572,2431,2433],{"className":2432},[1085],[572,2434,2436,2584],{"className":2435},[575],[572,2437,2439],{"className":2438},[579],[581,2440,2441],{"xmlns":583,"display":1094},[585,2442,2443,2581],{},[588,2444,2445,2451,2453,2455,2457,2459,2461,2463,2522,2524,2527,2533,2535,2537,2539,2541,2543,2545],{},[712,2446,2447,2449],{},[591,2448,1198],{},[591,2450,706],{},[633,2452,1104],{"stretchy":1103},[591,2454,716],{},[633,2456,1929],{"separator":602},[591,2458,1932],{},[633,2460,1126],{"stretchy":1103},[633,2462,1820],{},[588,2464,2465,2467,2520],{},[633,2466,1104],{"fence":602},[2468,2469,2473,2498],"mtable",{"rowspacing":2470,"columnalign":2471,"columnspacing":2472},"0.16em","center center","1em",[2474,2475,2476,2488],"mtr",{},[2477,2478,2479],"mtd",{},[1171,2480,2481],{"scriptlevel":1173,"displaystyle":1103},[588,2482,2483,2486],{},[701,2484,2485],{},"6",[591,2487,716],{},[2477,2489,2490],{},[1171,2491,2492],{"scriptlevel":1173,"displaystyle":1103},[588,2493,2494,2496],{},[701,2495,2323],{},[591,2497,1932],{},[2474,2499,2500,2510],{},[2477,2501,2502],{},[1171,2503,2504],{"scriptlevel":1173,"displaystyle":1103},[588,2505,2506,2508],{},[701,2507,2323],{},[591,2509,1932],{},[2477,2511,2512],{},[1171,2513,2514],{"scriptlevel":1173,"displaystyle":1103},[588,2515,2516,2518],{},[701,2517,2323],{},[591,2519,716],{},[633,2521,1126],{"fence":602},[633,2523,1929],{"separator":602},[657,2525],{"width":2526},"2em",[712,2528,2529,2531],{},[591,2530,1198],{},[591,2532,706],{},[633,2534,1104],{"stretchy":1103},[701,2536,1178],{},[633,2538,1929],{"separator":602},[701,2540,1178],{},[633,2542,1126],{"stretchy":1103},[633,2544,1820],{},[588,2546,2547,2549,2579],{},[633,2548,1104],{"fence":602},[2468,2550,2551,2565],{"rowspacing":2470,"columnalign":2471,"columnspacing":2472},[2474,2552,2553,2559],{},[2477,2554,2555],{},[1171,2556,2557],{"scriptlevel":1173,"displaystyle":1103},[701,2558,2485],{},[2477,2560,2561],{},[1171,2562,2563],{"scriptlevel":1173,"displaystyle":1103},[701,2564,2323],{},[2474,2566,2567,2573],{},[2477,2568,2569],{},[1171,2570,2571],{"scriptlevel":1173,"displaystyle":1103},[701,2572,2323],{},[2477,2574,2575],{},[1171,2576,2577],{"scriptlevel":1173,"displaystyle":1103},[701,2578,2323],{},[633,2580,1126],{"fence":602},[595,2582,2583],{"encoding":597},"H_f(x, y) = \\begin{pmatrix} 6x & 4y \\\\ 4y & 4x \\end{pmatrix}, \\qquad H_f(1, 1) = \\begin{pmatrix} 6 & 4 \\\\ 4 & 4 \\end{pmatrix}",[572,2585,2587,2661,2888],{"className":2586,"ariaHidden":602},[601],[572,2588,2590,2594,2634,2637,2640,2643,2646,2649,2652,2655,2658],{"className":2589},[606],[572,2591],{"className":2592,"style":2593},[610],"height:1.0361em;vertical-align:-0.2861em;",[572,2595,2597,2600],{"className":2596},[615],[572,2598,1198],{"className":2599,"style":1682},[615,616],[572,2601,2603],{"className":2602},[802],[572,2604,2606,2626],{"className":2605},[760,761],[572,2607,2609,2623],{"className":2608},[765],[572,2610,2612],{"className":2611,"style":1695},[769],[572,2613,2614,2617],{"style":1698},[572,2615],{"className":2616,"style":1027},[777],[572,2618,2620],{"className":2619},[782,783,784,785],[572,2621,706],{"className":2622,"style":948},[615,616,785],[572,2624,832],{"className":2625},[831],[572,2627,2629],{"className":2628},[765],[572,2630,2632],{"className":2631,"style":1717},[769],[572,2633],{},[572,2635,1104],{"className":2636},[752],[572,2638,716],{"className":2639},[615,616],[572,2641,1929],{"className":2642},[1985],[572,2644],{"className":2645,"style":1609},[657],[572,2647,1932],{"className":2648,"style":1992},[615,616],[572,2650,1126],{"className":2651},[964],[572,2653],{"className":2654,"style":1348},[657],[572,2656,1820],{"className":2657},[1352],[572,2659],{"className":2660,"style":1348},[657],[572,2662,2664,2668,2808,2811,2814,2818,2821,2861,2864,2867,2870,2873,2876,2879,2882,2885],{"className":2663},[606],[572,2665],{"className":2666,"style":2667},[610],"height:2.4em;vertical-align:-0.95em;",[572,2669,2672,2681,2802],{"className":2670},[2671],"minner",[572,2673,2677],{"className":2674,"style":2676},[752,2675],"delimcenter","top:0em;",[572,2678,1104],{"className":2679},[2680,784],"delimsizing",[572,2682,2684],{"className":2683},[615],[572,2685,2687,2743,2748,2751],{"className":2686},[2468],[572,2688,2691],{"className":2689},[2690],"col-align-c",[572,2692,2694,2734],{"className":2693},[760,761],[572,2695,2697,2731],{"className":2696},[765],[572,2698,2701,2716],{"className":2699,"style":2700},[769],"height:1.45em;",[572,2702,2704,2707],{"style":2703},"top:-3.61em;",[572,2705],{"className":2706,"style":778},[777],[572,2708,2710,2713],{"className":2709},[615],[572,2711,2485],{"className":2712},[615],[572,2714,716],{"className":2715},[615,616],[572,2717,2719,2722],{"style":2718},"top:-2.41em;",[572,2720],{"className":2721,"style":778},[777],[572,2723,2725,2728],{"className":2724},[615],[572,2726,2323],{"className":2727},[615],[572,2729,1932],{"className":2730,"style":1992},[615,616],[572,2732,832],{"className":2733},[831],[572,2735,2737],{"className":2736},[765],[572,2738,2741],{"className":2739,"style":2740},[769],"height:0.95em;",[572,2742],{},[572,2744],{"className":2745,"style":2747},[2746],"arraycolsep","width:0.5em;",[572,2749],{"className":2750,"style":2747},[2746],[572,2752,2754],{"className":2753},[2690],[572,2755,2757,2794],{"className":2756},[760,761],[572,2758,2760,2791],{"className":2759},[765],[572,2761,2763,2777],{"className":2762,"style":2700},[769],[572,2764,2765,2768],{"style":2703},[572,2766],{"className":2767,"style":778},[777],[572,2769,2771,2774],{"className":2770},[615],[572,2772,2323],{"className":2773},[615],[572,2775,1932],{"className":2776,"style":1992},[615,616],[572,2778,2779,2782],{"style":2718},[572,2780],{"className":2781,"style":778},[777],[572,2783,2785,2788],{"className":2784},[615],[572,2786,2323],{"className":2787},[615],[572,2789,716],{"className":2790},[615,616],[572,2792,832],{"className":2793},[831],[572,2795,2797],{"className":2796},[765],[572,2798,2800],{"className":2799,"style":2740},[769],[572,2801],{},[572,2803,2805],{"className":2804,"style":2676},[964,2675],[572,2806,1126],{"className":2807},[2680,784],[572,2809],{"className":2810,"style":1609},[657],[572,2812,1929],{"className":2813},[1985],[572,2815],{"className":2816,"style":2817},[657],"margin-right:2em;",[572,2819],{"className":2820,"style":1609},[657],[572,2822,2824,2827],{"className":2823},[615],[572,2825,1198],{"className":2826,"style":1682},[615,616],[572,2828,2830],{"className":2829},[802],[572,2831,2833,2853],{"className":2832},[760,761],[572,2834,2836,2850],{"className":2835},[765],[572,2837,2839],{"className":2838,"style":1695},[769],[572,2840,2841,2844],{"style":1698},[572,2842],{"className":2843,"style":1027},[777],[572,2845,2847],{"className":2846},[782,783,784,785],[572,2848,706],{"className":2849,"style":948},[615,616,785],[572,2851,832],{"className":2852},[831],[572,2854,2856],{"className":2855},[765],[572,2857,2859],{"className":2858,"style":1717},[769],[572,2860],{},[572,2862,1104],{"className":2863},[752],[572,2865,1178],{"className":2866},[615],[572,2868,1929],{"className":2869},[1985],[572,2871],{"className":2872,"style":1609},[657],[572,2874,1178],{"className":2875},[615],[572,2877,1126],{"className":2878},[964],[572,2880],{"className":2881,"style":1348},[657],[572,2883,1820],{"className":2884},[1352],[572,2886],{"className":2887,"style":1348},[657],[572,2889,2891,2894],{"className":2890},[606],[572,2892],{"className":2893,"style":2667},[610],[572,2895,2897,2903,3005],{"className":2896},[2671],[572,2898,2900],{"className":2899,"style":2676},[752,2675],[572,2901,1104],{"className":2902},[2680,784],[572,2904,2906],{"className":2905},[615],[572,2907,2909,2954,2957,2960],{"className":2908},[2468],[572,2910,2912],{"className":2911},[2690],[572,2913,2915,2946],{"className":2914},[760,761],[572,2916,2918,2943],{"className":2917},[765],[572,2919,2921,2932],{"className":2920,"style":2700},[769],[572,2922,2923,2926],{"style":2703},[572,2924],{"className":2925,"style":778},[777],[572,2927,2929],{"className":2928},[615],[572,2930,2485],{"className":2931},[615],[572,2933,2934,2937],{"style":2718},[572,2935],{"className":2936,"style":778},[777],[572,2938,2940],{"className":2939},[615],[572,2941,2323],{"className":2942},[615],[572,2944,832],{"className":2945},[831],[572,2947,2949],{"className":2948},[765],[572,2950,2952],{"className":2951,"style":2740},[769],[572,2953],{},[572,2955],{"className":2956,"style":2747},[2746],[572,2958],{"className":2959,"style":2747},[2746],[572,2961,2963],{"className":2962},[2690],[572,2964,2966,2997],{"className":2965},[760,761],[572,2967,2969,2994],{"className":2968},[765],[572,2970,2972,2983],{"className":2971,"style":2700},[769],[572,2973,2974,2977],{"style":2703},[572,2975],{"className":2976,"style":778},[777],[572,2978,2980],{"className":2979},[615],[572,2981,2323],{"className":2982},[615],[572,2984,2985,2988],{"style":2718},[572,2986],{"className":2987,"style":778},[777],[572,2989,2991],{"className":2990},[615],[572,2992,2323],{"className":2993},[615],[572,2995,832],{"className":2996},[831],[572,2998,3000],{"className":2999},[765],[572,3001,3003],{"className":3002,"style":2740},[769],[572,3004],{},[572,3006,3008],{"className":3007,"style":2676},[964,2675],[572,3009,1126],{"className":3010},[2680,784],[564,3012,3013,3014,3066,3067,2300,3120,3372,3373,3488,3489,2300,3529,3570],{},"Bei einer ",[572,3015,3017,3035],{"className":3016},[575],[572,3018,3020],{"className":3019},[579],[581,3021,3022],{"xmlns":583},[585,3023,3024,3032],{},[588,3025,3026,3028,3030],{},[701,3027,703],{},[633,3029,635],{},[701,3031,703],{},[595,3033,3034],{"encoding":597},"2 \\times 2",[572,3036,3038,3057],{"className":3037,"ariaHidden":602},[601],[572,3039,3041,3045,3048,3051,3054],{"className":3040},[606],[572,3042],{"className":3043,"style":3044},[610],"height:0.7278em;vertical-align:-0.0833em;",[572,3046,703],{"className":3047},[615],[572,3049],{"className":3050,"style":658},[657],[572,3052,635],{"className":3053},[662],[572,3055],{"className":3056,"style":658},[657],[572,3058,3060,3063],{"className":3059},[606],[572,3061],{"className":3062,"style":2425},[610],[572,3064,703],{"className":3065},[615],"-Matrix reicht für positive Definitheit: Der Eintrag links oben ist positiv und die Determinante ist positiv. Hier ist ",[572,3068,3070,3089],{"className":3069},[575],[572,3071,3073],{"className":3072},[579],[581,3074,3075],{"xmlns":583},[585,3076,3077,3086],{},[588,3078,3079,3081,3084],{},[701,3080,2485],{},[633,3082,3083],{},">",[701,3085,1173],{},[595,3087,3088],{"encoding":597},"6 > 0",[572,3090,3092,3111],{"className":3091,"ariaHidden":602},[601],[572,3093,3095,3099,3102,3105,3108],{"className":3094},[606],[572,3096],{"className":3097,"style":3098},[610],"height:0.6835em;vertical-align:-0.0391em;",[572,3100,2485],{"className":3101},[615],[572,3103],{"className":3104,"style":1348},[657],[572,3106,3083],{"className":3107},[1352],[572,3109],{"className":3110,"style":1348},[657],[572,3112,3114,3117],{"className":3113},[606],[572,3115],{"className":3116,"style":2425},[610],[572,3118,1173],{"className":3119},[615],[572,3121,3123,3182],{"className":3122},[575],[572,3124,3126],{"className":3125},[579],[581,3127,3128],{"xmlns":583},[585,3129,3130,3179],{},[588,3131,3132,3135,3138,3144,3146,3148,3150,3152,3154,3156,3158,3160,3162,3164,3170,3172,3175,3177],{},[591,3133,3134],{},"det",[633,3136,3137],{},"⁡",[712,3139,3140,3142],{},[591,3141,1198],{},[591,3143,706],{},[633,3145,1104],{"stretchy":1103},[701,3147,1178],{},[633,3149,1929],{"separator":602},[701,3151,1178],{},[633,3153,1126],{"stretchy":1103},[633,3155,1820],{},[701,3157,2485],{},[633,3159,1161],{},[701,3161,2323],{},[633,3163,1958],{},[694,3165,3166,3168],{},[701,3167,2323],{},[701,3169,703],{},[633,3171,1820],{},[701,3173,3174],{},"8",[633,3176,3083],{},[701,3178,1173],{},[595,3180,3181],{"encoding":597},"\\det H_f(1, 1) = 6 \\cdot 4 - 4^2 = 8 > 0",[572,3183,3185,3265,3283,3301,3345,3363],{"className":3184,"ariaHidden":602},[601],[572,3186,3188,3191,3195,3198,3238,3241,3244,3247,3250,3253,3256,3259,3262],{"className":3187},[606],[572,3189],{"className":3190,"style":2593},[610],[572,3192,3134],{"className":3193},[3194],"mop",[572,3196],{"className":3197,"style":1609},[657],[572,3199,3201,3204],{"className":3200},[615],[572,3202,1198],{"className":3203,"style":1682},[615,616],[572,3205,3207],{"className":3206},[802],[572,3208,3210,3230],{"className":3209},[760,761],[572,3211,3213,3227],{"className":3212},[765],[572,3214,3216],{"className":3215,"style":1695},[769],[572,3217,3218,3221],{"style":1698},[572,3219],{"className":3220,"style":1027},[777],[572,3222,3224],{"className":3223},[782,783,784,785],[572,3225,706],{"className":3226,"style":948},[615,616,785],[572,3228,832],{"className":3229},[831],[572,3231,3233],{"className":3232},[765],[572,3234,3236],{"className":3235,"style":1717},[769],[572,3237],{},[572,3239,1104],{"className":3240},[752],[572,3242,1178],{"className":3243},[615],[572,3245,1929],{"className":3246},[1985],[572,3248],{"className":3249,"style":1609},[657],[572,3251,1178],{"className":3252},[615],[572,3254,1126],{"className":3255},[964],[572,3257],{"className":3258,"style":1348},[657],[572,3260,1820],{"className":3261},[1352],[572,3263],{"className":3264,"style":1348},[657],[572,3266,3268,3271,3274,3277,3280],{"className":3267},[606],[572,3269],{"className":3270,"style":2425},[610],[572,3272,2485],{"className":3273},[615],[572,3275],{"className":3276,"style":658},[657],[572,3278,1161],{"className":3279},[662],[572,3281],{"className":3282,"style":658},[657],[572,3284,3286,3289,3292,3295,3298],{"className":3285},[606],[572,3287],{"className":3288,"style":3044},[610],[572,3290,2323],{"className":3291},[615],[572,3293],{"className":3294,"style":658},[657],[572,3296,1958],{"className":3297},[662],[572,3299],{"className":3300,"style":658},[657],[572,3302,3304,3307,3336,3339,3342],{"className":3303},[606],[572,3305],{"className":3306,"style":2030},[610],[572,3308,3310,3313],{"className":3309},[615],[572,3311,2323],{"className":3312},[615],[572,3314,3316],{"className":3315},[802],[572,3317,3319],{"className":3318},[760],[572,3320,3322],{"className":3321},[765],[572,3323,3325],{"className":3324,"style":2030},[769],[572,3326,3327,3330],{"style":1023},[572,3328],{"className":3329,"style":1027},[777],[572,3331,3333],{"className":3332},[782,783,784,785],[572,3334,703],{"className":3335},[615,785],[572,3337],{"className":3338,"style":1348},[657],[572,3340,1820],{"className":3341},[1352],[572,3343],{"className":3344,"style":1348},[657],[572,3346,3348,3351,3354,3357,3360],{"className":3347},[606],[572,3349],{"className":3350,"style":3098},[610],[572,3352,3174],{"className":3353},[615],[572,3355],{"className":3356,"style":1348},[657],[572,3358,3083],{"className":3359},[1352],[572,3361],{"className":3362,"style":1348},[657],[572,3364,3366,3369],{"className":3365},[606],[572,3367],{"className":3368,"style":2425},[610],[572,3370,1173],{"className":3371},[615],", die Matrix ist also positiv definit. Die Eigenwerte ",[572,3374,3376,3400],{"className":3375},[575],[572,3377,3379],{"className":3378},[579],[581,3380,3381],{"xmlns":583},[585,3382,3383,3397],{},[588,3384,3385,3388,3391],{},[701,3386,3387],{},"5",[633,3389,3390],{},"±",[3392,3393,3394],"msqrt",{},[701,3395,3396],{},"17",[595,3398,3399],{"encoding":597},"5 \\pm \\sqrt{17}",[572,3401,3403,3421],{"className":3402,"ariaHidden":602},[601],[572,3404,3406,3409,3412,3415,3418],{"className":3405},[606],[572,3407],{"className":3408,"style":3044},[610],[572,3410,3387],{"className":3411},[615],[572,3413],{"className":3414,"style":658},[657],[572,3416,3390],{"className":3417},[662],[572,3419],{"className":3420,"style":658},[657],[572,3422,3424,3428],{"className":3423},[606],[572,3425],{"className":3426,"style":3427},[610],"height:1.04em;vertical-align:-0.1328em;",[572,3429,3432],{"className":3430},[615,3431],"sqrt",[572,3433,3435,3479],{"className":3434},[760,761],[572,3436,3438,3476],{"className":3437},[765],[572,3439,3442,3456],{"className":3440,"style":3441},[769],"height:0.9072em;",[572,3443,3446,3449],{"className":3444,"style":1254},[3445],"svg-align",[572,3447],{"className":3448,"style":778},[777],[572,3450,3453],{"className":3451,"style":3452},[615],"padding-left:0.833em;",[572,3454,3396],{"className":3455},[615],[572,3457,3459,3462],{"style":3458},"top:-2.8672em;",[572,3460],{"className":3461,"style":778},[777],[572,3463,3467],{"className":3464,"style":3466},[3465],"hide-tail","min-width:0.853em;height:1.08em;",[1277,3468,3473],{"xmlns":1279,"width":3469,"height":3470,"viewBox":3471,"preserveAspectRatio":3472},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[1286,3474],{"d":3475},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[572,3477,832],{"className":3478},[831],[572,3480,3482],{"className":3481},[765],[572,3483,3486],{"className":3484,"style":3485},[769],"height:0.1328em;",[572,3487],{},", also etwa ",[572,3490,3492,3507],{"className":3491},[575],[572,3493,3495],{"className":3494},[579],[581,3496,3497],{"xmlns":583},[585,3498,3499,3504],{},[588,3500,3501],{},[701,3502,3503],{},"0,88",[595,3505,3506],{"encoding":597},"0{,}88",[572,3508,3510],{"className":3509,"ariaHidden":602},[601],[572,3511,3513,3516,3519,3525],{"className":3512},[606],[572,3514],{"className":3515,"style":2400},[610],[572,3517,1173],{"className":3518},[615],[572,3520,3522],{"className":3521},[615],[572,3523,1929],{"className":3524},[1985],[572,3526,3528],{"className":3527},[615],"88",[572,3530,3532,3547],{"className":3531},[575],[572,3533,3535],{"className":3534},[579],[581,3536,3537],{"xmlns":583},[585,3538,3539,3544],{},[588,3540,3541],{},[701,3542,3543],{},"9,12",[595,3545,3546],{"encoding":597},"9{,}12",[572,3548,3550],{"className":3549,"ariaHidden":602},[601],[572,3551,3553,3556,3560,3566],{"className":3552},[606],[572,3554],{"className":3555,"style":2400},[610],[572,3557,3559],{"className":3558},[615],"9",[572,3561,3563],{"className":3562},[615],[572,3564,1929],{"className":3565},[1985],[572,3567,3569],{"className":3568},[615],"12",", sind beide positiv.",[564,3572,3573,3574,3626,3627,3810],{},"Der Punkt ",[572,3575,3577,3599],{"className":3576},[575],[572,3578,3580],{"className":3579},[579],[581,3581,3582],{"xmlns":583},[585,3583,3584,3596],{},[588,3585,3586,3588,3590,3592,3594],{},[633,3587,1104],{"stretchy":1103},[701,3589,1178],{},[633,3591,1929],{"separator":602},[701,3593,1178],{},[633,3595,1126],{"stretchy":1103},[595,3597,3598],{"encoding":597},"(1, 1)",[572,3600,3602],{"className":3601,"ariaHidden":602},[601],[572,3603,3605,3608,3611,3614,3617,3620,3623],{"className":3604},[606],[572,3606],{"className":3607,"style":1231},[610],[572,3609,1104],{"className":3610},[752],[572,3612,1178],{"className":3613},[615],[572,3615,1929],{"className":3616},[1985],[572,3618],{"className":3619,"style":1609},[657],[572,3621,1178],{"className":3622},[615],[572,3624,1126],{"className":3625},[964]," ist allerdings kein kritischer Punkt, weil ",[572,3628,3630,3667],{"className":3629},[575],[572,3631,3633],{"className":3632},[579],[581,3634,3635],{"xmlns":583},[585,3636,3637,3664],{},[588,3638,3639,3645,3647,3649,3651,3653,3655,3657,3659,3662],{},[712,3640,3641,3643],{},[591,3642,706],{},[591,3644,716],{},[633,3646,1104],{"stretchy":1103},[701,3648,1178],{},[633,3650,1929],{"separator":602},[701,3652,1178],{},[633,3654,1126],{"stretchy":1103},[633,3656,1820],{},[701,3658,3387],{},[633,3660,3661],{"mathvariant":698},"≠",[701,3663,1173],{},[595,3665,3666],{"encoding":597},"f_x(1, 1) = 5 \\neq 0",[572,3668,3670,3743,3801],{"className":3669,"ariaHidden":602},[601],[572,3671,3673,3676,3716,3719,3722,3725,3728,3731,3734,3737,3740],{"className":3672},[606],[572,3674],{"className":3675,"style":1231},[610],[572,3677,3679,3682],{"className":3678},[615],[572,3680,706],{"className":3681,"style":948},[615,616],[572,3683,3685],{"className":3684},[802],[572,3686,3688,3708],{"className":3687},[760,761],[572,3689,3691,3705],{"className":3690},[765],[572,3692,3694],{"className":3693,"style":2181},[769],[572,3695,3696,3699],{"style":2184},[572,3697],{"className":3698,"style":1027},[777],[572,3700,3702],{"className":3701},[782,783,784,785],[572,3703,716],{"className":3704},[615,616,785],[572,3706,832],{"className":3707},[831],[572,3709,3711],{"className":3710},[765],[572,3712,3714],{"className":3713,"style":2203},[769],[572,3715],{},[572,3717,1104],{"className":3718},[752],[572,3720,1178],{"className":3721},[615],[572,3723,1929],{"className":3724},[1985],[572,3726],{"className":3727,"style":1609},[657],[572,3729,1178],{"className":3730},[615],[572,3732,1126],{"className":3733},[964],[572,3735],{"className":3736,"style":1348},[657],[572,3738,1820],{"className":3739},[1352],[572,3741],{"className":3742,"style":1348},[657],[572,3744,3746,3749,3752,3755,3798],{"className":3745},[606],[572,3747],{"className":3748,"style":1067},[610],[572,3750,3387],{"className":3751},[615],[572,3753],{"className":3754,"style":1348},[657],[572,3756,3758,3791,3795],{"className":3757},[1352],[572,3759,3761],{"className":3760},[1352],[572,3762,3765],{"className":3763},[615,3764],"vbox",[572,3766,3769],{"className":3767},[3768],"thinbox",[572,3770,3773,3776,3787],{"className":3771},[3772],"rlap",[572,3774],{"className":3775,"style":1067},[610],[572,3777,3780],{"className":3778},[3779],"inner",[572,3781,3783],{"className":3782},[615],[572,3784,3786],{"className":3785},[1352],"",[572,3788],{"className":3789},[3790],"fix",[572,3792],{"className":3793},[657,3794],"nobreak",[572,3796,1820],{"className":3797},[1352],[572,3799],{"className":3800,"style":1348},[657],[572,3802,3804,3807],{"className":3803},[606],[572,3805],{"className":3806,"style":2425},[610],[572,3808,1173],{"className":3809},[615],". Die Hesse-Matrix sagt dort nur, dass der Graph wie eine nach oben geöffnete Schale gekrümmt ist.",[559,3812,3814],{"id":3813},"typische-fehler","Typische Fehler",[3816,3817,3818,3822,3917],"ul",{},[3819,3820,3821],"li",{},"Die Hesse-Matrix mit der Jacobi-Matrix verwechseln. Die Jacobi-Matrix enthält erste Ableitungen, die Hesse-Matrix zweite.",[3819,3823,3824,3825,3916],{},"Die Definitheit an einem Punkt prüfen, der nicht kritisch ist, und daraus ein Extremum folgern. Zuerst muss ",[572,3826,3828,3851],{"className":3827},[575],[572,3829,3831],{"className":3830},[579],[581,3832,3833],{"xmlns":583},[585,3834,3835,3849],{},[588,3836,3837,3839,3841,3843],{},[591,3838,1146],{"mathvariant":698},[591,3840,706],{},[633,3842,1820],{},[1106,3844,3845,3847],{"accent":602},[701,3846,1173],{},[633,3848,1113],{},[595,3850,1829],{"encoding":597},[572,3852,3854,3875],{"className":3853,"ariaHidden":602},[601],[572,3855,3857,3860,3863,3866,3869,3872],{"className":3856},[606],[572,3858],{"className":3859,"style":1067},[610],[572,3861,1146],{"className":3862},[615],[572,3864,706],{"className":3865,"style":948},[615,616],[572,3867],{"className":3868,"style":1348},[657],[572,3870,1820],{"className":3871},[1352],[572,3873],{"className":3874,"style":1348},[657],[572,3876,3878,3881],{"className":3877},[606],[572,3879],{"className":3880,"style":1860},[610],[572,3882,3884],{"className":3883},[615,1241],[572,3885,3887],{"className":3886},[760],[572,3888,3890],{"className":3889},[765],[572,3891,3893,3901],{"className":3892,"style":1860},[769],[572,3894,3895,3898],{"style":1254},[572,3896],{"className":3897,"style":778},[777],[572,3899,1173],{"className":3900},[615],[572,3902,3903,3906],{"style":1883},[572,3904],{"className":3905,"style":778},[777],[572,3907,3909],{"className":3908,"style":1270},[1269],[572,3910,3912],{"className":3911,"style":1275},[1274],[1277,3913,3914],{"xmlns":1279,"width":1280,"height":1281,"style":1282,"viewBox":1283,"preserveAspectRatio":1284},[1286,3915],{"d":1288}," gelten.",[3819,3918,3919,3920,4095,4096,4130],{},"Nur die Diagonale anschauen. Positive Diagonaleinträge allein reichen nicht, z.B. ist ",[572,3921,3923,3969],{"className":3922},[575],[572,3924,3926],{"className":3925},[579],[581,3927,3928],{"xmlns":583},[585,3929,3930,3966],{},[588,3931,3932,3934,3964],{},[633,3933,1104],{"fence":602},[2468,3935,3936,3950],{"rowspacing":2470,"columnalign":2471,"columnspacing":2472},[2474,3937,3938,3944],{},[2477,3939,3940],{},[1171,3941,3942],{"scriptlevel":1173,"displaystyle":1103},[701,3943,1178],{},[2477,3945,3946],{},[1171,3947,3948],{"scriptlevel":1173,"displaystyle":1103},[701,3949,703],{},[2474,3951,3952,3958],{},[2477,3953,3954],{},[1171,3955,3956],{"scriptlevel":1173,"displaystyle":1103},[701,3957,703],{},[2477,3959,3960],{},[1171,3961,3962],{"scriptlevel":1173,"displaystyle":1103},[701,3963,1178],{},[633,3965,1126],{"fence":602},[595,3967,3968],{"encoding":597},"\\begin{pmatrix} 1 & 2 \\\\ 2 & 1 \\end{pmatrix}",[572,3970,3972],{"className":3971,"ariaHidden":602},[601],[572,3973,3975,3978],{"className":3974},[606],[572,3976],{"className":3977,"style":2667},[610],[572,3979,3981,3987,4089],{"className":3980},[2671],[572,3982,3984],{"className":3983,"style":2676},[752,2675],[572,3985,1104],{"className":3986},[2680,784],[572,3988,3990],{"className":3989},[615],[572,3991,3993,4038,4041,4044],{"className":3992},[2468],[572,3994,3996],{"className":3995},[2690],[572,3997,3999,4030],{"className":3998},[760,761],[572,4000,4002,4027],{"className":4001},[765],[572,4003,4005,4016],{"className":4004,"style":2700},[769],[572,4006,4007,4010],{"style":2703},[572,4008],{"className":4009,"style":778},[777],[572,4011,4013],{"className":4012},[615],[572,4014,1178],{"className":4015},[615],[572,4017,4018,4021],{"style":2718},[572,4019],{"className":4020,"style":778},[777],[572,4022,4024],{"className":4023},[615],[572,4025,703],{"className":4026},[615],[572,4028,832],{"className":4029},[831],[572,4031,4033],{"className":4032},[765],[572,4034,4036],{"className":4035,"style":2740},[769],[572,4037],{},[572,4039],{"className":4040,"style":2747},[2746],[572,4042],{"className":4043,"style":2747},[2746],[572,4045,4047],{"className":4046},[2690],[572,4048,4050,4081],{"className":4049},[760,761],[572,4051,4053,4078],{"className":4052},[765],[572,4054,4056,4067],{"className":4055,"style":2700},[769],[572,4057,4058,4061],{"style":2703},[572,4059],{"className":4060,"style":778},[777],[572,4062,4064],{"className":4063},[615],[572,4065,703],{"className":4066},[615],[572,4068,4069,4072],{"style":2718},[572,4070],{"className":4071,"style":778},[777],[572,4073,4075],{"className":4074},[615],[572,4076,1178],{"className":4077},[615],[572,4079,832],{"className":4080},[831],[572,4082,4084],{"className":4083},[765],[572,4085,4087],{"className":4086,"style":2740},[769],[572,4088],{},[572,4090,4092],{"className":4091,"style":2676},[964,2675],[572,4093,1126],{"className":4094},[2680,784]," mit Determinante ",[572,4097,4099,4115],{"className":4098},[575],[572,4100,4102],{"className":4101},[579],[581,4103,4104],{"xmlns":583},[585,4105,4106,4112],{},[588,4107,4108,4110],{},[633,4109,1958],{},[701,4111,1943],{},[595,4113,4114],{"encoding":597},"-3",[572,4116,4118],{"className":4117,"ariaHidden":602},[601],[572,4119,4121,4124,4127],{"className":4120},[606],[572,4122],{"className":4123,"style":3044},[610],[572,4125,1958],{"className":4126},[615],[572,4128,1943],{"className":4129},[615]," indefinit.",{"title":4132,"searchDepth":4133,"depth":4133,"links":4134},"",2,[4135,4136,4137],{"id":561,"depth":4133,"text":562},{"id":1904,"depth":4133,"text":1905},{"id":3813,"depth":4133,"text":3814},[4139,4141,4143,4145,4147,4149,4151,4153,4156,4159,4161],{"lhs":4140,"rhs":1173},"round(((x + 1e-4)^3 + 2*(x + 1e-4)*y^2 - y - 2*(x^3 + 2*x*y^2 - y) + (x - 1e-4)^3 + 2*(x - 1e-4)*y^2 - y) \u002F 1e-8 - 6*x, 4)",{"lhs":4142,"rhs":1173},"round((x^3 + 2*x*(y + 1e-4)^2 - (y + 1e-4) - 2*(x^3 + 2*x*y^2 - y) + x^3 + 2*x*(y - 1e-4)^2 - (y - 1e-4)) \u002F 1e-8 - 4*x, 4)",{"lhs":4144,"rhs":1173},"round((2*(x + 1e-4)*(y + 1e-4)^2 - 2*(x + 1e-4)*(y - 1e-4)^2 - 2*(x - 1e-4)*(y + 1e-4)^2 + 2*(x - 1e-4)*(y - 1e-4)^2) \u002F 4e-8 - 4*y, 4)",{"lhs":4146,"rhs":3174},"det([[6, 4], [4, 4]])",{"lhs":4148,"rhs":3174},"6*4 - 4^2",{"lhs":4150,"rhs":1173},"det([[6 - (5 + sqrt(17)), 4], [4, 4 - (5 + sqrt(17))]])",{"lhs":4152,"rhs":1173},"det([[6 - (5 - sqrt(17)), 4], [4, 4 - (5 - sqrt(17))]])",{"lhs":4154,"rhs":4155},"round(5 - sqrt(17), 2)","0.88",{"lhs":4157,"rhs":4158},"round(5 + sqrt(17), 2)","9.12",{"lhs":4160,"rhs":3387},"3*1^2 + 2*1^2",{"lhs":4162,"rhs":4114},"det([[1, 2], [2, 1]])","f \\text{ zweimal partiell differenzierbar; ist } f \\text{ zweimal stetig differenzierbar, gilt } f_{xy} = f_{yx}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">zweimal partiell differenzierbar; ist\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">zweimal stetig differenzierbar, gilt\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f_{xy} = f_{yx}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","hessian-matrix","H_f(x, y) = \\begin{pmatrix} f_{xx}(x, y) & f_{xy}(x, y) \\\\ f_{yx}(x, y) & f_{yy}(x, y) \\end{pmatrix}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>H\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">H_f(x, y) = \\begin{pmatrix} f_{xx}(x, y) & f_{xy}(x, y) \\\\ f_{yx}(x, y) & f_{yy}(x, y) \\end{pmatrix}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.4em;vertical-align:-0.95em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em;\">\u003Cspan style=\"top:-3.61em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">xx\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:0.5em;\">\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:0.5em;\">\u003C\u002Fspan>\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em;\">\u003Cspan style=\"top:-3.61em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fhesse-matrix",[],[4176,4177,4178,4179],"partial-derivative","multivariable-extrema","characteristic-polynomial","symmetric-matrix-definiteness",{"title":548,"description":4132},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fhesse-matrix","Die Hesse-Matrix enthält alle zweiten partiellen Ableitungen und beschreibt die Krümmung einer Funktion mehrerer Variablen.",[4184,4185,4186,4187,4188],"hesse-matrix","zweite ableitung","partielle ableitung","definitheit","extremstellen",[4190,4194,4198],{"symbol":4191,"meaning":4192,"symbolHtml":4193},"H_f","Hesse-Matrix der Funktion f","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>H\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">H_f\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4195,"meaning":4196,"symbolHtml":4197},"f_{xx}, f_{yy}","zweite partielle Ableitungen nach x bzw. nach y","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f_{xx}, f_{yy}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">xx\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4199,"meaning":4200,"symbolHtml":4201},"f_{xy}, f_{yx}","gemischte zweite partielle Ableitungen (erst nach x, dann nach y bzw. umgekehrt)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f_{xy}, f_{yx}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","ej-38eqYkbZtKiT0K3Pxq5O8jwMbaiOGpJAEmQxxTcQ",[4204,4207,4210,4213],{"stem":4205,"title":4206,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fdefinitheit-symmetrischer-matrizen","Definitheit symmetrischer Matrizen",{"stem":4208,"title":4209,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Feigenwerte-charakteristisches-polynom","Eigenwerte und charakteristisches Polynom",{"stem":4211,"title":4212,"category":368},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fextremstellen-zwei-variablen","Extremstellen bei zwei Variablen",{"stem":4214,"title":4215,"category":368},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fpartielle-ableitung","Partielle Ableitung",1793824266927]