[{"data":1,"prerenderedAt":4761},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fkugelkoordinaten":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fkugelkoordinaten":4748},[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":556,"category":368,"checks":4675,"conditions":4703,"conditionsHtml":4704,"deprecated":4705,"description":4669,"extension":4706,"figure":4705,"figureCaption":4705,"figureHtml":4705,"formulaId":4707,"latex":4708,"latexHtml":4709,"level":4710,"meta":4711,"navigation":4712,"path":4713,"redirectFrom":4714,"related":4715,"seo":4720,"stem":4721,"subject":581,"summary":4722,"tags":4723,"variables":4729,"__hash__":4747},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fkugelkoordinaten.md","Kugelkoordinaten",[550,551,552,553,554,555],"Kugelkoordinaten umrechnen","Volumenelement Kugelkoordinaten","räumliche Polarkoordinaten","dV = r² sin theta","Integral in Kugelkoordinaten","sphärische Koordinaten",{"type":557,"value":558,"toc":4668},"minimark",[559,564,895,1394,1669,1802,1806,2103,2779,2962,3611,4176,4180],[560,561,563],"h2",{"id":562},"erklärung","Erklärung",[565,566,567,570,571,618,619,622,623,654,655,685,686,745,746,799,800,622,803,834,835,864,865,894],"p",{},[568,569,548],"strong",{}," beschreiben einen Punkt im Raum durch seinen Abstand ",[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",{},"r",[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,"style":617},[615,616],"mord","mathnormal","margin-right:0.0278em;"," vom Ursprung und zwei Winkel. Der ",[568,620,621],{},"Polarwinkel"," ",[572,624,626,641],{"className":625},[575],[572,627,629],{"className":628},[579],[581,630,631],{"xmlns":583},[585,632,633,638],{},[588,634,635],{},[591,636,637],{},"θ",[595,639,640],{"encoding":597},"\\theta",[572,642,644],{"className":643,"ariaHidden":602},[601],[572,645,647,651],{"className":646},[606],[572,648],{"className":649,"style":650},[610],"height:0.6944em;",[572,652,637],{"className":653,"style":617},[615,616]," wird von der positiven ",[572,656,658,672],{"className":657},[575],[572,659,661],{"className":660},[579],[581,662,663],{"xmlns":583},[585,664,665,670],{},[588,666,667],{},[591,668,669],{},"z",[595,671,669],{"encoding":597},[572,673,675],{"className":674,"ariaHidden":602},[601],[572,676,678,681],{"className":677},[606],[572,679],{"className":680,"style":611},[610],[572,682,669],{"className":683,"style":684},[615,616],"margin-right:0.044em;","-Achse aus gemessen (",[572,687,689,711],{"className":688},[575],[572,690,692],{"className":691},[579],[581,693,694],{"xmlns":583},[585,695,696,708],{},[588,697,698,700,704],{},[591,699,637],{},[701,702,703],"mo",{},"=",[705,706,707],"mn",{},"0",[595,709,710],{"encoding":597},"\\theta = 0",[572,712,714,735],{"className":713,"ariaHidden":602},[601],[572,715,717,720,723,728,732],{"className":716},[606],[572,718],{"className":719,"style":650},[610],[572,721,637],{"className":722,"style":617},[615,616],[572,724],{"className":725,"style":727},[726],"mspace","margin-right:0.2778em;",[572,729,703],{"className":730},[731],"mrel",[572,733],{"className":734,"style":727},[726],[572,736,738,742],{"className":737},[606],[572,739],{"className":740,"style":741},[610],"height:0.6444em;",[572,743,707],{"className":744},[615]," am Nordpol, ",[572,747,749,768],{"className":748},[575],[572,750,752],{"className":751},[579],[581,753,754],{"xmlns":583},[585,755,756,765],{},[588,757,758,760,762],{},[591,759,637],{},[701,761,703],{},[591,763,764],{},"π",[595,766,767],{"encoding":597},"\\theta = \\pi",[572,769,771,789],{"className":770,"ariaHidden":602},[601],[572,772,774,777,780,783,786],{"className":773},[606],[572,775],{"className":776,"style":650},[610],[572,778,637],{"className":779,"style":617},[615,616],[572,781],{"className":782,"style":727},[726],[572,784,703],{"className":785},[731],[572,787],{"className":788,"style":727},[726],[572,790,792,795],{"className":791},[606],[572,793],{"className":794,"style":611},[610],[572,796,764],{"className":797,"style":798},[615,616],"margin-right:0.0359em;"," am Südpol), der ",[568,801,802],{},"Azimutwinkel",[572,804,806,821],{"className":805},[575],[572,807,809],{"className":808},[579],[581,810,811],{"xmlns":583},[585,812,813,818],{},[588,814,815],{},[591,816,817],{},"φ",[595,819,820],{"encoding":597},"\\varphi",[572,822,824],{"className":823,"ariaHidden":602},[601],[572,825,827,831],{"className":826},[606],[572,828],{"className":829,"style":830},[610],"height:0.625em;vertical-align:-0.1944em;",[572,832,817],{"className":833},[615,616]," wie bei Polarkoordinaten in der ",[572,836,838,852],{"className":837},[575],[572,839,841],{"className":840},[579],[581,842,843],{"xmlns":583},[585,844,845,850],{},[588,846,847],{},[591,848,849],{},"x",[595,851,849],{"encoding":597},[572,853,855],{"className":854,"ariaHidden":602},[601],[572,856,858,861],{"className":857},[606],[572,859],{"className":860,"style":611},[610],[572,862,849],{"className":863},[615,616],"-",[572,866,868,882],{"className":867},[575],[572,869,871],{"className":870},[579],[581,872,873],{"xmlns":583},[585,874,875,880],{},[588,876,877],{},[591,878,879],{},"y",[595,881,879],{"encoding":597},[572,883,885],{"className":884,"ariaHidden":602},[601],[572,886,888,891],{"className":887},[606],[572,889],{"className":890,"style":830},[610],[572,892,879],{"className":893,"style":798},[615,616],"-Ebene. Sie passen zu allem, was um den Ursprung kugelsymmetrisch ist.",[565,896,897,898,1159,1160,1307,1308,1336,1337,1159,1365,1393],{},"Die Rückrechnung: ",[572,899,901,947],{"className":900},[575],[572,902,904],{"className":903},[579],[581,905,906],{"xmlns":583},[585,907,908,944],{},[588,909,910,912,914],{},[591,911,593],{},[701,913,703],{},[915,916,917],"msqrt",{},[588,918,919,927,930,936,938],{},[920,921,922,924],"msup",{},[591,923,849],{},[705,925,926],{},"2",[701,928,929],{},"+",[920,931,932,934],{},[591,933,879],{},[705,935,926],{},[701,937,929],{},[920,939,940,942],{},[591,941,669],{},[705,943,926],{},[595,945,946],{"encoding":597},"r = \\sqrt{x^2 + y^2 + z^2}",[572,948,950,968],{"className":949,"ariaHidden":602},[601],[572,951,953,956,959,962,965],{"className":952},[606],[572,954],{"className":955,"style":611},[610],[572,957,593],{"className":958,"style":617},[615,616],[572,960],{"className":961,"style":727},[726],[572,963,703],{"className":964},[731],[572,966],{"className":967,"style":727},[726],[572,969,971,975],{"className":970},[606],[572,972],{"className":973,"style":974},[610],"height:1.24em;vertical-align:-0.2822em;",[572,976,979],{"className":977},[615,978],"sqrt",[572,980,984,1150],{"className":981},[982,983],"vlist-t","vlist-t2",[572,985,988,1145],{"className":986},[987],"vlist-r",[572,989,993,1122],{"className":990,"style":992},[991],"vlist","height:0.9578em;",[572,994,998,1003],{"className":995,"style":997},[996],"svg-align","top:-3.2em;",[572,999],{"className":1000,"style":1002},[1001],"pstrut","height:3.2em;",[572,1004,1007,1044,1048,1052,1055,1084,1087,1090,1093],{"className":1005,"style":1006},[615],"padding-left:1em;",[572,1008,1010,1013],{"className":1009},[615],[572,1011,849],{"className":1012},[615,616],[572,1014,1017],{"className":1015},[1016],"msupsub",[572,1018,1020],{"className":1019},[982],[572,1021,1023],{"className":1022},[987],[572,1024,1027],{"className":1025,"style":1026},[991],"height:0.7401em;",[572,1028,1030,1034],{"style":1029},"top:-2.989em;margin-right:0.05em;",[572,1031],{"className":1032,"style":1033},[1001],"height:2.7em;",[572,1035,1041],{"className":1036},[1037,1038,1039,1040],"sizing","reset-size6","size3","mtight",[572,1042,926],{"className":1043},[615,1040],[572,1045],{"className":1046,"style":1047},[726],"margin-right:0.2222em;",[572,1049,929],{"className":1050},[1051],"mbin",[572,1053],{"className":1054,"style":1047},[726],[572,1056,1058,1061],{"className":1057},[615],[572,1059,879],{"className":1060,"style":798},[615,616],[572,1062,1064],{"className":1063},[1016],[572,1065,1067],{"className":1066},[982],[572,1068,1070],{"className":1069},[987],[572,1071,1073],{"className":1072,"style":1026},[991],[572,1074,1075,1078],{"style":1029},[572,1076],{"className":1077,"style":1033},[1001],[572,1079,1081],{"className":1080},[1037,1038,1039,1040],[572,1082,926],{"className":1083},[615,1040],[572,1085],{"className":1086,"style":1047},[726],[572,1088,929],{"className":1089},[1051],[572,1091],{"className":1092,"style":1047},[726],[572,1094,1096,1099],{"className":1095},[615],[572,1097,669],{"className":1098,"style":684},[615,616],[572,1100,1102],{"className":1101},[1016],[572,1103,1105],{"className":1104},[982],[572,1106,1108],{"className":1107},[987],[572,1109,1111],{"className":1110,"style":1026},[991],[572,1112,1113,1116],{"style":1029},[572,1114],{"className":1115,"style":1033},[1001],[572,1117,1119],{"className":1118},[1037,1038,1039,1040],[572,1120,926],{"className":1121},[615,1040],[572,1123,1125,1128],{"style":1124},"top:-2.9178em;",[572,1126],{"className":1127,"style":1002},[1001],[572,1129,1133],{"className":1130,"style":1132},[1131],"hide-tail","min-width:1.02em;height:1.28em;",[1134,1135,1141],"svg",{"xmlns":1136,"width":1137,"height":1138,"viewBox":1139,"preserveAspectRatio":1140},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[1142,1143],"path",{"d":1144},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[572,1146,1149],{"className":1147},[1148],"vlist-s","​",[572,1151,1153],{"className":1152},[987],[572,1154,1157],{"className":1155,"style":1156},[991],"height:0.2822em;",[572,1158],{}," und ",[572,1161,1163,1192],{"className":1162},[575],[572,1164,1166],{"className":1165},[579],[581,1167,1168],{"xmlns":583},[585,1169,1170,1189],{},[588,1171,1172,1175,1178,1180,1182],{},[591,1173,1174],{},"cos",[701,1176,1177],{},"⁡",[591,1179,637],{},[701,1181,703],{},[1183,1184,1185,1187],"mfrac",{},[591,1186,669],{},[591,1188,593],{},[595,1190,1191],{"encoding":597},"\\cos\\theta = \\frac{z}{r}",[572,1193,1195,1221],{"className":1194,"ariaHidden":602},[601],[572,1196,1198,1201,1205,1209,1212,1215,1218],{"className":1197},[606],[572,1199],{"className":1200,"style":650},[610],[572,1202,1174],{"className":1203},[1204],"mop",[572,1206],{"className":1207,"style":1208},[726],"margin-right:0.1667em;",[572,1210,637],{"className":1211,"style":617},[615,616],[572,1213],{"className":1214,"style":727},[726],[572,1216,703],{"className":1217},[731],[572,1219],{"className":1220,"style":727},[726],[572,1222,1224,1228],{"className":1223},[606],[572,1225],{"className":1226,"style":1227},[610],"height:1.0404em;vertical-align:-0.345em;",[572,1229,1231,1236,1303],{"className":1230},[615],[572,1232],{"className":1233},[1234,1235],"mopen","nulldelimiter",[572,1237,1239],{"className":1238},[1183],[572,1240,1242,1294],{"className":1241},[982,983],[572,1243,1245,1291],{"className":1244},[987],[572,1246,1249,1265,1276],{"className":1247,"style":1248},[991],"height:0.6954em;",[572,1250,1252,1256],{"style":1251},"top:-2.655em;",[572,1253],{"className":1254,"style":1255},[1001],"height:3em;",[572,1257,1259],{"className":1258},[1037,1038,1039,1040],[572,1260,1262],{"className":1261},[615,1040],[572,1263,593],{"className":1264,"style":617},[615,616,1040],[572,1266,1268,1271],{"style":1267},"top:-3.23em;",[572,1269],{"className":1270,"style":1255},[1001],[572,1272],{"className":1273,"style":1275},[1274],"frac-line","border-bottom-width:0.04em;",[572,1277,1279,1282],{"style":1278},"top:-3.394em;",[572,1280],{"className":1281,"style":1255},[1001],[572,1283,1285],{"className":1284},[1037,1038,1039,1040],[572,1286,1288],{"className":1287},[615,1040],[572,1289,669],{"className":1290,"style":684},[615,616,1040],[572,1292,1149],{"className":1293},[1148],[572,1295,1297],{"className":1296},[987],[572,1298,1301],{"className":1299,"style":1300},[991],"height:0.345em;",[572,1302],{},[572,1304],{"className":1305},[1306,1235],"mclose","; ",[572,1309,1311,1324],{"className":1310},[575],[572,1312,1314],{"className":1313},[579],[581,1315,1316],{"xmlns":583},[585,1317,1318,1322],{},[588,1319,1320],{},[591,1321,817],{},[595,1323,820],{"encoding":597},[572,1325,1327],{"className":1326,"ariaHidden":602},[601],[572,1328,1330,1333],{"className":1329},[606],[572,1331],{"className":1332,"style":830},[610],[572,1334,817],{"className":1335},[615,616]," bestimmt man wie bei Polarkoordinaten aus ",[572,1338,1340,1353],{"className":1339},[575],[572,1341,1343],{"className":1342},[579],[581,1344,1345],{"xmlns":583},[585,1346,1347,1351],{},[588,1348,1349],{},[591,1350,849],{},[595,1352,849],{"encoding":597},[572,1354,1356],{"className":1355,"ariaHidden":602},[601],[572,1357,1359,1362],{"className":1358},[606],[572,1360],{"className":1361,"style":611},[610],[572,1363,849],{"className":1364},[615,616],[572,1366,1368,1381],{"className":1367},[575],[572,1369,1371],{"className":1370},[579],[581,1372,1373],{"xmlns":583},[585,1374,1375,1379],{},[588,1376,1377],{},[591,1378,879],{},[595,1380,879],{"encoding":597},[572,1382,1384],{"className":1383,"ariaHidden":602},[601],[572,1385,1387,1390],{"className":1386},[606],[572,1388],{"className":1389,"style":830},[610],[572,1391,879],{"className":1392,"style":798},[615,616],".",[565,1395,1396,1397,1526,1527,1561,1562,1159,1606,1393],{},"Das Volumenelement ",[572,1398,1400,1446],{"className":1399},[575],[572,1401,1403],{"className":1402},[579],[581,1404,1405],{"xmlns":583},[585,1406,1407,1443],{},[588,1408,1409,1415,1418,1420,1422,1426,1429,1431,1433,1435,1437,1439,1441],{},[920,1410,1411,1413],{},[591,1412,593],{},[705,1414,926],{},[591,1416,1417],{},"sin",[701,1419,1177],{},[591,1421,637],{},[1423,1424,1425],"mtext",{}," ",[591,1427,1428],{},"d",[591,1430,593],{},[1423,1432,1425],{},[591,1434,1428],{},[591,1436,637],{},[1423,1438,1425],{},[591,1440,1428],{},[591,1442,817],{},[595,1444,1445],{"encoding":597},"r^2 \\sin\\theta\\, dr\\, d\\theta\\, d\\varphi",[572,1447,1449],{"className":1448,"ariaHidden":602},[601],[572,1450,1452,1456,1487,1490,1493,1496,1499,1502,1505,1508,1511,1514,1517,1520,1523],{"className":1451},[606],[572,1453],{"className":1454,"style":1455},[610],"height:1.0085em;vertical-align:-0.1944em;",[572,1457,1459,1462],{"className":1458},[615],[572,1460,593],{"className":1461,"style":617},[615,616],[572,1463,1465],{"className":1464},[1016],[572,1466,1468],{"className":1467},[982],[572,1469,1471],{"className":1470},[987],[572,1472,1475],{"className":1473,"style":1474},[991],"height:0.8141em;",[572,1476,1478,1481],{"style":1477},"top:-3.063em;margin-right:0.05em;",[572,1479],{"className":1480,"style":1033},[1001],[572,1482,1484],{"className":1483},[1037,1038,1039,1040],[572,1485,926],{"className":1486},[615,1040],[572,1488],{"className":1489,"style":1208},[726],[572,1491,1417],{"className":1492},[1204],[572,1494],{"className":1495,"style":1208},[726],[572,1497,637],{"className":1498,"style":617},[615,616],[572,1500],{"className":1501,"style":1208},[726],[572,1503,1428],{"className":1504},[615,616],[572,1506,593],{"className":1507,"style":617},[615,616],[572,1509],{"className":1510,"style":1208},[726],[572,1512,1428],{"className":1513},[615,616],[572,1515,637],{"className":1516,"style":617},[615,616],[572,1518],{"className":1519,"style":1208},[726],[572,1521,1428],{"className":1522},[615,616],[572,1524,817],{"className":1525},[615,616]," ist der Betrag der Jacobi-Determinante. Anschaulich ist ein kleines Volumenstück ein Quader mit den Kanten ",[572,1528,1530,1546],{"className":1529},[575],[572,1531,1533],{"className":1532},[579],[581,1534,1535],{"xmlns":583},[585,1536,1537,1543],{},[588,1538,1539,1541],{},[591,1540,1428],{},[591,1542,593],{},[595,1544,1545],{"encoding":597},"dr",[572,1547,1549],{"className":1548,"ariaHidden":602},[601],[572,1550,1552,1555,1558],{"className":1551},[606],[572,1553],{"className":1554,"style":650},[610],[572,1556,1428],{"className":1557},[615,616],[572,1559,593],{"className":1560,"style":617},[615,616],", ",[572,1563,1565,1585],{"className":1564},[575],[572,1566,1568],{"className":1567},[579],[581,1569,1570],{"xmlns":583},[585,1571,1572,1582],{},[588,1573,1574,1576,1578,1580],{},[591,1575,593],{},[1423,1577,1425],{},[591,1579,1428],{},[591,1581,637],{},[595,1583,1584],{"encoding":597},"r\\, d\\theta",[572,1586,1588],{"className":1587,"ariaHidden":602},[601],[572,1589,1591,1594,1597,1600,1603],{"className":1590},[606],[572,1592],{"className":1593,"style":650},[610],[572,1595,593],{"className":1596,"style":617},[615,616],[572,1598],{"className":1599,"style":1208},[726],[572,1601,1428],{"className":1602},[615,616],[572,1604,637],{"className":1605,"style":617},[615,616],[572,1607,1609,1635],{"className":1608},[575],[572,1610,1612],{"className":1611},[579],[581,1613,1614],{"xmlns":583},[585,1615,1616,1632],{},[588,1617,1618,1620,1622,1624,1626,1628,1630],{},[591,1619,593],{},[591,1621,1417],{},[701,1623,1177],{},[591,1625,637],{},[1423,1627,1425],{},[591,1629,1428],{},[591,1631,817],{},[595,1633,1634],{"encoding":597},"r \\sin\\theta\\, d\\varphi",[572,1636,1638],{"className":1637,"ariaHidden":602},[601],[572,1639,1641,1645,1648,1651,1654,1657,1660,1663,1666],{"className":1640},[606],[572,1642],{"className":1643,"style":1644},[610],"height:0.8889em;vertical-align:-0.1944em;",[572,1646,593],{"className":1647,"style":617},[615,616],[572,1649],{"className":1650,"style":1208},[726],[572,1652,1417],{"className":1653},[1204],[572,1655],{"className":1656,"style":1208},[726],[572,1658,637],{"className":1659,"style":617},[615,616],[572,1661],{"className":1662,"style":1208},[726],[572,1664,1428],{"className":1665},[615,616],[572,1667,817],{"className":1668},[615,616],[565,1670,1671,1672,1700,1701,1729,1730,1769,1770,1801],{},"Die Bezeichnungen sind nicht einheitlich. In Teilen der Mathematik heißt der Polarwinkel ",[572,1673,1675,1688],{"className":1674},[575],[572,1676,1678],{"className":1677},[579],[581,1679,1680],{"xmlns":583},[585,1681,1682,1686],{},[588,1683,1684],{},[591,1685,817],{},[595,1687,820],{"encoding":597},[572,1689,1691],{"className":1690,"ariaHidden":602},[601],[572,1692,1694,1697],{"className":1693},[606],[572,1695],{"className":1696,"style":830},[610],[572,1698,817],{"className":1699},[615,616]," und der Azimut ",[572,1702,1704,1717],{"className":1703},[575],[572,1705,1707],{"className":1706},[579],[581,1708,1709],{"xmlns":583},[585,1710,1711,1715],{},[588,1712,1713],{},[591,1714,637],{},[595,1716,640],{"encoding":597},[572,1718,1720],{"className":1719,"ariaHidden":602},[601],[572,1721,1723,1726],{"className":1722},[606],[572,1724],{"className":1725,"style":650},[610],[572,1727,637],{"className":1728,"style":617},[615,616],", in der Geografie misst man die Breite vom Äquator aus. Dann ändern sich die Formeln, z.B. wird ",[572,1731,1733,1751],{"className":1732},[575],[572,1734,1736],{"className":1735},[579],[581,1737,1738],{"xmlns":583},[585,1739,1740,1748],{},[588,1741,1742,1744,1746],{},[591,1743,1417],{},[701,1745,1177],{},[591,1747,637],{},[595,1749,1750],{"encoding":597},"\\sin\\theta",[572,1752,1754],{"className":1753,"ariaHidden":602},[601],[572,1755,1757,1760,1763,1766],{"className":1756},[606],[572,1758],{"className":1759,"style":650},[610],[572,1761,1417],{"className":1762},[1204],[572,1764],{"className":1765,"style":1208},[726],[572,1767,637],{"className":1768,"style":617},[615,616]," zu ",[572,1771,1773,1789],{"className":1772},[575],[572,1774,1776],{"className":1775},[579],[581,1777,1778],{"xmlns":583},[585,1779,1780,1786],{},[588,1781,1782,1784],{},[591,1783,1174],{},[701,1785,1177],{},[595,1787,1788],{"encoding":597},"\\cos",[572,1790,1792],{"className":1791,"ariaHidden":602},[601],[572,1793,1795,1798],{"className":1794},[606],[572,1796],{"className":1797,"style":611},[610],[572,1799,1174],{"className":1800},[1204]," der Breite.",[560,1803,1805],{"id":1804},"beispiele","Beispiele",[565,1807,1808,1809,1561,1860,1159,1981,2102],{},"Der Punkt mit ",[572,1810,1812,1830],{"className":1811},[575],[572,1813,1815],{"className":1814},[579],[581,1816,1817],{"xmlns":583},[585,1818,1819,1827],{},[588,1820,1821,1823,1825],{},[591,1822,593],{},[701,1824,703],{},[705,1826,926],{},[595,1828,1829],{"encoding":597},"r = 2",[572,1831,1833,1851],{"className":1832,"ariaHidden":602},[601],[572,1834,1836,1839,1842,1845,1848],{"className":1835},[606],[572,1837],{"className":1838,"style":611},[610],[572,1840,593],{"className":1841,"style":617},[615,616],[572,1843],{"className":1844,"style":727},[726],[572,1846,703],{"className":1847},[731],[572,1849],{"className":1850,"style":727},[726],[572,1852,1854,1857],{"className":1853},[606],[572,1855],{"className":1856,"style":741},[610],[572,1858,926],{"className":1859},[615],[572,1861,1863,1886],{"className":1862},[575],[572,1864,1866],{"className":1865},[579],[581,1867,1868],{"xmlns":583},[585,1869,1870,1883],{},[588,1871,1872,1874,1876],{},[591,1873,637],{},[701,1875,703],{},[1183,1877,1878,1880],{},[591,1879,764],{},[705,1881,1882],{},"3",[595,1884,1885],{"encoding":597},"\\theta = \\frac{\\pi}{3}",[572,1887,1889,1907],{"className":1888,"ariaHidden":602},[601],[572,1890,1892,1895,1898,1901,1904],{"className":1891},[606],[572,1893],{"className":1894,"style":650},[610],[572,1896,637],{"className":1897,"style":617},[615,616],[572,1899],{"className":1900,"style":727},[726],[572,1902,703],{"className":1903},[731],[572,1905],{"className":1906,"style":727},[726],[572,1908,1910,1913],{"className":1909},[606],[572,1911],{"className":1912,"style":1227},[610],[572,1914,1916,1919,1978],{"className":1915},[615],[572,1917],{"className":1918},[1234,1235],[572,1920,1922],{"className":1921},[1183],[572,1923,1925,1970],{"className":1924},[982,983],[572,1926,1928,1967],{"className":1927},[987],[572,1929,1931,1945,1953],{"className":1930,"style":1248},[991],[572,1932,1933,1936],{"style":1251},[572,1934],{"className":1935,"style":1255},[1001],[572,1937,1939],{"className":1938},[1037,1038,1039,1040],[572,1940,1942],{"className":1941},[615,1040],[572,1943,1882],{"className":1944},[615,1040],[572,1946,1947,1950],{"style":1267},[572,1948],{"className":1949,"style":1255},[1001],[572,1951],{"className":1952,"style":1275},[1274],[572,1954,1955,1958],{"style":1278},[572,1956],{"className":1957,"style":1255},[1001],[572,1959,1961],{"className":1960},[1037,1038,1039,1040],[572,1962,1964],{"className":1963},[615,1040],[572,1965,764],{"className":1966,"style":798},[615,616,1040],[572,1968,1149],{"className":1969},[1148],[572,1971,1973],{"className":1972},[987],[572,1974,1976],{"className":1975,"style":1300},[991],[572,1977],{},[572,1979],{"className":1980},[1306,1235],[572,1982,1984,2007],{"className":1983},[575],[572,1985,1987],{"className":1986},[579],[581,1988,1989],{"xmlns":583},[585,1990,1991,2004],{},[588,1992,1993,1995,1997],{},[591,1994,817],{},[701,1996,703],{},[1183,1998,1999,2001],{},[591,2000,764],{},[705,2002,2003],{},"4",[595,2005,2006],{"encoding":597},"\\varphi = \\frac{\\pi}{4}",[572,2008,2010,2028],{"className":2009,"ariaHidden":602},[601],[572,2011,2013,2016,2019,2022,2025],{"className":2012},[606],[572,2014],{"className":2015,"style":830},[610],[572,2017,817],{"className":2018},[615,616],[572,2020],{"className":2021,"style":727},[726],[572,2023,703],{"className":2024},[731],[572,2026],{"className":2027,"style":727},[726],[572,2029,2031,2034],{"className":2030},[606],[572,2032],{"className":2033,"style":1227},[610],[572,2035,2037,2040,2099],{"className":2036},[615],[572,2038],{"className":2039},[1234,1235],[572,2041,2043],{"className":2042},[1183],[572,2044,2046,2091],{"className":2045},[982,983],[572,2047,2049,2088],{"className":2048},[987],[572,2050,2052,2066,2074],{"className":2051,"style":1248},[991],[572,2053,2054,2057],{"style":1251},[572,2055],{"className":2056,"style":1255},[1001],[572,2058,2060],{"className":2059},[1037,1038,1039,1040],[572,2061,2063],{"className":2062},[615,1040],[572,2064,2003],{"className":2065},[615,1040],[572,2067,2068,2071],{"style":1267},[572,2069],{"className":2070,"style":1255},[1001],[572,2072],{"className":2073,"style":1275},[1274],[572,2075,2076,2079],{"style":1278},[572,2077],{"className":2078,"style":1255},[1001],[572,2080,2082],{"className":2081},[1037,1038,1039,1040],[572,2083,2085],{"className":2084},[615,1040],[572,2086,764],{"className":2087,"style":798},[615,616,1040],[572,2089,1149],{"className":2090},[1148],[572,2092,2094],{"className":2093},[987],[572,2095,2097],{"className":2096,"style":1300},[991],[572,2098],{},[572,2100],{"className":2101},[1306,1235],":",[572,2104,2107],{"className":2105},[2106],"katex-display",[572,2108,2110,2196],{"className":2109},[575],[572,2111,2113],{"className":2112},[579],[581,2114,2116],{"xmlns":583,"display":2115},"block",[585,2117,2118,2193],{},[588,2119,2120,2122,2124,2126,2128,2130,2133,2141,2143,2151,2153,2162,2165,2168,2171,2174,2176,2178,2180,2182,2189,2191],{},[591,2121,849],{},[701,2123,703],{},[591,2125,879],{},[701,2127,703],{},[705,2129,926],{},[701,2131,2132],{},"⋅",[1183,2134,2135,2139],{},[915,2136,2137],{},[705,2138,1882],{},[705,2140,926],{},[701,2142,2132],{},[1183,2144,2145,2149],{},[915,2146,2147],{},[705,2148,926],{},[705,2150,926],{},[701,2152,703],{},[1183,2154,2155,2160],{},[915,2156,2157],{},[705,2158,2159],{},"6",[705,2161,926],{},[701,2163,2164],{},"≈",[705,2166,2167],{},"1,2247",[701,2169,2170],{"separator":602},",",[726,2172],{"width":2173},"2em",[591,2175,669],{},[701,2177,703],{},[705,2179,926],{},[701,2181,2132],{},[1183,2183,2184,2187],{},[705,2185,2186],{},"1",[705,2188,926],{},[701,2190,703],{},[705,2192,2186],{},[595,2194,2195],{"encoding":597},"x = y = 2 \\cdot \\frac{\\sqrt{3}}{2} \\cdot \\frac{\\sqrt{2}}{2} = \\frac{\\sqrt{6}}{2} \\approx 1{,}2247, \\qquad z = 2 \\cdot \\frac{1}{2} = 1",[572,2197,2199,2217,2235,2253,2388,2509,2630,2673,2691,2770],{"className":2198,"ariaHidden":602},[601],[572,2200,2202,2205,2208,2211,2214],{"className":2201},[606],[572,2203],{"className":2204,"style":611},[610],[572,2206,849],{"className":2207},[615,616],[572,2209],{"className":2210,"style":727},[726],[572,2212,703],{"className":2213},[731],[572,2215],{"className":2216,"style":727},[726],[572,2218,2220,2223,2226,2229,2232],{"className":2219},[606],[572,2221],{"className":2222,"style":830},[610],[572,2224,879],{"className":2225,"style":798},[615,616],[572,2227],{"className":2228,"style":727},[726],[572,2230,703],{"className":2231},[731],[572,2233],{"className":2234,"style":727},[726],[572,2236,2238,2241,2244,2247,2250],{"className":2237},[606],[572,2239],{"className":2240,"style":741},[610],[572,2242,926],{"className":2243},[615],[572,2245],{"className":2246,"style":1047},[726],[572,2248,2132],{"className":2249},[1051],[572,2251],{"className":2252,"style":1047},[726],[572,2254,2256,2260,2379,2382,2385],{"className":2255},[606],[572,2257],{"className":2258,"style":2259},[610],"height:2.2702em;vertical-align:-0.686em;",[572,2261,2263,2266,2376],{"className":2262},[615],[572,2264],{"className":2265},[1234,1235],[572,2267,2269],{"className":2268},[1183],[572,2270,2272,2367],{"className":2271},[982,983],[572,2273,2275,2364],{"className":2274},[987],[572,2276,2279,2291,2299],{"className":2277,"style":2278},[991],"height:1.5842em;",[572,2280,2282,2285],{"style":2281},"top:-2.314em;",[572,2283],{"className":2284,"style":1255},[1001],[572,2286,2288],{"className":2287},[615],[572,2289,926],{"className":2290},[615],[572,2292,2293,2296],{"style":1267},[572,2294],{"className":2295,"style":1255},[1001],[572,2297],{"className":2298,"style":1275},[1274],[572,2300,2302,2305],{"style":2301},"top:-3.677em;",[572,2303],{"className":2304,"style":1255},[1001],[572,2306,2308],{"className":2307},[615],[572,2309,2311],{"className":2310},[615,978],[572,2312,2314,2355],{"className":2313},[982,983],[572,2315,2317,2352],{"className":2316},[987],[572,2318,2321,2335],{"className":2319,"style":2320},[991],"height:0.9072em;",[572,2322,2325,2328],{"className":2323,"style":2324},[996],"top:-3em;",[572,2326],{"className":2327,"style":1255},[1001],[572,2329,2332],{"className":2330,"style":2331},[615],"padding-left:0.833em;",[572,2333,1882],{"className":2334},[615],[572,2336,2338,2341],{"style":2337},"top:-2.8672em;",[572,2339],{"className":2340,"style":1255},[1001],[572,2342,2345],{"className":2343,"style":2344},[1131],"min-width:0.853em;height:1.08em;",[1134,2346,2349],{"xmlns":1136,"width":1137,"height":2347,"viewBox":2348,"preserveAspectRatio":1140},"1.08em","0 0 400000 1080",[1142,2350],{"d":2351},"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",[572,2353,1149],{"className":2354},[1148],[572,2356,2358],{"className":2357},[987],[572,2359,2362],{"className":2360,"style":2361},[991],"height:0.1328em;",[572,2363],{},[572,2365,1149],{"className":2366},[1148],[572,2368,2370],{"className":2369},[987],[572,2371,2374],{"className":2372,"style":2373},[991],"height:0.686em;",[572,2375],{},[572,2377],{"className":2378},[1306,1235],[572,2380],{"className":2381,"style":1047},[726],[572,2383,2132],{"className":2384},[1051],[572,2386],{"className":2387,"style":1047},[726],[572,2389,2391,2394,2500,2503,2506],{"className":2390},[606],[572,2392],{"className":2393,"style":2259},[610],[572,2395,2397,2400,2497],{"className":2396},[615],[572,2398],{"className":2399},[1234,1235],[572,2401,2403],{"className":2402},[1183],[572,2404,2406,2489],{"className":2405},[982,983],[572,2407,2409,2486],{"className":2408},[987],[572,2410,2412,2423,2431],{"className":2411,"style":2278},[991],[572,2413,2414,2417],{"style":2281},[572,2415],{"className":2416,"style":1255},[1001],[572,2418,2420],{"className":2419},[615],[572,2421,926],{"className":2422},[615],[572,2424,2425,2428],{"style":1267},[572,2426],{"className":2427,"style":1255},[1001],[572,2429],{"className":2430,"style":1275},[1274],[572,2432,2433,2436],{"style":2301},[572,2434],{"className":2435,"style":1255},[1001],[572,2437,2439],{"className":2438},[615],[572,2440,2442],{"className":2441},[615,978],[572,2443,2445,2478],{"className":2444},[982,983],[572,2446,2448,2475],{"className":2447},[987],[572,2449,2451,2463],{"className":2450,"style":2320},[991],[572,2452,2454,2457],{"className":2453,"style":2324},[996],[572,2455],{"className":2456,"style":1255},[1001],[572,2458,2460],{"className":2459,"style":2331},[615],[572,2461,926],{"className":2462},[615],[572,2464,2465,2468],{"style":2337},[572,2466],{"className":2467,"style":1255},[1001],[572,2469,2471],{"className":2470,"style":2344},[1131],[1134,2472,2473],{"xmlns":1136,"width":1137,"height":2347,"viewBox":2348,"preserveAspectRatio":1140},[1142,2474],{"d":2351},[572,2476,1149],{"className":2477},[1148],[572,2479,2481],{"className":2480},[987],[572,2482,2484],{"className":2483,"style":2361},[991],[572,2485],{},[572,2487,1149],{"className":2488},[1148],[572,2490,2492],{"className":2491},[987],[572,2493,2495],{"className":2494,"style":2373},[991],[572,2496],{},[572,2498],{"className":2499},[1306,1235],[572,2501],{"className":2502,"style":727},[726],[572,2504,703],{"className":2505},[731],[572,2507],{"className":2508,"style":727},[726],[572,2510,2512,2515,2621,2624,2627],{"className":2511},[606],[572,2513],{"className":2514,"style":2259},[610],[572,2516,2518,2521,2618],{"className":2517},[615],[572,2519],{"className":2520},[1234,1235],[572,2522,2524],{"className":2523},[1183],[572,2525,2527,2610],{"className":2526},[982,983],[572,2528,2530,2607],{"className":2529},[987],[572,2531,2533,2544,2552],{"className":2532,"style":2278},[991],[572,2534,2535,2538],{"style":2281},[572,2536],{"className":2537,"style":1255},[1001],[572,2539,2541],{"className":2540},[615],[572,2542,926],{"className":2543},[615],[572,2545,2546,2549],{"style":1267},[572,2547],{"className":2548,"style":1255},[1001],[572,2550],{"className":2551,"style":1275},[1274],[572,2553,2554,2557],{"style":2301},[572,2555],{"className":2556,"style":1255},[1001],[572,2558,2560],{"className":2559},[615],[572,2561,2563],{"className":2562},[615,978],[572,2564,2566,2599],{"className":2565},[982,983],[572,2567,2569,2596],{"className":2568},[987],[572,2570,2572,2584],{"className":2571,"style":2320},[991],[572,2573,2575,2578],{"className":2574,"style":2324},[996],[572,2576],{"className":2577,"style":1255},[1001],[572,2579,2581],{"className":2580,"style":2331},[615],[572,2582,2159],{"className":2583},[615],[572,2585,2586,2589],{"style":2337},[572,2587],{"className":2588,"style":1255},[1001],[572,2590,2592],{"className":2591,"style":2344},[1131],[1134,2593,2594],{"xmlns":1136,"width":1137,"height":2347,"viewBox":2348,"preserveAspectRatio":1140},[1142,2595],{"d":2351},[572,2597,1149],{"className":2598},[1148],[572,2600,2602],{"className":2601},[987],[572,2603,2605],{"className":2604,"style":2361},[991],[572,2606],{},[572,2608,1149],{"className":2609},[1148],[572,2611,2613],{"className":2612},[987],[572,2614,2616],{"className":2615,"style":2373},[991],[572,2617],{},[572,2619],{"className":2620},[1306,1235],[572,2622],{"className":2623,"style":727},[726],[572,2625,2164],{"className":2626},[731],[572,2628],{"className":2629,"style":727},[726],[572,2631,2633,2637,2640,2647,2651,2654,2658,2661,2664,2667,2670],{"className":2632},[606],[572,2634],{"className":2635,"style":2636},[610],"height:0.8389em;vertical-align:-0.1944em;",[572,2638,2186],{"className":2639},[615],[572,2641,2643],{"className":2642},[615],[572,2644,2170],{"className":2645},[2646],"mpunct",[572,2648,2650],{"className":2649},[615],"2247",[572,2652,2170],{"className":2653},[2646],[572,2655],{"className":2656,"style":2657},[726],"margin-right:2em;",[572,2659],{"className":2660,"style":1208},[726],[572,2662,669],{"className":2663,"style":684},[615,616],[572,2665],{"className":2666,"style":727},[726],[572,2668,703],{"className":2669},[731],[572,2671],{"className":2672,"style":727},[726],[572,2674,2676,2679,2682,2685,2688],{"className":2675},[606],[572,2677],{"className":2678,"style":741},[610],[572,2680,926],{"className":2681},[615],[572,2683],{"className":2684,"style":1047},[726],[572,2686,2132],{"className":2687},[1051],[572,2689],{"className":2690,"style":1047},[726],[572,2692,2694,2698,2761,2764,2767],{"className":2693},[606],[572,2695],{"className":2696,"style":2697},[610],"height:2.0074em;vertical-align:-0.686em;",[572,2699,2701,2704,2758],{"className":2700},[615],[572,2702],{"className":2703},[1234,1235],[572,2705,2707],{"className":2706},[1183],[572,2708,2710,2750],{"className":2709},[982,983],[572,2711,2713,2747],{"className":2712},[987],[572,2714,2717,2728,2736],{"className":2715,"style":2716},[991],"height:1.3214em;",[572,2718,2719,2722],{"style":2281},[572,2720],{"className":2721,"style":1255},[1001],[572,2723,2725],{"className":2724},[615],[572,2726,926],{"className":2727},[615],[572,2729,2730,2733],{"style":1267},[572,2731],{"className":2732,"style":1255},[1001],[572,2734],{"className":2735,"style":1275},[1274],[572,2737,2738,2741],{"style":2301},[572,2739],{"className":2740,"style":1255},[1001],[572,2742,2744],{"className":2743},[615],[572,2745,2186],{"className":2746},[615],[572,2748,1149],{"className":2749},[1148],[572,2751,2753],{"className":2752},[987],[572,2754,2756],{"className":2755,"style":2373},[991],[572,2757],{},[572,2759],{"className":2760},[1306,1235],[572,2762],{"className":2763,"style":727},[726],[572,2765,703],{"className":2766},[731],[572,2768],{"className":2769,"style":727},[726],[572,2771,2773,2776],{"className":2772},[606],[572,2774],{"className":2775,"style":741},[610],[572,2777,2186],{"className":2778},[615],[565,2780,2781,2782,2813,2814,2102],{},"Das Volumen einer Kugel mit Radius ",[572,2783,2785,2799],{"className":2784},[575],[572,2786,2788],{"className":2787},[579],[581,2789,2790],{"xmlns":583},[585,2791,2792,2797],{},[588,2793,2794],{},[591,2795,2796],{},"R",[595,2798,2796],{"encoding":597},[572,2800,2802],{"className":2801,"ariaHidden":602},[601],[572,2803,2805,2809],{"className":2804},[606],[572,2806],{"className":2807,"style":2808},[610],"height:0.6833em;",[572,2810,2796],{"className":2811,"style":2812},[615,616],"margin-right:0.0077em;",", mit ",[572,2815,2817,2855],{"className":2816},[575],[572,2818,2820],{"className":2819},[579],[581,2821,2822],{"xmlns":583},[585,2823,2824,2852],{},[588,2825,2826,2836,2838,2840,2842,2844,2846,2848,2850],{},[2827,2828,2829,2832,2834],"msubsup",{},[701,2830,2831],{},"∫",[705,2833,707],{},[591,2835,764],{},[591,2837,1417],{},[701,2839,1177],{},[591,2841,637],{},[1423,2843,1425],{},[591,2845,1428],{},[591,2847,637],{},[701,2849,703],{},[705,2851,926],{},[595,2853,2854],{"encoding":597},"\\int_0^\\pi \\sin\\theta\\, d\\theta = 2",[572,2856,2858,2953],{"className":2857,"ariaHidden":602},[601],[572,2859,2861,2865,2923,2926,2929,2932,2935,2938,2941,2944,2947,2950],{"className":2860},[606],[572,2862],{"className":2863,"style":2864},[610],"height:1.2151em;vertical-align:-0.3558em;",[572,2866,2868,2874],{"className":2867},[1204],[572,2869,2831],{"className":2870,"style":2873},[1204,2871,2872],"op-symbol","small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[572,2875,2877],{"className":2876},[1016],[572,2878,2880,2914],{"className":2879},[982,983],[572,2881,2883,2911],{"className":2882},[987],[572,2884,2887,2899],{"className":2885,"style":2886},[991],"height:0.8593em;",[572,2888,2890,2893],{"style":2889},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[572,2891],{"className":2892,"style":1033},[1001],[572,2894,2896],{"className":2895},[1037,1038,1039,1040],[572,2897,707],{"className":2898},[615,1040],[572,2900,2902,2905],{"style":2901},"top:-3.2579em;margin-right:0.05em;",[572,2903],{"className":2904,"style":1033},[1001],[572,2906,2908],{"className":2907},[1037,1038,1039,1040],[572,2909,764],{"className":2910,"style":798},[615,616,1040],[572,2912,1149],{"className":2913},[1148],[572,2915,2917],{"className":2916},[987],[572,2918,2921],{"className":2919,"style":2920},[991],"height:0.3558em;",[572,2922],{},[572,2924],{"className":2925,"style":1208},[726],[572,2927,1417],{"className":2928},[1204],[572,2930],{"className":2931,"style":1208},[726],[572,2933,637],{"className":2934,"style":617},[615,616],[572,2936],{"className":2937,"style":1208},[726],[572,2939,1428],{"className":2940},[615,616],[572,2942,637],{"className":2943,"style":617},[615,616],[572,2945],{"className":2946,"style":727},[726],[572,2948,703],{"className":2949},[731],[572,2951],{"className":2952,"style":727},[726],[572,2954,2956,2959],{"className":2955},[606],[572,2957],{"className":2958,"style":741},[610],[572,2960,926],{"className":2961},[615],[572,2963,2965],{"className":2964},[2106],[572,2966,2968,3081],{"className":2967},[575],[572,2969,2971],{"className":2970},[579],[581,2972,2973],{"xmlns":583,"display":2115},[585,2974,2975,3078],{},[588,2976,2977,2980,2982,2994,3002,3010,3016,3018,3020,3022,3024,3026,3028,3030,3032,3034,3036,3038,3040,3042,3052,3054,3056,3058,3060,3062,3064,3070,3072],{},[591,2978,2979],{},"V",[701,2981,703],{},[2827,2983,2984,2986,2988],{},[701,2985,2831],{},[705,2987,707],{},[588,2989,2990,2992],{},[705,2991,926],{},[591,2993,764],{},[2827,2995,2996,2998,3000],{},[701,2997,2831],{},[705,2999,707],{},[591,3001,764],{},[2827,3003,3004,3006,3008],{},[701,3005,2831],{},[705,3007,707],{},[591,3009,2796],{},[920,3011,3012,3014],{},[591,3013,593],{},[705,3015,926],{},[591,3017,1417],{},[701,3019,1177],{},[591,3021,637],{},[1423,3023,1425],{},[591,3025,1428],{},[591,3027,593],{},[1423,3029,1425],{},[591,3031,1428],{},[591,3033,637],{},[1423,3035,1425],{},[591,3037,1428],{},[591,3039,817],{},[701,3041,703],{},[1183,3043,3044,3050],{},[920,3045,3046,3048],{},[591,3047,2796],{},[705,3049,1882],{},[705,3051,1882],{},[701,3053,2132],{},[705,3055,926],{},[701,3057,2132],{},[705,3059,926],{},[591,3061,764],{},[701,3063,703],{},[1183,3065,3066,3068],{},[705,3067,2003],{},[705,3069,1882],{},[591,3071,764],{},[920,3073,3074,3076],{},[591,3075,2796],{},[705,3077,1882],{},[595,3079,3080],{"encoding":597},"V = \\int_0^{2\\pi} \\int_0^{\\pi} \\int_0^R r^2 \\sin\\theta\\, dr\\, d\\theta\\, d\\varphi = \\frac{R^3}{3} \\cdot 2 \\cdot 2\\pi = \\frac{4}{3}\\pi R^3",[572,3082,3084,3102,3367,3472,3490,3511],{"className":3083,"ariaHidden":602},[601],[572,3085,3087,3090,3093,3096,3099],{"className":3086},[606],[572,3088],{"className":3089,"style":2808},[610],[572,3091,2979],{"className":3092,"style":1047},[615,616],[572,3094],{"className":3095,"style":727},[726],[572,3097,703],{"className":3098},[731],[572,3100],{"className":3101,"style":727},[726],[572,3103,3105,3109,3172,3175,3230,3233,3285,3288,3319,3322,3325,3328,3331,3334,3337,3340,3343,3346,3349,3352,3355,3358,3361,3364],{"className":3104},[606],[572,3106],{"className":3107,"style":3108},[610],"height:2.5032em;vertical-align:-0.9119em;",[572,3110,3112,3117],{"className":3111},[1204],[572,3113,2831],{"className":3114,"style":3116},[1204,2871,3115],"large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[572,3118,3120],{"className":3119},[1016],[572,3121,3123,3163],{"className":3122},[982,983],[572,3124,3126,3160],{"className":3125},[987],[572,3127,3130,3142],{"className":3128,"style":3129},[991],"height:1.564em;",[572,3131,3133,3136],{"style":3132},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[572,3134],{"className":3135,"style":1033},[1001],[572,3137,3139],{"className":3138},[1037,1038,1039,1040],[572,3140,707],{"className":3141},[615,1040],[572,3143,3145,3148],{"style":3144},"top:-3.8129em;margin-right:0.05em;",[572,3146],{"className":3147,"style":1033},[1001],[572,3149,3151],{"className":3150},[1037,1038,1039,1040],[572,3152,3154,3157],{"className":3153},[615,1040],[572,3155,926],{"className":3156},[615,1040],[572,3158,764],{"className":3159,"style":798},[615,616,1040],[572,3161,1149],{"className":3162},[1148],[572,3164,3166],{"className":3165},[987],[572,3167,3170],{"className":3168,"style":3169},[991],"height:0.9119em;",[572,3171],{},[572,3173],{"className":3174,"style":1208},[726],[572,3176,3178,3181],{"className":3177},[1204],[572,3179,2831],{"className":3180,"style":3116},[1204,2871,3115],[572,3182,3184],{"className":3183},[1016],[572,3185,3187,3222],{"className":3186},[982,983],[572,3188,3190,3219],{"className":3189},[987],[572,3191,3194,3205],{"className":3192,"style":3193},[991],"height:1.4143em;",[572,3195,3196,3199],{"style":3132},[572,3197],{"className":3198,"style":1033},[1001],[572,3200,3202],{"className":3201},[1037,1038,1039,1040],[572,3203,707],{"className":3204},[615,1040],[572,3206,3207,3210],{"style":3144},[572,3208],{"className":3209,"style":1033},[1001],[572,3211,3213],{"className":3212},[1037,1038,1039,1040],[572,3214,3216],{"className":3215},[615,1040],[572,3217,764],{"className":3218,"style":798},[615,616,1040],[572,3220,1149],{"className":3221},[1148],[572,3223,3225],{"className":3224},[987],[572,3226,3228],{"className":3227,"style":3169},[991],[572,3229],{},[572,3231],{"className":3232,"style":1208},[726],[572,3234,3236,3239],{"className":3235},[1204],[572,3237,2831],{"className":3238,"style":3116},[1204,2871,3115],[572,3240,3242],{"className":3241},[1016],[572,3243,3245,3277],{"className":3244},[982,983],[572,3246,3248,3274],{"className":3247},[987],[572,3249,3252,3263],{"className":3250,"style":3251},[991],"height:1.5912em;",[572,3253,3254,3257],{"style":3132},[572,3255],{"className":3256,"style":1033},[1001],[572,3258,3260],{"className":3259},[1037,1038,1039,1040],[572,3261,707],{"className":3262},[615,1040],[572,3264,3265,3268],{"style":3144},[572,3266],{"className":3267,"style":1033},[1001],[572,3269,3271],{"className":3270},[1037,1038,1039,1040],[572,3272,2796],{"className":3273,"style":2812},[615,616,1040],[572,3275,1149],{"className":3276},[1148],[572,3278,3280],{"className":3279},[987],[572,3281,3283],{"className":3282,"style":3169},[991],[572,3284],{},[572,3286],{"className":3287,"style":1208},[726],[572,3289,3291,3294],{"className":3290},[615],[572,3292,593],{"className":3293,"style":617},[615,616],[572,3295,3297],{"className":3296},[1016],[572,3298,3300],{"className":3299},[982],[572,3301,3303],{"className":3302},[987],[572,3304,3307],{"className":3305,"style":3306},[991],"height:0.8641em;",[572,3308,3310,3313],{"style":3309},"top:-3.113em;margin-right:0.05em;",[572,3311],{"className":3312,"style":1033},[1001],[572,3314,3316],{"className":3315},[1037,1038,1039,1040],[572,3317,926],{"className":3318},[615,1040],[572,3320],{"className":3321,"style":1208},[726],[572,3323,1417],{"className":3324},[1204],[572,3326],{"className":3327,"style":1208},[726],[572,3329,637],{"className":3330,"style":617},[615,616],[572,3332],{"className":3333,"style":1208},[726],[572,3335,1428],{"className":3336},[615,616],[572,3338,593],{"className":3339,"style":617},[615,616],[572,3341],{"className":3342,"style":1208},[726],[572,3344,1428],{"className":3345},[615,616],[572,3347,637],{"className":3348,"style":617},[615,616],[572,3350],{"className":3351,"style":1208},[726],[572,3353,1428],{"className":3354},[615,616],[572,3356,817],{"className":3357},[615,616],[572,3359],{"className":3360,"style":727},[726],[572,3362,703],{"className":3363},[731],[572,3365],{"className":3366,"style":727},[726],[572,3368,3370,3374,3463,3466,3469],{"className":3369},[606],[572,3371],{"className":3372,"style":3373},[610],"height:2.1771em;vertical-align:-0.686em;",[572,3375,3377,3380,3460],{"className":3376},[615],[572,3378],{"className":3379},[1234,1235],[572,3381,3383],{"className":3382},[1183],[572,3384,3386,3452],{"className":3385},[982,983],[572,3387,3389,3449],{"className":3388},[987],[572,3390,3393,3404,3412],{"className":3391,"style":3392},[991],"height:1.4911em;",[572,3394,3395,3398],{"style":2281},[572,3396],{"className":3397,"style":1255},[1001],[572,3399,3401],{"className":3400},[615],[572,3402,1882],{"className":3403},[615],[572,3405,3406,3409],{"style":1267},[572,3407],{"className":3408,"style":1255},[1001],[572,3410],{"className":3411,"style":1275},[1274],[572,3413,3414,3417],{"style":2301},[572,3415],{"className":3416,"style":1255},[1001],[572,3418,3420],{"className":3419},[615],[572,3421,3423,3426],{"className":3422},[615],[572,3424,2796],{"className":3425,"style":2812},[615,616],[572,3427,3429],{"className":3428},[1016],[572,3430,3432],{"className":3431},[982],[572,3433,3435],{"className":3434},[987],[572,3436,3438],{"className":3437,"style":1474},[991],[572,3439,3440,3443],{"style":1477},[572,3441],{"className":3442,"style":1033},[1001],[572,3444,3446],{"className":3445},[1037,1038,1039,1040],[572,3447,1882],{"className":3448},[615,1040],[572,3450,1149],{"className":3451},[1148],[572,3453,3455],{"className":3454},[987],[572,3456,3458],{"className":3457,"style":2373},[991],[572,3459],{},[572,3461],{"className":3462},[1306,1235],[572,3464],{"className":3465,"style":1047},[726],[572,3467,2132],{"className":3468},[1051],[572,3470],{"className":3471,"style":1047},[726],[572,3473,3475,3478,3481,3484,3487],{"className":3474},[606],[572,3476],{"className":3477,"style":741},[610],[572,3479,926],{"className":3480},[615],[572,3482],{"className":3483,"style":1047},[726],[572,3485,2132],{"className":3486},[1051],[572,3488],{"className":3489,"style":1047},[726],[572,3491,3493,3496,3499,3502,3505,3508],{"className":3492},[606],[572,3494],{"className":3495,"style":741},[610],[572,3497,926],{"className":3498},[615],[572,3500,764],{"className":3501,"style":798},[615,616],[572,3503],{"className":3504,"style":727},[726],[572,3506,703],{"className":3507},[731],[572,3509],{"className":3510,"style":727},[726],[572,3512,3514,3517,3579,3582],{"className":3513},[606],[572,3515],{"className":3516,"style":2697},[610],[572,3518,3520,3523,3576],{"className":3519},[615],[572,3521],{"className":3522},[1234,1235],[572,3524,3526],{"className":3525},[1183],[572,3527,3529,3568],{"className":3528},[982,983],[572,3530,3532,3565],{"className":3531},[987],[572,3533,3535,3546,3554],{"className":3534,"style":2716},[991],[572,3536,3537,3540],{"style":2281},[572,3538],{"className":3539,"style":1255},[1001],[572,3541,3543],{"className":3542},[615],[572,3544,1882],{"className":3545},[615],[572,3547,3548,3551],{"style":1267},[572,3549],{"className":3550,"style":1255},[1001],[572,3552],{"className":3553,"style":1275},[1274],[572,3555,3556,3559],{"style":2301},[572,3557],{"className":3558,"style":1255},[1001],[572,3560,3562],{"className":3561},[615],[572,3563,2003],{"className":3564},[615],[572,3566,1149],{"className":3567},[1148],[572,3569,3571],{"className":3570},[987],[572,3572,3574],{"className":3573,"style":2373},[991],[572,3575],{},[572,3577],{"className":3578},[1306,1235],[572,3580,764],{"className":3581,"style":798},[615,616],[572,3583,3585,3588],{"className":3584},[615],[572,3586,2796],{"className":3587,"style":2812},[615,616],[572,3589,3591],{"className":3590},[1016],[572,3592,3594],{"className":3593},[982],[572,3595,3597],{"className":3596},[987],[572,3598,3600],{"className":3599,"style":3306},[991],[572,3601,3602,3605],{"style":3309},[572,3603],{"className":3604,"style":1033},[1001],[572,3606,3608],{"className":3607},[1037,1038,1039,1040],[572,3609,1882],{"className":3610},[615,1040],[565,3612,3613,3614,3854,3855,1393],{},"Für ",[572,3615,3617,3664],{"className":3616},[575],[572,3618,3620],{"className":3619},[579],[581,3621,3622],{"xmlns":583},[585,3623,3624,3661],{},[588,3625,3626,3629,3631,3637,3639,3645,3647,3653,3655],{},[591,3627,3628],{},"f",[701,3630,703],{},[920,3632,3633,3635],{},[591,3634,849],{},[705,3636,926],{},[701,3638,929],{},[920,3640,3641,3643],{},[591,3642,879],{},[705,3644,926],{},[701,3646,929],{},[920,3648,3649,3651],{},[591,3650,669],{},[705,3652,926],{},[701,3654,703],{},[920,3656,3657,3659],{},[591,3658,593],{},[705,3660,926],{},[595,3662,3663],{"encoding":597},"f = x^2 + y^2 + z^2 = r^2",[572,3665,3667,3686,3731,3775,3819],{"className":3666,"ariaHidden":602},[601],[572,3668,3670,3673,3677,3680,3683],{"className":3669},[606],[572,3671],{"className":3672,"style":1644},[610],[572,3674,3628],{"className":3675,"style":3676},[615,616],"margin-right:0.1076em;",[572,3678],{"className":3679,"style":727},[726],[572,3681,703],{"className":3682},[731],[572,3684],{"className":3685,"style":727},[726],[572,3687,3689,3693,3722,3725,3728],{"className":3688},[606],[572,3690],{"className":3691,"style":3692},[610],"height:0.8974em;vertical-align:-0.0833em;",[572,3694,3696,3699],{"className":3695},[615],[572,3697,849],{"className":3698},[615,616],[572,3700,3702],{"className":3701},[1016],[572,3703,3705],{"className":3704},[982],[572,3706,3708],{"className":3707},[987],[572,3709,3711],{"className":3710,"style":1474},[991],[572,3712,3713,3716],{"style":1477},[572,3714],{"className":3715,"style":1033},[1001],[572,3717,3719],{"className":3718},[1037,1038,1039,1040],[572,3720,926],{"className":3721},[615,1040],[572,3723],{"className":3724,"style":1047},[726],[572,3726,929],{"className":3727},[1051],[572,3729],{"className":3730,"style":1047},[726],[572,3732,3734,3737,3766,3769,3772],{"className":3733},[606],[572,3735],{"className":3736,"style":1455},[610],[572,3738,3740,3743],{"className":3739},[615],[572,3741,879],{"className":3742,"style":798},[615,616],[572,3744,3746],{"className":3745},[1016],[572,3747,3749],{"className":3748},[982],[572,3750,3752],{"className":3751},[987],[572,3753,3755],{"className":3754,"style":1474},[991],[572,3756,3757,3760],{"style":1477},[572,3758],{"className":3759,"style":1033},[1001],[572,3761,3763],{"className":3762},[1037,1038,1039,1040],[572,3764,926],{"className":3765},[615,1040],[572,3767],{"className":3768,"style":1047},[726],[572,3770,929],{"className":3771},[1051],[572,3773],{"className":3774,"style":1047},[726],[572,3776,3778,3781,3810,3813,3816],{"className":3777},[606],[572,3779],{"className":3780,"style":1474},[610],[572,3782,3784,3787],{"className":3783},[615],[572,3785,669],{"className":3786,"style":684},[615,616],[572,3788,3790],{"className":3789},[1016],[572,3791,3793],{"className":3792},[982],[572,3794,3796],{"className":3795},[987],[572,3797,3799],{"className":3798,"style":1474},[991],[572,3800,3801,3804],{"style":1477},[572,3802],{"className":3803,"style":1033},[1001],[572,3805,3807],{"className":3806},[1037,1038,1039,1040],[572,3808,926],{"className":3809},[615,1040],[572,3811],{"className":3812,"style":727},[726],[572,3814,703],{"className":3815},[731],[572,3817],{"className":3818,"style":727},[726],[572,3820,3822,3825],{"className":3821},[606],[572,3823],{"className":3824,"style":1474},[610],[572,3826,3828,3831],{"className":3827},[615],[572,3829,593],{"className":3830,"style":617},[615,616],[572,3832,3834],{"className":3833},[1016],[572,3835,3837],{"className":3836},[982],[572,3838,3840],{"className":3839},[987],[572,3841,3843],{"className":3842,"style":1474},[991],[572,3844,3845,3848],{"style":1477},[572,3846],{"className":3847,"style":1033},[1001],[572,3849,3851],{"className":3850},[1037,1038,1039,1040],[572,3852,926],{"className":3853},[615,1040]," über die Einheitskugel ergibt sich ",[572,3856,3858,3918],{"className":3857},[575],[572,3859,3861],{"className":3860},[579],[581,3862,3863],{"xmlns":583},[585,3864,3865,3915],{},[588,3866,3867,3875,3881,3883,3885,3887,3889,3891,3893,3895,3897,3899,3910,3912],{},[2827,3868,3869,3871,3873],{},[701,3870,2831],{},[705,3872,707],{},[705,3874,2186],{},[920,3876,3877,3879],{},[591,3878,593],{},[705,3880,2003],{},[1423,3882,1425],{},[591,3884,1428],{},[591,3886,593],{},[701,3888,2132],{},[705,3890,926],{},[701,3892,2132],{},[705,3894,926],{},[591,3896,764],{},[701,3898,703],{},[1183,3900,3901,3907],{},[588,3902,3903,3905],{},[705,3904,2003],{},[591,3906,764],{},[705,3908,3909],{},"5",[701,3911,2164],{},[705,3913,3914],{},"2,5133",[595,3916,3917],{"encoding":597},"\\int_0^1 r^4\\, dr \\cdot 2 \\cdot 2\\pi = \\frac{4\\pi}{5} \\approx 2{,}5133",[572,3919,3921,4030,4048,4069,4157],{"className":3920,"ariaHidden":602},[601],[572,3922,3924,3928,3980,3983,4012,4015,4018,4021,4024,4027],{"className":3923},[606],[572,3925],{"className":3926,"style":3927},[610],"height:1.3648em;vertical-align:-0.3558em;",[572,3929,3931,3934],{"className":3930},[1204],[572,3932,2831],{"className":3933,"style":2873},[1204,2871,2872],[572,3935,3937],{"className":3936},[1016],[572,3938,3940,3972],{"className":3939},[982,983],[572,3941,3943,3969],{"className":3942},[987],[572,3944,3947,3958],{"className":3945,"style":3946},[991],"height:1.009em;",[572,3948,3949,3952],{"style":2889},[572,3950],{"className":3951,"style":1033},[1001],[572,3953,3955],{"className":3954},[1037,1038,1039,1040],[572,3956,707],{"className":3957},[615,1040],[572,3959,3960,3963],{"style":2901},[572,3961],{"className":3962,"style":1033},[1001],[572,3964,3966],{"className":3965},[1037,1038,1039,1040],[572,3967,2186],{"className":3968},[615,1040],[572,3970,1149],{"className":3971},[1148],[572,3973,3975],{"className":3974},[987],[572,3976,3978],{"className":3977,"style":2920},[991],[572,3979],{},[572,3981],{"className":3982,"style":1208},[726],[572,3984,3986,3989],{"className":3985},[615],[572,3987,593],{"className":3988,"style":617},[615,616],[572,3990,3992],{"className":3991},[1016],[572,3993,3995],{"className":3994},[982],[572,3996,3998],{"className":3997},[987],[572,3999,4001],{"className":4000,"style":1474},[991],[572,4002,4003,4006],{"style":1477},[572,4004],{"className":4005,"style":1033},[1001],[572,4007,4009],{"className":4008},[1037,1038,1039,1040],[572,4010,2003],{"className":4011},[615,1040],[572,4013],{"className":4014,"style":1208},[726],[572,4016,1428],{"className":4017},[615,616],[572,4019,593],{"className":4020,"style":617},[615,616],[572,4022],{"className":4023,"style":1047},[726],[572,4025,2132],{"className":4026},[1051],[572,4028],{"className":4029,"style":1047},[726],[572,4031,4033,4036,4039,4042,4045],{"className":4032},[606],[572,4034],{"className":4035,"style":741},[610],[572,4037,926],{"className":4038},[615],[572,4040],{"className":4041,"style":1047},[726],[572,4043,2132],{"className":4044},[1051],[572,4046],{"className":4047,"style":1047},[726],[572,4049,4051,4054,4057,4060,4063,4066],{"className":4050},[606],[572,4052],{"className":4053,"style":741},[610],[572,4055,926],{"className":4056},[615],[572,4058,764],{"className":4059,"style":798},[615,616],[572,4061],{"className":4062,"style":727},[726],[572,4064,703],{"className":4065},[731],[572,4067],{"className":4068,"style":727},[726],[572,4070,4072,4076,4148,4151,4154],{"className":4071},[606],[572,4073],{"className":4074,"style":4075},[610],"height:1.1901em;vertical-align:-0.345em;",[572,4077,4079,4082,4145],{"className":4078},[615],[572,4080],{"className":4081},[1234,1235],[572,4083,4085],{"className":4084},[1183],[572,4086,4088,4137],{"className":4087},[982,983],[572,4089,4091,4134],{"className":4090},[987],[572,4092,4095,4109,4117],{"className":4093,"style":4094},[991],"height:0.8451em;",[572,4096,4097,4100],{"style":1251},[572,4098],{"className":4099,"style":1255},[1001],[572,4101,4103],{"className":4102},[1037,1038,1039,1040],[572,4104,4106],{"className":4105},[615,1040],[572,4107,3909],{"className":4108},[615,1040],[572,4110,4111,4114],{"style":1267},[572,4112],{"className":4113,"style":1255},[1001],[572,4115],{"className":4116,"style":1275},[1274],[572,4118,4119,4122],{"style":1278},[572,4120],{"className":4121,"style":1255},[1001],[572,4123,4125],{"className":4124},[1037,1038,1039,1040],[572,4126,4128,4131],{"className":4127},[615,1040],[572,4129,2003],{"className":4130},[615,1040],[572,4132,764],{"className":4133,"style":798},[615,616,1040],[572,4135,1149],{"className":4136},[1148],[572,4138,4140],{"className":4139},[987],[572,4141,4143],{"className":4142,"style":1300},[991],[572,4144],{},[572,4146],{"className":4147},[1306,1235],[572,4149],{"className":4150,"style":727},[726],[572,4152,2164],{"className":4153},[731],[572,4155],{"className":4156,"style":727},[726],[572,4158,4160,4163,4166,4172],{"className":4159},[606],[572,4161],{"className":4162,"style":2636},[610],[572,4164,926],{"className":4165},[615],[572,4167,4169],{"className":4168},[615],[572,4170,2170],{"className":4171},[2646],[572,4173,4175],{"className":4174},[615],"5133",[560,4177,4179],{"id":4178},"typische-fehler","Typische Fehler",[4181,4182,4183,4294,4446],"ul",{},[4184,4185,4186,4187,4264,4265,4293],"li",{},"Den Faktor ",[572,4188,4190,4214],{"className":4189},[575],[572,4191,4193],{"className":4192},[579],[581,4194,4195],{"xmlns":583},[585,4196,4197,4211],{},[588,4198,4199,4205,4207,4209],{},[920,4200,4201,4203],{},[591,4202,593],{},[705,4204,926],{},[591,4206,1417],{},[701,4208,1177],{},[591,4210,637],{},[595,4212,4213],{"encoding":597},"r^2 \\sin\\theta",[572,4215,4217],{"className":4216,"ariaHidden":602},[601],[572,4218,4220,4223,4252,4255,4258,4261],{"className":4219},[606],[572,4221],{"className":4222,"style":1474},[610],[572,4224,4226,4229],{"className":4225},[615],[572,4227,593],{"className":4228,"style":617},[615,616],[572,4230,4232],{"className":4231},[1016],[572,4233,4235],{"className":4234},[982],[572,4236,4238],{"className":4237},[987],[572,4239,4241],{"className":4240,"style":1474},[991],[572,4242,4243,4246],{"style":1477},[572,4244],{"className":4245,"style":1033},[1001],[572,4247,4249],{"className":4248},[1037,1038,1039,1040],[572,4250,926],{"className":4251},[615,1040],[572,4253],{"className":4254,"style":1208},[726],[572,4256,1417],{"className":4257},[1204],[572,4259],{"className":4260,"style":1208},[726],[572,4262,637],{"className":4263,"style":617},[615,616]," vergessen oder nur ",[572,4266,4268,4281],{"className":4267},[575],[572,4269,4271],{"className":4270},[579],[581,4272,4273],{"xmlns":583},[585,4274,4275,4279],{},[588,4276,4277],{},[591,4278,593],{},[595,4280,593],{"encoding":597},[572,4282,4284],{"className":4283,"ariaHidden":602},[601],[572,4285,4287,4290],{"className":4286},[606],[572,4288],{"className":4289,"style":611},[610],[572,4291,593],{"className":4292,"style":617},[615,616]," verwenden wie bei Zylinderkoordinaten.",[4184,4295,4296,4324,4325,4353,4354,4388,4389,4353,4417,1393],{},[572,4297,4299,4312],{"className":4298},[575],[572,4300,4302],{"className":4301},[579],[581,4303,4304],{"xmlns":583},[585,4305,4306,4310],{},[588,4307,4308],{},[591,4309,637],{},[595,4311,640],{"encoding":597},[572,4313,4315],{"className":4314,"ariaHidden":602},[601],[572,4316,4318,4321],{"className":4317},[606],[572,4319],{"className":4320,"style":650},[610],[572,4322,637],{"className":4323,"style":617},[615,616]," von ",[572,4326,4328,4341],{"className":4327},[575],[572,4329,4331],{"className":4330},[579],[581,4332,4333],{"xmlns":583},[585,4334,4335,4339],{},[588,4336,4337],{},[705,4338,707],{},[595,4340,707],{"encoding":597},[572,4342,4344],{"className":4343,"ariaHidden":602},[601],[572,4345,4347,4350],{"className":4346},[606],[572,4348],{"className":4349,"style":741},[610],[572,4351,707],{"className":4352},[615]," bis ",[572,4355,4357,4373],{"className":4356},[575],[572,4358,4360],{"className":4359},[579],[581,4361,4362],{"xmlns":583},[585,4363,4364,4370],{},[588,4365,4366,4368],{},[705,4367,926],{},[591,4369,764],{},[595,4371,4372],{"encoding":597},"2\\pi",[572,4374,4376],{"className":4375,"ariaHidden":602},[601],[572,4377,4379,4382,4385],{"className":4378},[606],[572,4380],{"className":4381,"style":741},[610],[572,4383,926],{"className":4384},[615],[572,4386,764],{"className":4387,"style":798},[615,616]," laufen lassen. Dann wird jeder Punkt doppelt erfasst; der Polarwinkel läuft nur von ",[572,4390,4392,4405],{"className":4391},[575],[572,4393,4395],{"className":4394},[579],[581,4396,4397],{"xmlns":583},[585,4398,4399,4403],{},[588,4400,4401],{},[705,4402,707],{},[595,4404,707],{"encoding":597},[572,4406,4408],{"className":4407,"ariaHidden":602},[601],[572,4409,4411,4414],{"className":4410},[606],[572,4412],{"className":4413,"style":741},[610],[572,4415,707],{"className":4416},[615],[572,4418,4420,4434],{"className":4419},[575],[572,4421,4423],{"className":4422},[579],[581,4424,4425],{"xmlns":583},[585,4426,4427,4431],{},[588,4428,4429],{},[591,4430,764],{},[595,4432,4433],{"encoding":597},"\\pi",[572,4435,4437],{"className":4436,"ariaHidden":602},[601],[572,4438,4440,4443],{"className":4439},[606],[572,4441],{"className":4442,"style":611},[610],[572,4444,764],{"className":4445,"style":798},[615,616],[4184,4447,4448,4449,4518,4519,1393],{},"Formeln aus Quellen mit anderer Winkelkonvention mischen, z.B. ",[572,4450,4452,4476],{"className":4451},[575],[572,4453,4455],{"className":4454},[579],[581,4456,4457],{"xmlns":583},[585,4458,4459,4473],{},[588,4460,4461,4463,4465,4467,4469,4471],{},[591,4462,669],{},[701,4464,703],{},[591,4466,593],{},[591,4468,1417],{},[701,4470,1177],{},[591,4472,637],{},[595,4474,4475],{"encoding":597},"z = r \\sin\\theta",[572,4477,4479,4497],{"className":4478,"ariaHidden":602},[601],[572,4480,4482,4485,4488,4491,4494],{"className":4481},[606],[572,4483],{"className":4484,"style":611},[610],[572,4486,669],{"className":4487,"style":684},[615,616],[572,4489],{"className":4490,"style":727},[726],[572,4492,703],{"className":4493},[731],[572,4495],{"className":4496,"style":727},[726],[572,4498,4500,4503,4506,4509,4512,4515],{"className":4499},[606],[572,4501],{"className":4502,"style":650},[610],[572,4504,593],{"className":4505,"style":617},[615,616],[572,4507],{"className":4508,"style":1208},[726],[572,4510,1417],{"className":4511},[1204],[572,4513],{"className":4514,"style":1208},[726],[572,4516,637],{"className":4517,"style":617},[615,616]," mit ",[572,4520,4522,4570],{"className":4521},[575],[572,4523,4525],{"className":4524},[579],[581,4526,4527],{"xmlns":583},[585,4528,4529,4567],{},[588,4530,4531,4533,4535,4537,4543,4545,4547,4549,4551,4553,4555,4557,4559,4561,4563,4565],{},[591,4532,1428],{},[591,4534,2979],{},[701,4536,703],{},[920,4538,4539,4541],{},[591,4540,593],{},[705,4542,926],{},[591,4544,1417],{},[701,4546,1177],{},[591,4548,637],{},[1423,4550,1425],{},[591,4552,1428],{},[591,4554,593],{},[1423,4556,1425],{},[591,4558,1428],{},[591,4560,637],{},[1423,4562,1425],{},[591,4564,1428],{},[591,4566,817],{},[595,4568,4569],{"encoding":597},"dV = r^2 \\sin\\theta\\, dr\\, d\\theta\\, d\\varphi",[572,4571,4573,4594],{"className":4572,"ariaHidden":602},[601],[572,4574,4576,4579,4582,4585,4588,4591],{"className":4575},[606],[572,4577],{"className":4578,"style":650},[610],[572,4580,1428],{"className":4581},[615,616],[572,4583,2979],{"className":4584,"style":1047},[615,616],[572,4586],{"className":4587,"style":727},[726],[572,4589,703],{"className":4590},[731],[572,4592],{"className":4593,"style":727},[726],[572,4595,4597,4600,4629,4632,4635,4638,4641,4644,4647,4650,4653,4656,4659,4662,4665],{"className":4596},[606],[572,4598],{"className":4599,"style":1455},[610],[572,4601,4603,4606],{"className":4602},[615],[572,4604,593],{"className":4605,"style":617},[615,616],[572,4607,4609],{"className":4608},[1016],[572,4610,4612],{"className":4611},[982],[572,4613,4615],{"className":4614},[987],[572,4616,4618],{"className":4617,"style":1474},[991],[572,4619,4620,4623],{"style":1477},[572,4621],{"className":4622,"style":1033},[1001],[572,4624,4626],{"className":4625},[1037,1038,1039,1040],[572,4627,926],{"className":4628},[615,1040],[572,4630],{"className":4631,"style":1208},[726],[572,4633,1417],{"className":4634},[1204],[572,4636],{"className":4637,"style":1208},[726],[572,4639,637],{"className":4640,"style":617},[615,616],[572,4642],{"className":4643,"style":1208},[726],[572,4645,1428],{"className":4646},[615,616],[572,4648,593],{"className":4649,"style":617},[615,616],[572,4651],{"className":4652,"style":1208},[726],[572,4654,1428],{"className":4655},[615,616],[572,4657,637],{"className":4658,"style":617},[615,616],[572,4660],{"className":4661,"style":1208},[726],[572,4663,1428],{"className":4664},[615,616],[572,4666,817],{"className":4667},[615,616],{"title":4669,"searchDepth":4670,"depth":4670,"links":4671},"",2,[4672,4673,4674],{"id":562,"depth":4670,"text":563},{"id":1804,"depth":4670,"text":1805},{"id":4178,"depth":4670,"text":4179},[4676,4679,4682,4685,4687,4689,4692,4694,4697,4700],{"lhs":4677,"rhs":4678},"(r*sin(t)*cos(p))^2 + (r*sin(t)*sin(p))^2 + (r*cos(t))^2","r^2",{"lhs":4680,"rhs":4681},"det([[sin(t)*cos(p), r*cos(t)*cos(p), -r*sin(t)*sin(p)], [sin(t)*sin(p), r*cos(t)*sin(p), r*sin(t)*cos(p)], [cos(t), -r*sin(t), 0]])","r^2*sin(t)",{"lhs":4683,"rhs":4684},"2*sin(pi\u002F3)*cos(pi\u002F4)","sqrt(6)\u002F2",{"lhs":4686,"rhs":4684},"2*sin(pi\u002F3)*sin(pi\u002F4)",{"lhs":4688,"rhs":2186},"2*cos(pi\u002F3)",{"lhs":4690,"rhs":4691},"round(sqrt(6)\u002F2, 4)","1.2247",{"lhs":4693,"rhs":926},"round(sum(map(0:999, f(k) = sin((k + 0.5)*pi\u002F1000))) * pi\u002F1000, 4)",{"lhs":4695,"rhs":4696},"R^3\u002F3 * 2 * 2*pi","4\u002F3*pi*R^3",{"lhs":4698,"rhs":4699},"1\u002F5 * 2 * 2*pi","4*pi\u002F5",{"lhs":4701,"rhs":4702},"round(4*pi\u002F5, 4)","2.5133","r \\geq 0,\\ 0 \\leq \\theta \\leq \\pi,\\ 0 \\leq \\varphi \u003C 2\\pi","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r \\geq 0,\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:0.7719em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\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:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>π\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">0 \\leq \\theta \\leq \\pi,\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:0.7804em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\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:0.8304em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo><\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>π\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">0 \\leq \\varphi < 2\\pi\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:0.7804em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\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:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","spherical-coordinates","x = r\\sin\\theta\\cos\\varphi, \\quad y = r\\sin\\theta\\sin\\varphi, \\quad z = r\\cos\\theta, \\qquad dV = r^2 \\sin\\theta\\, dr\\, d\\theta\\, d\\varphi","\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>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmi>cos\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmi>y\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmi>z\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>cos\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"2em\"\u002F>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>V\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>r\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>r\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>θ\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>φ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x = r\\sin\\theta\\cos\\varphi, \\quad y = r\\sin\\theta\\sin\\varphi, \\quad z = r\\cos\\theta, \\qquad dV = r^2 \\sin\\theta\\, dr\\, d\\theta\\, d\\varphi\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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">cos\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\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=\"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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">cos\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:2em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\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:1.0585em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8641em;\">\u003Cspan style=\"top:-3.113em;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\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fkugelkoordinaten",[],[4716,4717,4718,4719],"cylindrical-coordinates","triple-integral","change-of-variables","sphere-volume",{"title":548,"description":4669},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fkugelkoordinaten","Kugelkoordinaten beschreiben einen Punkt durch Abstand zum Ursprung und zwei Winkel; das Volumenelement ist r² sin θ dr dθ dφ.",[4724,4725,4726,4727,4728],"kugelkoordinaten","koordinatensystem","volumenelement","dreifachintegral","kugel",[4730,4733,4736,4739,4743],{"symbol":593,"meaning":4731,"symbolHtml":4732},"Abstand des Punkts vom Ursprung","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r\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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":640,"meaning":4734,"symbolHtml":4735},"Polarwinkel, gemessen von der positiven z-Achse","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>θ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\theta\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:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":820,"meaning":4737,"symbolHtml":4738},"Azimutwinkel in der x-y-Ebene, gemessen von der positiven x-Achse","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>φ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4740,"meaning":4741,"symbolHtml":4742},"x, y, z","kartesische Koordinaten","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>z\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x, y, z\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:0.625em;vertical-align:-0.1944em;\">\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=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4744,"meaning":4745,"symbolHtml":4746},"dV","Volumenelement für Dreifachintegrale","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>V\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">dV\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:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","E8JkpWjl62jgkXexpKbSCrpgeMbiX6wUUjKreVOmAvw",[4749,4752,4755,4758],{"stem":4750,"title":4751,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fvolumen-kugel","Volumen einer Kugel",{"stem":4753,"title":4754,"category":368},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fdreifachintegral","Dreifachintegral",{"stem":4756,"title":4757,"category":368},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Ftransformationssatz","Transformationssatz",{"stem":4759,"title":4760,"category":368},"de\u002Fformulas\u002Fmath\u002Fmultivariable-calculus\u002Fzylinderkoordinaten","Zylinderkoordinaten",1793824267088]