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v",[573,1034,1036,1147,1170],{"className":1035,"ariaHidden":608},[607],[573,1037,1039,1042,1135,1138,1141,1144],{"className":1038},[612],[573,1040],{"className":1041,"style":617},[616],[573,1043,1045,1049],{"className":1044},[621],[573,1046,1005],{"className":1047,"style":1048},[621],"margin-right:0.0556em;",[573,1050,1053],{"className":1051},[1052],"msupsub",[573,1054,1057,1126],{"className":1055},[779,1056],"vlist-t2",[573,1058,1060,1121],{"className":1059},[783],[573,1061,1064],{"className":1062,"style":1063},[787],"height:0.3448em;",[573,1065,1067,1071],{"style":1066},"top:-2.545em;margin-left:-0.0556em;margin-right:0.05em;",[573,1068],{"className":1069,"style":1070},[794],"height:2.7em;",[573,1072,1078],{"className":1073},[1074,1075,1076,1077],"sizing","reset-size6","size3","mtight",[573,1079,1081],{"className":1080},[621,1077],[573,1082,1084],{"className":1083},[621,775,1077],[573,1085,1087],{"className":1086},[779],[573,1088,1090],{"className":1089},[783],[573,1091,1094,1105],{"className":1092,"style":1093},[787],"height:0.714em;",[573,1095,1097,1101],{"style":1096},"top:-2.714em;",[573,1098],{"className":1099,"style":1100},[794],"height:2.714em;",[573,1102,1010],{"className":1103,"style":1104},[621,625,1077],"margin-right:0.0359em;",[573,1106,1107,1110],{"style":1096},[573,1108],{"className":1109,"style":1100},[794],[573,1111,1114],{"className":1112,"style":1113},[808],"left:-0.2077em;",[573,1115,1117],{"className":1116,"style":814},[813,1077],[816,1118,1119],{"xmlns":818,"width":819,"height":820,"style":821,"viewBox":822,"preserveAspectRatio":823},[825,1120],{"d":827},[573,1122,1125],{"className":1123},[1124],"vlist-s","​",[573,1127,1129],{"className":1128},[783],[573,1130,1133],{"className":1131,"style":1132},[787],"height:0.155em;",[573,1134],{},[573,1136,598],{"className":1137,"style":626},[621,625],[573,1139],{"className":1140,"style":716},[715],[573,1142,761],{"className":1143},[720],[573,1145],{"className":1146,"style":716},[715],[573,1148,1150,1153,1156,1159,1163,1167],{"className":1149},[612],[573,1151],{"className":1152,"style":617},[616],[573,1154,595],{"className":1155},[621],[573,1157,598],{"className":1158,"style":626},[621,625],[573,1160],{"className":1161,"style":1162},[715],"margin-right:0.2222em;",[573,1164,1023],{"className":1165},[1166],"mbin",[573,1168],{"className":1169,"style":1162},[715],[573,1171,1173,1176],{"className":1172},[612],[573,1174],{"className":1175,"style":1093},[616],[573,1177,1179],{"className":1178},[621,775],[573,1180,1182],{"className":1181},[779],[573,1183,1185],{"className":1184},[783],[573,1186,1188,1196],{"className":1187,"style":1093},[787],[573,1189,1190,1193],{"style":790},[573,1191],{"className":1192,"style":795},[794],[573,1194,1010],{"className":1195,"style":1104},[621,625],[573,1197,1198,1201],{"style":790},[573,1199],{"className":1200,"style":795},[794],[573,1202,1204],{"className":1203,"style":1113},[808],[573,1205,1207],{"className":1206,"style":814},[813],[816,1208,1209],{"xmlns":818,"width":819,"height":820,"style":821,"viewBox":822,"preserveAspectRatio":823},[825,1210],{"d":827},". Punkte mit ",[573,1213,1215,1239],{"className":1214},[576],[573,1216,1218],{"className":1217},[580],[582,1219,1220],{"xmlns":584},[586,1221,1222,1236],{},[589,1223,1224,1226,1228,1230],{},[592,1225,595],{"mathvariant":594},[592,1227,598],{},[681,1229,761],{},[685,1231,1232,1234],{"accent":608},[688,1233,690],{},[681,1235,693],{},[600,1237,1238],{"encoding":602},"\\nabla f = \\vec{0}",[573,1240,1242,1263],{"className":1241,"ariaHidden":608},[607],[573,1243,1245,1248,1251,1254,1257,1260],{"className":1244},[612],[573,1246],{"className":1247,"style":617},[616],[573,1249,595],{"className":1250},[621],[573,1252,598],{"className":1253,"style":626},[621,625],[573,1255],{"className":1256,"style":716},[715],[573,1258,761],{"className":1259},[720],[573,1261],{"className":1262,"style":716},[715],[573,1264,1266,1269],{"className":1265},[612],[573,1267],{"className":1268,"style":771},[616],[573,1270,1272],{"className":1271},[621,775],[573,1273,1275],{"className":1274},[779],[573,1276,1278],{"className":1277},[783],[573,1279,1281,1289],{"className":1280,"style":771},[787],[573,1282,1283,1286],{"style":790},[573,1284],{"className":1285,"style":795},[794],[573,1287,690],{"className":1288},[621],[573,1290,1291,1294],{"style":801},[573,1292],{"className":1293,"style":795},[794],[573,1295,1297],{"className":1296,"style":809},[808],[573,1298,1300],{"className":1299,"style":814},[813],[816,1301,1302],{"xmlns":818,"width":819,"height":820,"style":821,"viewBox":822,"preserveAspectRatio":823},[825,1303],{"d":827}," heißen ",[569,1306,1307],{},"kritische"," oder ",[569,1310,1311],{},"stationäre Punkte","; dort können Extrema liegen.",[560,1314,1316],{"id":1315},"beispiele","Beispiele",[565,1318,1319,1320,1504],{},"Für ",[573,1321,1323,1375],{"className":1322},[576],[573,1324,1326],{"className":1325},[580],[582,1327,1328],{"xmlns":584},[586,1329,1330,1372],{},[589,1331,1332,1334,1338,1341,1344,1347,1350,1352,1360,1363,1366],{},[592,1333,598],{},[681,1335,1337],{"stretchy":1336},"false","(",[592,1339,1340],{},"x",[681,1342,1343],{"separator":608},",",[592,1345,1346],{},"y",[681,1348,1349],{"stretchy":1336},")",[681,1351,761],{},[1353,1354,1355,1357],"msup",{},[592,1356,1340],{},[688,1358,1359],{},"2",[681,1361,1362],{},"+",[688,1364,1365],{},"3",[1353,1367,1368,1370],{},[592,1369,1346],{},[688,1371,1359],{},[600,1373,1374],{"encoding":602},"f(x, y) = x^2 + 3y^2",[573,1376,1378,1418,1465],{"className":1377,"ariaHidden":608},[607],[573,1379,1381,1384,1387,1391,1394,1398,1402,1405,1409,1412,1415],{"className":1380},[612],[573,1382],{"className":1383,"style":933},[616],[573,1385,598],{"className":1386,"style":626},[621,625],[573,1388,1337],{"className":1389},[1390],"mopen",[573,1392,1340],{"className":1393},[621,625],[573,1395,1343],{"className":1396},[1397],"mpunct",[573,1399],{"className":1400,"style":1401},[715],"margin-right:0.1667em;",[573,1403,1346],{"className":1404,"style":1104},[621,625],[573,1406,1349],{"className":1407},[1408],"mclose",[573,1410],{"className":1411,"style":716},[715],[573,1413,761],{"className":1414},[720],[573,1416],{"className":1417,"style":716},[715],[573,1419,1421,1425,1456,1459,1462],{"className":1420},[612],[573,1422],{"className":1423,"style":1424},[616],"height:0.8974em;vertical-align:-0.0833em;",[573,1426,1428,1431],{"className":1427},[621],[573,1429,1340],{"className":1430},[621,625],[573,1432,1434],{"className":1433},[1052],[573,1435,1437],{"className":1436},[779],[573,1438,1440],{"className":1439},[783],[573,1441,1444],{"className":1442,"style":1443},[787],"height:0.8141em;",[573,1445,1447,1450],{"style":1446},"top:-3.063em;margin-right:0.05em;",[573,1448],{"className":1449,"style":1070},[794],[573,1451,1453],{"className":1452},[1074,1075,1076,1077],[573,1454,1359],{"className":1455},[621,1077],[573,1457],{"className":1458,"style":1162},[715],[573,1460,1362],{"className":1461},[1166],[573,1463],{"className":1464,"style":1162},[715],[573,1466,1468,1472,1475],{"className":1467},[612],[573,1469],{"className":1470,"style":1471},[616],"height:1.0085em;vertical-align:-0.1944em;",[573,1473,1365],{"className":1474},[621],[573,1476,1478,1481],{"className":1477},[621],[573,1479,1346],{"className":1480,"style":1104},[621,625],[573,1482,1484],{"className":1483},[1052],[573,1485,1487],{"className":1486},[779],[573,1488,1490],{"className":1489},[783],[573,1491,1493],{"className":1492,"style":1443},[787],[573,1494,1495,1498],{"style":1446},[573,1496],{"className":1497,"style":1070},[794],[573,1499,1501],{"className":1500},[1074,1075,1076,1077],[573,1502,1359],{"className":1503},[621,1077]," ist",[573,1506,1509],{"className":1507},[1508],"katex-display",[573,1510,1512,1628],{"className":1511},[576],[573,1513,1515],{"className":1514},[580],[582,1516,1518],{"xmlns":584,"display":1517},"block",[586,1519,1520,1625],{},[589,1521,1522,1524,1526,1528,1530,1532,1534,1536,1538,1578,1580,1583,1585,1587,1589,1591,1593,1596,1598,1600],{},[592,1523,595],{"mathvariant":594},[592,1525,598],{},[681,1527,1337],{"stretchy":1336},[592,1529,1340],{},[681,1531,1343],{"separator":608},[592,1533,1346],{},[681,1535,1349],{"stretchy":1336},[681,1537,761],{},[589,1539,1540,1542,1576],{},[681,1541,1337],{"fence":608},[1543,1544,1548,1563],"mtable",{"rowspacing":1545,"columnalign":1546,"columnspacing":1547},"0.16em","center","1em",[1549,1550,1551],"mtr",{},[1552,1553,1554],"mtd",{},[1555,1556,1557],"mstyle",{"scriptlevel":690,"displaystyle":1336},[589,1558,1559,1561],{},[688,1560,1359],{},[592,1562,1340],{},[1549,1564,1565],{},[1552,1566,1567],{},[1555,1568,1569],{"scriptlevel":690,"displaystyle":1336},[589,1570,1571,1574],{},[688,1572,1573],{},"6",[592,1575,1346],{},[681,1577,1349],{"fence":608},[681,1579,1343],{"separator":608},[715,1581],{"width":1582},"2em",[592,1584,595],{"mathvariant":594},[592,1586,598],{},[681,1588,1337],{"stretchy":1336},[688,1590,1359],{},[681,1592,1343],{"separator":608},[688,1594,1595],{},"1",[681,1597,1349],{"stretchy":1336},[681,1599,761],{},[589,1601,1602,1604,1623],{},[681,1603,1337],{"fence":608},[1543,1605,1606,1615],{"rowspacing":1545,"columnalign":1546,"columnspacing":1547},[1549,1607,1608],{},[1552,1609,1610],{},[1555,1611,1612],{"scriptlevel":690,"displaystyle":1336},[688,1613,1614],{},"4",[1549,1616,1617],{},[1552,1618,1619],{},[1555,1620,1621],{"scriptlevel":690,"displaystyle":1336},[688,1622,1573],{},[681,1624,1349],{"fence":608},[600,1626,1627],{"encoding":602},"\\nabla f(x, y) = \\begin{pmatrix} 2x \\\\ 6y \\end{pmatrix}, \\qquad \\nabla f(2, 1) = \\begin{pmatrix} 4 \\\\ 6 \\end{pmatrix}",[573,1629,1631,1670,1804],{"className":1630,"ariaHidden":608},[607],[573,1632,1634,1637,1640,1643,1646,1649,1652,1655,1658,1661,1664,1667],{"className":1633},[612],[573,1635],{"className":1636,"style":933},[616],[573,1638,595],{"className":1639},[621],[573,1641,598],{"className":1642,"style":626},[621,625],[573,1644,1337],{"className":1645},[1390],[573,1647,1340],{"className":1648},[621,625],[573,1650,1343],{"className":1651},[1397],[573,1653],{"className":1654,"style":1401},[715],[573,1656,1346],{"className":1657,"style":1104},[621,625],[573,1659,1349],{"className":1660},[1408],[573,1662],{"className":1663,"style":716},[715],[573,1665,761],{"className":1666},[720],[573,1668],{"className":1669,"style":716},[715],[573,1671,1673,1677,1758,1761,1764,1768,1771,1774,1777,1780,1783,1786,1789,1792,1795,1798,1801],{"className":1672},[612],[573,1674],{"className":1675,"style":1676},[616],"height:2.4em;vertical-align:-0.95em;",[573,1678,1681,1690,1752],{"className":1679},[1680],"minner",[573,1682,1686],{"className":1683,"style":1685},[1390,1684],"delimcenter","top:0em;",[573,1687,1337],{"className":1688},[1689,1076],"delimsizing",[573,1691,1693],{"className":1692},[621],[573,1694,1696],{"className":1695},[1543],[573,1697,1700],{"className":1698},[1699],"col-align-c",[573,1701,1703,1743],{"className":1702},[779,1056],[573,1704,1706,1740],{"className":1705},[783],[573,1707,1710,1725],{"className":1708,"style":1709},[787],"height:1.45em;",[573,1711,1713,1716],{"style":1712},"top:-3.61em;",[573,1714],{"className":1715,"style":795},[794],[573,1717,1719,1722],{"className":1718},[621],[573,1720,1359],{"className":1721},[621],[573,1723,1340],{"className":1724},[621,625],[573,1726,1728,1731],{"style":1727},"top:-2.41em;",[573,1729],{"className":1730,"style":795},[794],[573,1732,1734,1737],{"className":1733},[621],[573,1735,1573],{"className":1736},[621],[573,1738,1346],{"className":1739,"style":1104},[621,625],[573,1741,1125],{"className":1742},[1124],[573,1744,1746],{"className":1745},[783],[573,1747,1750],{"className":1748,"style":1749},[787],"height:0.95em;",[573,1751],{},[573,1753,1755],{"className":1754,"style":1685},[1408,1684],[573,1756,1349],{"className":1757},[1689,1076],[573,1759],{"className":1760,"style":1401},[715],[573,1762,1343],{"className":1763},[1397],[573,1765],{"className":1766,"style":1767},[715],"margin-right:2em;",[573,1769],{"className":1770,"style":1401},[715],[573,1772,595],{"className":1773},[621],[573,1775,598],{"className":1776,"style":626},[621,625],[573,1778,1337],{"className":1779},[1390],[573,1781,1359],{"className":1782},[621],[573,1784,1343],{"className":1785},[1397],[573,1787],{"className":1788,"style":1401},[715],[573,1790,1595],{"className":1791},[621],[573,1793,1349],{"className":1794},[1408],[573,1796],{"className":1797,"style":716},[715],[573,1799,761],{"className":1800},[720],[573,1802],{"className":1803,"style":716},[715],[573,1805,1807,1810],{"className":1806},[612],[573,1808],{"className":1809,"style":1676},[616],[573,1811,1813,1819,1870],{"className":1812},[1680],[573,1814,1816],{"className":1815,"style":1685},[1390,1684],[573,1817,1337],{"className":1818},[1689,1076],[573,1820,1822],{"className":1821},[621],[573,1823,1825],{"className":1824},[1543],[573,1826,1828],{"className":1827},[1699],[573,1829,1831,1862],{"className":1830},[779,1056],[573,1832,1834,1859],{"className":1833},[783],[573,1835,1837,1848],{"className":1836,"style":1709},[787],[573,1838,1839,1842],{"style":1712},[573,1840],{"className":1841,"style":795},[794],[573,1843,1845],{"className":1844},[621],[573,1846,1614],{"className":1847},[621],[573,1849,1850,1853],{"style":1727},[573,1851],{"className":1852,"style":795},[794],[573,1854,1856],{"className":1855},[621],[573,1857,1573],{"className":1858},[621],[573,1860,1125],{"className":1861},[1124],[573,1863,1865],{"className":1864},[783],[573,1866,1868],{"className":1867,"style":1749},[787],[573,1869],{},[573,1871,1873],{"className":1872,"style":1685},[1408,1684],[573,1874,1349],{"className":1875},[1689,1076],[565,1877,1878,1879,1931,1932,1960,1961,2013,2014,631],{},"Im Punkt ",[573,1880,1882,1904],{"className":1881},[576],[573,1883,1885],{"className":1884},[580],[582,1886,1887],{"xmlns":584},[586,1888,1889,1901],{},[589,1890,1891,1893,1895,1897,1899],{},[681,1892,1337],{"stretchy":1336},[688,1894,1359],{},[681,1896,1343],{"separator":608},[688,1898,1595],{},[681,1900,1349],{"stretchy":1336},[600,1902,1903],{"encoding":602},"(2, 1)",[573,1905,1907],{"className":1906,"ariaHidden":608},[607],[573,1908,1910,1913,1916,1919,1922,1925,1928],{"className":1909},[612],[573,1911],{"className":1912,"style":933},[616],[573,1914,1337],{"className":1915},[1390],[573,1917,1359],{"className":1918},[621],[573,1920,1343],{"className":1921},[1397],[573,1923],{"className":1924,"style":1401},[715],[573,1926,1595],{"className":1927},[621],[573,1929,1349],{"className":1930},[1408]," steigt ",[573,1933,1935,1948],{"className":1934},[576],[573,1936,1938],{"className":1937},[580],[582,1939,1940],{"xmlns":584},[586,1941,1942,1946],{},[589,1943,1944],{},[592,1945,598],{},[600,1947,598],{"encoding":602},[573,1949,1951],{"className":1950,"ariaHidden":608},[607],[573,1952,1954,1957],{"className":1953},[612],[573,1955],{"className":1956,"style":617},[616],[573,1958,598],{"className":1959,"style":626},[621,625]," also am stärksten in Richtung ",[573,1962,1964,1986],{"className":1963},[576],[573,1965,1967],{"className":1966},[580],[582,1968,1969],{"xmlns":584},[586,1970,1971,1983],{},[589,1972,1973,1975,1977,1979,1981],{},[681,1974,1337],{"stretchy":1336},[688,1976,1614],{},[681,1978,1343],{"separator":608},[688,1980,1573],{},[681,1982,1349],{"stretchy":1336},[600,1984,1985],{"encoding":602},"(4, 6)",[573,1987,1989],{"className":1988,"ariaHidden":608},[607],[573,1990,1992,1995,1998,2001,2004,2007,2010],{"className":1991},[612],[573,1993],{"className":1994,"style":933},[616],[573,1996,1337],{"className":1997},[1390],[573,1999,1614],{"className":2000},[621],[573,2002,1343],{"className":2003},[1397],[573,2005],{"className":2006,"style":1401},[715],[573,2008,1573],{"className":2009},[621],[573,2011,1349],{"className":2012},[1408],", und zwar mit der Steigung ",[573,2015,2017,2063],{"className":2016},[576],[573,2018,2020],{"className":2019},[580],[582,2021,2022],{"xmlns":584},[586,2023,2024,2060],{},[589,2025,2026,2045,2047,2049,2054,2057],{},[2027,2028,2029],"msqrt",{},[589,2030,2031,2037,2039],{},[1353,2032,2033,2035],{},[688,2034,1614],{},[688,2036,1359],{},[681,2038,1362],{},[1353,2040,2041,2043],{},[688,2042,1573],{},[688,2044,1359],{},[681,2046,761],{},[688,2048,1359],{},[2027,2050,2051],{},[688,2052,2053],{},"13",[681,2055,2056],{},"≈",[688,2058,2059],{},"7,21",[600,2061,2062],{"encoding":602},"\\sqrt{4^2 + 6^2} = 2\\sqrt{13} \\approx 7{,}21",[573,2064,2066,2208,2277],{"className":2065,"ariaHidden":608},[607],[573,2067,2069,2073,2199,2202,2205],{"className":2068},[612],[573,2070],{"className":2071,"style":2072},[616],"height:1.04em;vertical-align:-0.1266em;",[573,2074,2077],{"className":2075},[621,2076],"sqrt",[573,2078,2080,2190],{"className":2079},[779,1056],[573,2081,2083,2187],{"className":2082},[783],[573,2084,2087,2167],{"className":2085,"style":2086},[787],"height:0.9134em;",[573,2088,2091,2094],{"className":2089,"style":790},[2090],"svg-align",[573,2092],{"className":2093,"style":795},[794],[573,2095,2098,2129,2132,2135,2138],{"className":2096,"style":2097},[621],"padding-left:0.833em;",[573,2099,2101,2104],{"className":2100},[621],[573,2102,1614],{"className":2103},[621],[573,2105,2107],{"className":2106},[1052],[573,2108,2110],{"className":2109},[779],[573,2111,2113],{"className":2112},[783],[573,2114,2117],{"className":2115,"style":2116},[787],"height:0.7401em;",[573,2118,2120,2123],{"style":2119},"top:-2.989em;margin-right:0.05em;",[573,2121],{"className":2122,"style":1070},[794],[573,2124,2126],{"className":2125},[1074,1075,1076,1077],[573,2127,1359],{"className":2128},[621,1077],[573,2130],{"className":2131,"style":1162},[715],[573,2133,1362],{"className":2134},[1166],[573,2136],{"className":2137,"style":1162},[715],[573,2139,2141,2144],{"className":2140},[621],[573,2142,1573],{"className":2143},[621],[573,2145,2147],{"className":2146},[1052],[573,2148,2150],{"className":2149},[779],[573,2151,2153],{"className":2152},[783],[573,2154,2156],{"className":2155,"style":2116},[787],[573,2157,2158,2161],{"style":2119},[573,2159],{"className":2160,"style":1070},[794],[573,2162,2164],{"className":2163},[1074,1075,1076,1077],[573,2165,1359],{"className":2166},[621,1077],[573,2168,2170,2173],{"style":2169},"top:-2.8734em;",[573,2171],{"className":2172,"style":795},[794],[573,2174,2178],{"className":2175,"style":2177},[2176],"hide-tail","min-width:0.853em;height:1.08em;",[816,2179,2184],{"xmlns":818,"width":2180,"height":2181,"viewBox":2182,"preserveAspectRatio":2183},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[825,2185],{"d":2186},"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",[573,2188,1125],{"className":2189},[1124],[573,2191,2193],{"className":2192},[783],[573,2194,2197],{"className":2195,"style":2196},[787],"height:0.1266em;",[573,2198],{},[573,2200],{"className":2201,"style":716},[715],[573,2203,761],{"className":2204},[720],[573,2206],{"className":2207,"style":716},[715],[573,2209,2211,2215,2218,2268,2271,2274],{"className":2210},[612],[573,2212],{"className":2213,"style":2214},[616],"height:1.04em;vertical-align:-0.1328em;",[573,2216,1359],{"className":2217},[621],[573,2219,2221],{"className":2220},[621,2076],[573,2222,2224,2259],{"className":2223},[779,1056],[573,2225,2227,2256],{"className":2226},[783],[573,2228,2231,2243],{"className":2229,"style":2230},[787],"height:0.9072em;",[573,2232,2234,2237],{"className":2233,"style":790},[2090],[573,2235],{"className":2236,"style":795},[794],[573,2238,2240],{"className":2239,"style":2097},[621],[573,2241,2053],{"className":2242},[621],[573,2244,2246,2249],{"style":2245},"top:-2.8672em;",[573,2247],{"className":2248,"style":795},[794],[573,2250,2252],{"className":2251,"style":2177},[2176],[816,2253,2254],{"xmlns":818,"width":2180,"height":2181,"viewBox":2182,"preserveAspectRatio":2183},[825,2255],{"d":2186},[573,2257,1125],{"className":2258},[1124],[573,2260,2262],{"className":2261},[783],[573,2263,2266],{"className":2264,"style":2265},[787],"height:0.1328em;",[573,2267],{},[573,2269],{"className":2270,"style":716},[715],[573,2272,2056],{"className":2273},[720],[573,2275],{"className":2276,"style":716},[715],[573,2278,2280,2284,2288,2294],{"className":2279},[612],[573,2281],{"className":2282,"style":2283},[616],"height:0.8389em;vertical-align:-0.1944em;",[573,2285,2287],{"className":2286},[621],"7",[573,2289,2291],{"className":2290},[621],[573,2292,1343],{"className":2293},[1397],[573,2295,2297],{"className":2296},[621],"21",[565,2299,2300,2301,2440,2441,2499,2500,631],{},"Der Punkt liegt auf der Höhenlinie ",[573,2302,2304,2336],{"className":2303},[576],[573,2305,2307],{"className":2306},[580],[582,2308,2309],{"xmlns":584},[586,2310,2311,2333],{},[589,2312,2313,2319,2321,2323,2329,2331],{},[1353,2314,2315,2317],{},[592,2316,1340],{},[688,2318,1359],{},[681,2320,1362],{},[688,2322,1365],{},[1353,2324,2325,2327],{},[592,2326,1346],{},[688,2328,1359],{},[681,2330,761],{},[688,2332,2287],{},[600,2334,2335],{"encoding":602},"x^2 + 3y^2 = 7",[573,2337,2339,2383,2430],{"className":2338,"ariaHidden":608},[607],[573,2340,2342,2345,2374,2377,2380],{"className":2341},[612],[573,2343],{"className":2344,"style":1424},[616],[573,2346,2348,2351],{"className":2347},[621],[573,2349,1340],{"className":2350},[621,625],[573,2352,2354],{"className":2353},[1052],[573,2355,2357],{"className":2356},[779],[573,2358,2360],{"className":2359},[783],[573,2361,2363],{"className":2362,"style":1443},[787],[573,2364,2365,2368],{"style":1446},[573,2366],{"className":2367,"style":1070},[794],[573,2369,2371],{"className":2370},[1074,1075,1076,1077],[573,2372,1359],{"className":2373},[621,1077],[573,2375],{"className":2376,"style":1162},[715],[573,2378,1362],{"className":2379},[1166],[573,2381],{"className":2382,"style":1162},[715],[573,2384,2386,2389,2392,2421,2424,2427],{"className":2385},[612],[573,2387],{"className":2388,"style":1471},[616],[573,2390,1365],{"className":2391},[621],[573,2393,2395,2398],{"className":2394},[621],[573,2396,1346],{"className":2397,"style":1104},[621,625],[573,2399,2401],{"className":2400},[1052],[573,2402,2404],{"className":2403},[779],[573,2405,2407],{"className":2406},[783],[573,2408,2410],{"className":2409,"style":1443},[787],[573,2411,2412,2415],{"style":1446},[573,2413],{"className":2414,"style":1070},[794],[573,2416,2418],{"className":2417},[1074,1075,1076,1077],[573,2419,1359],{"className":2420},[621,1077],[573,2422],{"className":2423,"style":716},[715],[573,2425,761],{"className":2426},[720],[573,2428],{"className":2429,"style":716},[715],[573,2431,2433,2437],{"className":2432},[612],[573,2434],{"className":2435,"style":2436},[616],"height:0.6444em;",[573,2438,2287],{"className":2439},[621],". Deren Tangentenrichtung ",[573,2442,2444,2469],{"className":2443},[576],[573,2445,2447],{"className":2446},[580],[582,2448,2449],{"xmlns":584},[586,2450,2451,2466],{},[589,2452,2453,2455,2457,2459,2462,2464],{},[681,2454,1337],{"stretchy":1336},[688,2456,1365],{},[681,2458,1343],{"separator":608},[681,2460,2461],{},"−",[688,2463,1359],{},[681,2465,1349],{"stretchy":1336},[600,2467,2468],{"encoding":602},"(3, -2)",[573,2470,2472],{"className":2471,"ariaHidden":608},[607],[573,2473,2475,2478,2481,2484,2487,2490,2493,2496],{"className":2474},[612],[573,2476],{"className":2477,"style":933},[616],[573,2479,1337],{"className":2480},[1390],[573,2482,1365],{"className":2483},[621],[573,2485,1343],{"className":2486},[1397],[573,2488],{"className":2489,"style":1401},[715],[573,2491,2461],{"className":2492},[621],[573,2494,1359],{"className":2495},[621],[573,2497,1349],{"className":2498},[1408]," steht senkrecht auf dem Gradienten: ",[573,2501,2503,2539],{"className":2502},[576],[573,2504,2506],{"className":2505},[580],[582,2507,2508],{"xmlns":584},[586,2509,2510,2536],{},[589,2511,2512,2514,2516,2518,2520,2522,2524,2526,2528,2530,2532,2534],{},[688,2513,1614],{},[681,2515,1023],{},[688,2517,1365],{},[681,2519,1362],{},[688,2521,1573],{},[681,2523,1023],{},[681,2525,1337],{"stretchy":1336},[681,2527,2461],{},[688,2529,1359],{},[681,2531,1349],{"stretchy":1336},[681,2533,761],{},[688,2535,690],{},[600,2537,2538],{"encoding":602},"4 \\cdot 3 + 6 \\cdot (-2) = 0",[573,2540,2542,2560,2579,2597,2624],{"className":2541,"ariaHidden":608},[607],[573,2543,2545,2548,2551,2554,2557],{"className":2544},[612],[573,2546],{"className":2547,"style":2436},[616],[573,2549,1614],{"className":2550},[621],[573,2552],{"className":2553,"style":1162},[715],[573,2555,1023],{"className":2556},[1166],[573,2558],{"className":2559,"style":1162},[715],[573,2561,2563,2567,2570,2573,2576],{"className":2562},[612],[573,2564],{"className":2565,"style":2566},[616],"height:0.7278em;vertical-align:-0.0833em;",[573,2568,1365],{"className":2569},[621],[573,2571],{"className":2572,"style":1162},[715],[573,2574,1362],{"className":2575},[1166],[573,2577],{"className":2578,"style":1162},[715],[573,2580,2582,2585,2588,2591,2594],{"className":2581},[612],[573,2583],{"className":2584,"style":2436},[616],[573,2586,1573],{"className":2587},[621],[573,2589],{"className":2590,"style":1162},[715],[573,2592,1023],{"className":2593},[1166],[573,2595],{"className":2596,"style":1162},[715],[573,2598,2600,2603,2606,2609,2612,2615,2618,2621],{"className":2599},[612],[573,2601],{"className":2602,"style":933},[616],[573,2604,1337],{"className":2605},[1390],[573,2607,2461],{"className":2608},[621],[573,2610,1359],{"className":2611},[621],[573,2613,1349],{"className":2614},[1408],[573,2616],{"className":2617,"style":716},[715],[573,2619,761],{"className":2620},[720],[573,2622],{"className":2623,"style":716},[715],[573,2625,2627,2630],{"className":2626},[612],[573,2628],{"className":2629,"style":2436},[616],[573,2631,690],{"className":2632},[621],[565,2634,1319,2635,2740,2741,2882,2883,2948,2949,631],{},[573,2636,2638,2675],{"className":2637},[576],[573,2639,2641],{"className":2640},[580],[582,2642,2643],{"xmlns":584},[586,2644,2645,2672],{},[589,2646,2647,2649,2651,2653,2655,2657,2659,2662,2664,2666,2668,2670],{},[592,2648,598],{},[681,2650,1337],{"stretchy":1336},[592,2652,1340],{},[681,2654,1343],{"separator":608},[592,2656,1346],{},[681,2658,1343],{"separator":608},[592,2660,2661],{},"z",[681,2663,1349],{"stretchy":1336},[681,2665,761],{},[592,2667,1340],{},[592,2669,1346],{},[592,2671,2661],{},[600,2673,2674],{"encoding":602},"f(x, y, z) = xyz",[573,2676,2678,2724],{"className":2677,"ariaHidden":608},[607],[573,2679,2681,2684,2687,2690,2693,2696,2699,2702,2705,2708,2712,2715,2718,2721],{"className":2680},[612],[573,2682],{"className":2683,"style":933},[616],[573,2685,598],{"className":2686,"style":626},[621,625],[573,2688,1337],{"className":2689},[1390],[573,2691,1340],{"className":2692},[621,625],[573,2694,1343],{"className":2695},[1397],[573,2697],{"className":2698,"style":1401},[715],[573,2700,1346],{"className":2701,"style":1104},[621,625],[573,2703,1343],{"className":2704},[1397],[573,2706],{"className":2707,"style":1401},[715],[573,2709,2661],{"className":2710,"style":2711},[621,625],"margin-right:0.044em;",[573,2713,1349],{"className":2714},[1408],[573,2716],{"className":2717,"style":716},[715],[573,2719,761],{"className":2720},[720],[573,2722],{"className":2723,"style":716},[715],[573,2725,2727,2731,2734,2737],{"className":2726},[612],[573,2728],{"className":2729,"style":2730},[616],"height:0.625em;vertical-align:-0.1944em;",[573,2732,1340],{"className":2733},[621,625],[573,2735,1346],{"className":2736,"style":1104},[621,625],[573,2738,2661],{"className":2739,"style":2711},[621,625]," ist ",[573,2742,2744,2787],{"className":2743},[576],[573,2745,2747],{"className":2746},[580],[582,2748,2749],{"xmlns":584},[586,2750,2751,2784],{},[589,2752,2753,2755,2757,2759,2761,2763,2765,2767,2769,2771,2773,2775,2777],{},[592,2754,595],{"mathvariant":594},[592,2756,598],{},[681,2758,761],{},[681,2760,1337],{"stretchy":1336},[592,2762,1346],{},[592,2764,2661],{},[681,2766,1343],{"separator":608},[592,2768,1340],{},[592,2770,2661],{},[681,2772,1343],{"separator":608},[592,2774,1340],{},[592,2776,1346],{},[1353,2778,2779,2781],{},[681,2780,1349],{"stretchy":1336},[592,2782,2783],{},"T",[600,2785,2786],{"encoding":602},"\\nabla f = (yz, xz, xy)^T",[573,2788,2790,2811],{"className":2789,"ariaHidden":608},[607],[573,2791,2793,2796,2799,2802,2805,2808],{"className":2792},[612],[573,2794],{"className":2795,"style":617},[616],[573,2797,595],{"className":2798},[621],[573,2800,598],{"className":2801,"style":626},[621,625],[573,2803],{"className":2804,"style":716},[715],[573,2806,761],{"className":2807},[720],[573,2809],{"className":2810,"style":716},[715],[573,2812,2814,2818,2821,2824,2827,2830,2833,2836,2839,2842,2845,2848,2851],{"className":2813},[612],[573,2815],{"className":2816,"style":2817},[616],"height:1.0913em;vertical-align:-0.25em;",[573,2819,1337],{"className":2820},[1390],[573,2822,1346],{"className":2823,"style":1104},[621,625],[573,2825,2661],{"className":2826,"style":2711},[621,625],[573,2828,1343],{"className":2829},[1397],[573,2831],{"className":2832,"style":1401},[715],[573,2834,1340],{"className":2835},[621,625],[573,2837,2661],{"className":2838,"style":2711},[621,625],[573,2840,1343],{"className":2841},[1397],[573,2843],{"className":2844,"style":1401},[715],[573,2846,1340],{"className":2847},[621,625],[573,2849,1346],{"className":2850,"style":1104},[621,625],[573,2852,2854,2857],{"className":2853},[1408],[573,2855,1349],{"className":2856},[1408],[573,2858,2860],{"className":2859},[1052],[573,2861,2863],{"className":2862},[779],[573,2864,2866],{"className":2865},[783],[573,2867,2870],{"className":2868,"style":2869},[787],"height:0.8413em;",[573,2871,2872,2875],{"style":1446},[573,2873],{"className":2874,"style":1070},[794],[573,2876,2878],{"className":2877},[1074,1075,1076,1077],[573,2879,2783],{"className":2880,"style":2881},[621,625,1077],"margin-right:0.1389em;",", im Punkt ",[573,2884,2886,2912],{"className":2885},[576],[573,2887,2889],{"className":2888},[580],[582,2890,2891],{"xmlns":584},[586,2892,2893,2909],{},[589,2894,2895,2897,2899,2901,2903,2905,2907],{},[681,2896,1337],{"stretchy":1336},[688,2898,1595],{},[681,2900,1343],{"separator":608},[688,2902,1359],{},[681,2904,1343],{"separator":608},[688,2906,1365],{},[681,2908,1349],{"stretchy":1336},[600,2910,2911],{"encoding":602},"(1, 2, 3)",[573,2913,2915],{"className":2914,"ariaHidden":608},[607],[573,2916,2918,2921,2924,2927,2930,2933,2936,2939,2942,2945],{"className":2917},[612],[573,2919],{"className":2920,"style":933},[616],[573,2922,1337],{"className":2923},[1390],[573,2925,1595],{"className":2926},[621],[573,2928,1343],{"className":2929},[1397],[573,2931],{"className":2932,"style":1401},[715],[573,2934,1359],{"className":2935},[621],[573,2937,1343],{"className":2938},[1397],[573,2940],{"className":2941,"style":1401},[715],[573,2943,1365],{"className":2944},[621],[573,2946,1349],{"className":2947},[1408]," also ",[573,2950,2952,2982],{"className":2951},[576],[573,2953,2955],{"className":2954},[580],[582,2956,2957],{"xmlns":584},[586,2958,2959,2979],{},[589,2960,2961,2963,2965,2967,2969,2971,2973],{},[681,2962,1337],{"stretchy":1336},[688,2964,1573],{},[681,2966,1343],{"separator":608},[688,2968,1365],{},[681,2970,1343],{"separator":608},[688,2972,1359],{},[1353,2974,2975,2977],{},[681,2976,1349],{"stretchy":1336},[592,2978,2783],{},[600,2980,2981],{"encoding":602},"(6, 3, 2)^T",[573,2983,2985],{"className":2984,"ariaHidden":608},[607],[573,2986,2988,2991,2994,2997,3000,3003,3006,3009,3012,3015],{"className":2987},[612],[573,2989],{"className":2990,"style":2817},[616],[573,2992,1337],{"className":2993},[1390],[573,2995,1573],{"className":2996},[621],[573,2998,1343],{"className":2999},[1397],[573,3001],{"className":3002,"style":1401},[715],[573,3004,1365],{"className":3005},[621],[573,3007,1343],{"className":3008},[1397],[573,3010],{"className":3011,"style":1401},[715],[573,3013,1359],{"className":3014},[621],[573,3016,3018,3021],{"className":3017},[1408],[573,3019,1349],{"className":3020},[1408],[573,3022,3024],{"className":3023},[1052],[573,3025,3027],{"className":3026},[779],[573,3028,3030],{"className":3029},[783],[573,3031,3033],{"className":3032,"style":2869},[787],[573,3034,3035,3038],{"style":1446},[573,3036],{"className":3037,"style":1070},[794],[573,3039,3041],{"className":3040},[1074,1075,1076,1077],[573,3042,2783],{"className":3043,"style":2881},[621,625,1077],[560,3045,3047],{"id":3046},"typische-fehler","Typische Fehler",[830,3049,3050,3082,3085],{},[833,3051,3052,3053,3081],{},"Den Gradienten als Zahl angeben. Er ist ein Vektor mit so vielen Komponenten, wie ",[573,3054,3056,3069],{"className":3055},[576],[573,3057,3059],{"className":3058},[580],[582,3060,3061],{"xmlns":584},[586,3062,3063,3067],{},[589,3064,3065],{},[592,3066,598],{},[600,3068,598],{"encoding":602},[573,3070,3072],{"className":3071,"ariaHidden":608},[607],[573,3073,3075,3078],{"className":3074},[612],[573,3076],{"className":3077,"style":617},[616],[573,3079,598],{"className":3080,"style":626},[621,625]," Variablen hat.",[833,3083,3084],{},"Annehmen, der Gradient zeige zum Maximum. Er zeigt nur lokal in die Richtung des steilsten Anstiegs.",[833,3086,3087,3120,3121,3213],{},[573,3088,3090,3105],{"className":3089},[576],[573,3091,3093],{"className":3092},[580],[582,3094,3095],{"xmlns":584},[586,3096,3097,3103],{},[589,3098,3099,3101],{},[592,3100,595],{"mathvariant":594},[592,3102,598],{},[600,3104,603],{"encoding":602},[573,3106,3108],{"className":3107,"ariaHidden":608},[607],[573,3109,3111,3114,3117],{"className":3110},[612],[573,3112],{"className":3113,"style":617},[616],[573,3115,595],{"className":3116},[621],[573,3118,598],{"className":3119,"style":626},[621,625]," mit der Divergenz ",[573,3122,3124,3147],{"className":3123},[576],[573,3125,3127],{"className":3126},[580],[582,3128,3129],{"xmlns":584},[586,3130,3131,3144],{},[589,3132,3133,3135,3137],{},[592,3134,595],{"mathvariant":594},[681,3136,1023],{},[685,3138,3139,3142],{"accent":608},[592,3140,3141],{},"F",[681,3143,693],{},[600,3145,3146],{"encoding":602},"\\nabla \\cdot \\vec F",[573,3148,3150,3169],{"className":3149,"ariaHidden":608},[607],[573,3151,3153,3157,3160,3163,3166],{"className":3152},[612],[573,3154],{"className":3155,"style":3156},[616],"height:0.6833em;",[573,3158,595],{"className":3159},[621],[573,3161],{"className":3162,"style":1162},[715],[573,3164,1023],{"className":3165},[1166],[573,3167],{"className":3168,"style":1162},[715],[573,3170,3172,3176],{"className":3171},[612],[573,3173],{"className":3174,"style":3175},[616],"height:0.9663em;",[573,3177,3179],{"className":3178},[621,775],[573,3180,3182],{"className":3181},[779],[573,3183,3185],{"className":3184},[783],[573,3186,3188,3196],{"className":3187,"style":3175},[787],[573,3189,3190,3193],{"style":790},[573,3191],{"className":3192,"style":795},[794],[573,3194,3141],{"className":3195,"style":2881},[621,625],[573,3197,3199,3202],{"style":3198},"top:-3.2523em;",[573,3200],{"className":3201,"style":795},[794],[573,3203,3206],{"className":3204,"style":3205},[808],"left:-0.1522em;",[573,3207,3209],{"className":3208,"style":814},[813],[816,3210,3211],{"xmlns":818,"width":819,"height":820,"style":821,"viewBox":822,"preserveAspectRatio":823},[825,3212],{"d":827}," verwechseln. Der Gradient wirkt auf eine skalare Funktion und liefert einen Vektor.",{"title":3215,"searchDepth":3216,"depth":3216,"links":3217},"",2,[3218,3219,3220],{"id":562,"depth":3216,"text":563},{"id":1315,"depth":3216,"text":1316},{"id":3046,"depth":3216,"text":3047},[3222,3224,3226,3229,3233,3236,3239,3241,3243,3247,3250],{"lhs":3223,"rhs":690},"round(((x + 1e-5)^2 + 3*y^2 - ((x - 1e-5)^2 + 3*y^2)) \u002F (2e-5) - 2*x, 6)",{"lhs":3225,"rhs":690},"round((x^2 + 3*(y + 1e-5)^2 - (x^2 + 3*(y - 1e-5)^2)) \u002F (2e-5) - 6*y, 6)",{"lhs":3227,"rhs":1614,"where":3228},"2*x",{"x":3216},{"lhs":3230,"rhs":1573,"where":3231},"6*y",{"y":3232},1,{"lhs":3234,"rhs":3235},"sqrt(4^2 + 6^2)","2*sqrt(13)",{"lhs":3237,"rhs":3238},"round(sqrt(52), 2)","7.21",{"lhs":3240,"rhs":2287},"2^2 + 3*1^2",{"lhs":3242,"rhs":690},"dot([4, 6], [3, -2])",{"lhs":3244,"rhs":1573,"where":3245},"y*z",{"y":3216,"z":3246},3,{"lhs":3248,"rhs":1365,"where":3249},"x*z",{"x":3232,"z":3246},{"lhs":3251,"rhs":1359,"where":3252},"x*y",{"x":3232,"y":3216},"f\\colon D \\subseteq \\mathbb{R}^n \\to \\mathbb{R} \\text{ partiell differenzierbar}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003Cmspace>\u003C\u002Fmspace>\u003Cmspace width=\"0.1111em\"\u002F>\u003Cmo lspace=\"0em\" rspace=\"0.17em\">\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmo lspace=\"0em\" rspace=\"0em\">:\u003C\u002Fmo>\u003Cmspace width=\"0.3333em\"\u002F>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>⊆\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f\\colon D \\subseteq \\mathbb{R}^n \\to \\mathbb{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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1111em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">:\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.3333em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\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:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">partiell differenzierbar\u003C\u002Fspan>",null,"md","gradient","\\nabla f(x_1, \\ldots, x_n) = \\left( \\frac{\\partial f}{\\partial x_1}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right)^{T}","\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 mathvariant=\"normal\">∇\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\nabla f(x_1, \\ldots, x_n) = \\left( \\frac{\\partial f}{\\partial x_1}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right)^{T}\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:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∇\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.6313em;vertical-align:-0.95em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3714em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.836em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3714em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" 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style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3290,"meaning":3291,"symbolHtml":3292},"\\frac{\\partial f}{\\partial x_k}","partielle Ableitung von f nach der k-ten Variablen","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmfrac>\u003Cmrow>\u003Cmi 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size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.3488em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" 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