Задачи по векторному анализу. Михайлов В.К - 3 стр.

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Ñîäåðæàíèå
ÏÐÅÄÈÑËÎÂÈÅ ............................................................. 5
ÎÁÎÇÍÀ×ÅÍÈß ............................................................. 6
1. ÑÊÀËßÐÍÎÅ ÏÎËÅ .................................................... 8
1.1. Ïîâåðõíîñòè óðîâíÿ, ëèíèè óðîâíÿ ................ 8
1.2. Ïðîèçâîäíàÿ ïî íàïðàâëåíèþ ....................... 12
1.3. Ãðàäèåíò .......................................................... 16
2. ÂÅÊÒÎÐÍÎÅ ÏÎËÅ .................................................. 23
2.1. Âåêòîðíûå ëèíèè ............................................ 23
2.2. Ïîòîê .............................................................. 28
2.3. Äèâåðãåíöèÿ ................................................... 33
2.4. Òåîðåìà Îñòðîãðàäñêîãî ................................. 36
2.5. Ðàáîòà .............................................................. 42
2.6. Ðîòîð. Ïðàâèëà ðàáîòû ñ îïåðàòîðîì ∇ ........... 51
2.7. Òåîðåìà Ñòîêñà. Ôîðìóëà Ãðèíà .................... 57
3. ÒÈÏÛ ÂÅÊÒÎÐÍÛÕ ÏÎËÅÉ ................................. 63
3.1. Ïîòåíöèàëüíîå ïîëå.
Ñêàëÿðíûé ïîòåíöèàë .................................... 63
3.2. Ñîëåíîèäàëüíîå ïîëå.
Âåêòîðíûé ïîòåíöèàë ..................................... 71
3.3. Ëàïëàñîâî ïîëå. Ãàðìîíè÷åñêèå ôóíêöèè ..... 80
4. ÊÐÈÂÎËÈÍÅÉÍÛÅ ÎÐÒÎÃÎÍÀËÜÍÛÅ
ÊÎÎÐÄÈÍÀÒÛ ......................................................... 84
4.1. Ïîíÿòèå î êðèâîëèíåéíûõ êîîðäèíàòàõ ....... 84
4.2. Ïðåîáðàçîâàíèÿ áàçèñà ................................... 88
4.3. Îïåðàöèè âåêòîðíîãî àíàëèçà
â êðèâîëèíåéíûõ êîîðäèíàòàõ...................... 90
4.3.1.Óðàâíåíèÿ âåêòîðíûõ ëèíèé ................ 90
4.3.2. Ãðàäèåíò ................................................ 91
4.3.3. Äèâåðãåíöèÿ .......................................... 92
4.3.4. Ðîòîð ..................................................... 92
4.3.5. Âû÷èñëåíèå ïîòîêà ............................... 94
4.3.6. Âû÷èñëåíèå ðàáîòû ................................. 9
                           Ñîäåðæàíèå
ÏÐÅÄÈÑËÎÂÈÅ ............................................................. 5
ÎÁÎÇÍÀ×ÅÍÈß ............................................................. 6
1. ÑÊÀËßÐÍÎÅ ÏÎËÅ .................................................... 8
     1.1. Ïîâåðõíîñòè óðîâíÿ, ëèíèè óðîâíÿ ................ 8
     1.2. Ïðîèçâîäíàÿ ïî íàïðàâëåíèþ ....................... 12
     1.3. Ãðàäèåíò .......................................................... 16
2. ÂÅÊÒÎÐÍÎÅ ÏÎËÅ .................................................. 23
     2.1. Âåêòîðíûå ëèíèè ............................................ 23
     2.2. Ïîòîê .............................................................. 28
     2.3. Äèâåðãåíöèÿ ................................................... 33
     2.4. Òåîðåìà Îñòðîãðàäñêîãî ................................. 36
     2.5. Ðàáîòà .............................................................. 42
     2.6. Ðîòîð. Ïðàâèëà ðàáîòû ñ îïåðàòîðîì ∇ ........... 51
     2.7. Òåîðåìà Ñòîêñà. Ôîðìóëà Ãðèíà .................... 57
3. ÒÈÏÛ ÂÅÊÒÎÐÍÛÕ ÏÎËÅÉ ................................. 63
     3.1. Ïîòåíöèàëüíîå ïîëå.
          Ñêàëÿðíûé ïîòåíöèàë .................................... 63
     3.2. Ñîëåíîèäàëüíîå ïîëå.
          Âåêòîðíûé ïîòåíöèàë ..................................... 71
     3.3. Ëàïëàñîâî ïîëå. Ãàðìîíè÷åñêèå ôóíêöèè ..... 80
4. ÊÐÈÂÎËÈÍÅÉÍÛÅ ÎÐÒÎÃÎÍÀËÜÍÛÅ
   ÊÎÎÐÄÈÍÀÒÛ ......................................................... 84
     4.1. Ïîíÿòèå î êðèâîëèíåéíûõ êîîðäèíàòàõ ....... 84
     4.2. Ïðåîáðàçîâàíèÿ áàçèñà ................................... 88
     4.3. Îïåðàöèè âåêòîðíîãî àíàëèçà
          â êðèâîëèíåéíûõ êîîðäèíàòàõ ...................... 90
          4.3.1.Óðàâíåíèÿ âåêòîðíûõ ëèíèé ................ 90
          4.3.2. Ãðàäèåíò ................................................ 91
          4.3.3. Äèâåðãåíöèÿ .......................................... 92
          4.3.4. Ðîòîð ..................................................... 92
          4.3.5. Âû÷èñëåíèå ïîòîêà ............................... 94
          4.3.6. Âû÷èñëåíèå ðàáîòû ................................. 9

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