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ÃÀÐÌÎÍÈ×ÅÑÊÀß ÔÓÍÊÖÈß
ÃÀÐÌÎÍÈ×ÅÑÊÀß ÔÓÍÊÖÈß. Äåéñòâèòåëüíàÿ
ôóíêöèÿ ,
îïðåäåëåííàÿ â íåêîòîðîé îáëàñòè
ïðîñòðàíñòâà ïåðåìåííûõ
, íàçûâàåòñÿ
Ã. ô. â
, åñëè âñå åå
âòîðûå ÷àñòíûå ïðîèçâîäíûå íåïðåðûâíû â
è
,
Ã. ô. îáëàäàåò ðÿäîì çàìå÷àòåëüíûõ ñâîéñòâ, ñðåäè íèõ:
1.
Ñðåäíåå çíà÷åíèå Ã, ô. ïî ñôåðå
ðàäèóñà
ñ öåíòðîì â òî÷êå
ðàâíÿåòñÿ
. Çäåñü ïðåäïîëàãàåòñÿ, ÷òî
ãàðìîíè÷íà âíóòðè ñôåðû
; ñðåäíåå çíà÷åíèå
åñòü èíòåãðàë
, äåëåííûé íà
ïëîùàäü ñôåðû.
2.
Êàê ñëåäñòâèå ñâîéñòâà 1 èìååò ìåñòî ïðèíöèï ìàêñèìóìà: ìàêñèìàëüíîå çíà÷åíèå,
à òàêæå ìèíèìàëüíîå ôóíêöèè â çàìûêàíèè
îãðàíè÷åííîé îáëàñòè
ïðè óñëîâèè, ÷òî
ãàðìîíè÷íà â
îáëàñòè
äîñòèãàåòñÿ â
òî÷êàõ ãðàíèöû îáëàñòè
.
3.
Äëÿ ôóíêöèè êîìïëåêñíîãî ïåðåìåííîãî,
àíàëèòè÷åñêîé â îáëàñòè
, èìååò ìåñòî
ñëåäóþùåå óòâåðæäåíèå: âåùåñòâåííàÿ è ìíèìàÿ ÷àñòè
, ò. å.
,
, ãäå
,
, ÿâëÿþòñÿ
Ã. ô.; áîëåå òîãî,
,
— ïàðà ñîïðÿæåííûõ Ã. ô. (ñì. Êîøè—Ðèìàíà
óðàâíåíèÿ).
4.
Ïóñòü ,
— çàìåíà ïåðåìåííûõ
íà
îðòîãîíàëüíûì
ïðåîáðàçîâàíèåì.
Åñëè
— Ã. ô.,
òî ôóíêöèÿ
áóäåò
Ã. ô. ïåðåìåííûõ ,
ò. å.
.
 n-ìåðíîì
ïðîñòðàíñòâå ïåðåìåííûõ çàìå÷àòåëüíû Ã. ô. âèäà
, ãäå
;
— ïðîèçâîëüíàÿ, íî
ôèêñèðîâàííàÿ òî÷êà. Äëÿ
çàìå÷àòåëüíà
Ã. ô.
,
— ïðîèçâîëüíà, íî
ôèêñèðîâàíà.
Ýòè ôóíêöèè ïîçâîëÿþò ïîñòðîèòü íîâûå Ã.
ô. ñëåäóþùèì îáðàçîì: åñëè — íåêîòîðàÿ ïîâåðõíîñòü â
è
— ôóíêöèÿ íà
, òî
ÿâëÿåòñÿ Ã. ô. âñþäó â
êðîìå
ïîâåðõíîñòè
.