ùî íàçèâàºòüñÿ ãðà䳺íòîì ôóíêö³¿ z = f(x, y) â òî÷ö³ M 0 (x 0 , y 0 ). Òóò i r , r j â³äîì³ îðòè. Ïîìíîæèìî òåïåð ñêàëÿðíî ö³ äâà âåêòîðè, ïðè÷îìó äâîìà ñïîñîáàìè (äèâ. ï. 2.1.6). Ó ðåçóëüòàò³ îòðèìàºìî r grad z ⋅ e = z′ x ( x0, y0) ⋅cos α + z′ y ( x0, y0) ⋅sin α M = M , (10.5.2) 0 r r grad z ⋅ e = grad z ⋅ e ⋅cos ϕ M = M0 M = M , 0 (10.5.3) äå ϕ — êóò ì³æ öèìè âåêòîðàìè. Îñê³ëüêè r e = 1 , òî ³ç (10.5.2) – (10.5.3) áóäåìî ìàòè ∂z ( x 0, y 0) = grad z ( x 0, y 0) ⋅cosϕ. (10.5.4) ∂e Ïðîñòèé àíàë³ç ôîðìóëè (10.5.4) ïîêàçóº, ùî íàéá³ëüøå çíà÷åííÿ ïîõ³äíî¿ çà íàïðÿìîì e r â òî÷ö³ M 0 áóäå òîä³, êîëè grad z ≠ 0 ³ íàïðÿì âåêòîðà e r ñï³âïàäຠç íàïðÿìîì ãðà䳺íòà. Ïðè öüîìó âåëè÷èíà íàéá³ëüøîãî çðîñ- M = M 0 òàííÿ ôóíêö³¿ z îá÷èñëþºòüñÿ çà ôîðìóëîþ ∂z x y z x y z x y z x y ∂e ( ) ( )( ) ( ( )) 2 ( ( )) 2 0, 0 = grad 0, 0 = ′ x 0, 0 + ′ y 0, 0 . (10.5.5) Çàóâàæåííÿ 1. Çã³äíî ç ôîðìóëîþ (10.5.1) ãðà䳺íòîì ôóíêö³¿ z = f(x, y) â òî÷ö³ (x 0 , y 0 ) º âåêòîð, êîîðäèíàòè ÿêîãî äîð³âíþþòü â³äïîâ³äíî z′ x ( x0, y0) , z′ y ( x0, y0) . Àíàëîã³÷íî îçíà÷àºòüñÿ ãðà䳺íò ôóíêö³¿ â³ä n íåçàëåæíèõ çì³ííèõ. Ñôîðìóëþºìî, íàïðèêëàä, îçíà÷åííÿ ãðà䳺íòà äëÿ ôóíêö³¿ â³ä òðüîõ íåçàëåæíèõ çì³ííèõ. Íåõàé ôóíêö³ÿ u = f(x, y, z) â òî÷ö³ Q 0 (x 0 , y 0 , z 0 ) äèôåðåíö³éîâíà. Òîä³ â ö³é òî÷ö³ ³ñíóº ãðà䳺íò, ÿêèé âèçíà÷àºòüñÿ çà ôîðìóëîþ grad u x , y , z = u x , y , z ⋅ i r + u x , y , z ⋅ j r + u x , y , z ⋅k, r ( ) ′ ( ) ′ ( ) ′( ) 0 0 0 x 0 0 0 y 0 0 0 z 0 0 0 äå i r , j r ³ k r çíàéîì³ íàì îðòè. Íàðåøò³ ìîæíà ñêàçàòè, ùî ïðîáëåìà ïîøóêó êëàñó ôóíêö³é (äèâ. ï. 10.3), äëÿ ÿêèõ ìîæíà ãàðàíòîâàíî çíàéòè íàïðÿì íàéá³ëüøîãî çðîñòàííÿ ôóíêö³¿ ³ ñàìó øâèäê³ñòü ¿¿ çðîñòàííÿ, âèð³øåíà, öå êëàñ äèôåðåíö³éîâíèõ ôóíêö³é. Äèôåðåíö³éîâí³ ôóíêö³¿ ìàþòü äóæå âàæëèâ³ âëàñòèâîñò³. Çîêðåìà, ÿêùî ôóíêö³ÿ äèôåðåíö³éîâíà, òî âîíà íåïåðåðâíà ³ ³ñíóþòü ÷àñòèíí³ ïîõ³äí³ çà áóäü-ÿêèì íàïðÿìîì. Ó çâ’ÿçêó ç öèì ïîñòຠïèòàííÿ ïðî âèÿâëåííÿ äîñòàòí³õ óìîâ, ÿê³ çàáåçïå÷óþòü äèôåðåíö³éîâí³ñòü ôóíêö³¿. ³äïîâ³äü íà çàïèòàííÿ äຠòàêà òåîðåìà. Òåîðåìà 10.5.1 (äîñòàòí³ óìîâè äèôåðåíö³éîâíîñò³ ôóíêö³¿). Íåõàé ôóíêö³ÿ z = f(x, y) â äåÿêîìó δ-îêîë³ òî÷êè M 0 (x 0 , y 0 ) ìຠñê³í÷åíí³ ïîõ³äí³ z′ ( x, y) ³ z ( x, y) x ′ , ÿê³ íåïåðåðâí³ â ñàì³é òî÷ö³ M 0 (x 0 , y 0 ). Òîä³ ôóíêö³ÿ z = f(x, y) äèôåðåíö³éîâíà â òî÷ö³ M 0 (x 0 , y 0 ). Äîâåäåííÿ. Ðîçãëÿíåìî ïîâíèé ïðèð³ñò ôóíêö³¿ â òî÷ö³, ÿêèé ìîæåìî çîáðàçèòè ó âèãëÿä³: ( 0, 0) ( 0 , 0 ) ( 0, 0) ∆ zx y = fx +∆ xy +∆y − fx y = ( f( x0 x, y0 y) f( x0, y0 y) ) = +∆ +∆ − +∆ + ( f( x0, y0 y) f( x0, y0) ) + +∆ − , (10.5.6) äå ∆x ≠ 0 ³ ∆y ≠ 0. Âèðàç, ùî ñòî¿òü ó ïåðøèõ äóæêàõ, º ïðèðîñòîì ôóíêö³¿ f(x, y 0 + ∆y) îäí³º¿ çì³ííî¿ x â òî÷ö³ x 0 . Îñê³ëüêè öÿ ôóíêö³ÿ äèôåðåíö³éîâíà â δ-îêîë³ òî÷êè x 0 (âíàñë³äîê ³ñíóâàííÿ ÷àñòèííî¿ ïîõ³äíî¿ f′ x ( x, y) â δ-îêîë³ òî÷êè M 0 (x 0 , y 0 )), òî äî ïðèðîñòó ö³º¿ ôóíêö³¿ ìîæíà çàñòîñóâàòè òåîðåìó Ëàãðàíæà. Çã³äíî ç ö³ºþ òåîðåìîþ áóäåìî ìàòè ( , ) ( , ) ′ ( , ) f x +∆ x y +∆y − f x y +∆ y = f x +θ∆ x y +∆y ∆ x, (10.5.7) 0 0 0 0 x 0 1 0 äå 0 < θ 1
Àíàëîã³÷íî ( 0, 0 ) ( 0, 0) ′ ( 0, 0 2 ) f x y +∆y − f x y = f x y +θ∆y ∆ y, (10.5.8) äå 0 < θ 2
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̲ͲÑÒÅÐÑÒÂÎ ÎѲÒ
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äåòüñÿ” âèêîðèñòî
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Ê36 Êåðåêåøà Ï. Â. Ëå