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∂u<br />
äå ( i = 1, 2, K , n)<br />
∂x<br />
÷àñòèíí³ ïîõ³äí³ â òî÷ö³ M 0 , à<br />
i<br />
2 2 2<br />
cos α<br />
1<br />
+ cos α<br />
2<br />
+ K + cos α<br />
n<br />
= 1.<br />
Çàóâàæåííÿ 3. Î÷åâèäíî, ùî âñ³ ïîíÿòòÿ, ÿê³ áóëî<br />
ââåäåíî äëÿ ôóíêö³¿ äâîõ çì³ííèõ, ïåðåíîñÿòüñÿ íà äîâ³ëüíèé<br />
n-âèì³ðíèé ïðîñò³ð äîñòàòíüî ëåãêî. Ïðè öüîìó òðåáà<br />
ðîçóì³òè, ùî ïåðåõ³ä â³ä n = 2 äî n = 3 ñóòòºâèé. Ñóòòºâ³ñòü<br />
ïîëÿãຠâ òîìó, ùî â äâîâèì³ðíîìó ïðîñòîð³ íàïðÿìëåí³<br />
⎛π<br />
⎞<br />
êîñèíóñè ïîâ’ÿçàí³ ð³âí³ñòþ cosβ= cos⎜<br />
−α ⎟= sin α (β êóò,<br />
⎝2<br />
⎠<br />
ï³ä ÿêèì âåêòîð e r ïåðåòèíຠîñü Oy), à â òðèâèì³ðíîìó<br />
ïðîñòîð³ àíàëîã³÷íî¿ çàëåæíîñò³ íå ³ñíóº.<br />
Ùîäî n-âèì³ðíîãî ïðîñòîðó (n > 3), òî ïðèíöèïîâî â³í<br />
â³äíîñíî ââåäåíèõ ïîíÿòü í³÷èì íå â³äð³çíÿºòüñÿ. Âñ³ ïîíÿòòÿ<br />
ïîâí³ñòþ ïîâòîðþþòüñÿ, ò³ëüêè á³ëüø ãðîì³çäêî.<br />
Çàóâàæåííÿ 4. Âèíèêຠïèòàííÿ: äëÿ ÷îãî òðåáà ðîçãëÿäàòè<br />
n-âèì³ðíèé ïðîñò³ð? ³äïîâ³äü îäíîçíà÷íà: äëÿ òîãî,<br />
ùîá ìîæíà áóëî åôåêòèâíî äîñë³äæóâàòè çàëåæí³ñòü îäí³º¿<br />
âåëè÷èíè â³ä áàãàòüîõ ³íøèõ.  åêîíîì³ö³ òàê³ çàëåæíîñò³<br />
³ñíóþòü (äóìàþ, ùî ÷èòà÷ äîáðå öå ðîçó쳺), ³ òîìó ïîòðåáà<br />
â àáñòðàãóâàíí³ ö³ëêîì î÷åâèäíà.<br />
Çàóâàæåííÿ 5. Ðîçãëÿäàííÿ äâîâèì³ðíîãî ïðîñòîðó, à<br />
³íêîëè òðèâèì³ðíîãî ïîâ’ÿçàíî ò³ëüêè ç åëåìåíòàìè íàî÷íîñò³.<br />
×èòà÷, ÿêèé áóäå äîáðå âîëîä³òè ïîíÿòòÿì äëÿ äâîâèì³ðíîãî<br />
³ òðèâèì³ðíîãî ïðîñòîðó, áåç âñÿêî¿ íàïðóãè ìîæå<br />
ïðàöþâàòè ³ â ïðîñòîð³ ðîçì³ðó á³ëüøå 3. Ïåâíà ð³÷, ÿêùî<br />
â öüîìó áóäå ïîòðåáà.<br />
10.5. ÃÐÀIJªÍÒ ÔÓÍÊÖ²¯<br />
10.5.1. Ïîíÿòòÿ ïðî ñêàëÿðí³ òà âåêòîðí³ ïîëÿ<br />
Ïðè äîñë³äæåíí³ ð³çíèõ ÿâèù, çîêðåìà â åêîíîì³ö³, ÷àñòî<br />
âèíèêຠïîòðåáà ðîçãëÿäàííÿ ñêàëÿðíîãî ïîëÿ, àáî âåêòîðíîãî,<br />
àáî â ñóêóïíîñò³. Íàïðèêëàä, íåð³âíîì³ðíå îá’ºìíå<br />
ò³ëî óòâîðþº â ÷àñòèí³ ïðîñòîðó, ÿêå çàéìຠò³ëî, ñêàëÿðíå<br />
³ âåêòîðíå ïîëå (ðèñ. 10.12) (òåìïåðàòóðà º ñêàëÿð, à íàïðÿì<br />
òåìïåðàòóðíîãî ïîòîêó — âåêòîð).<br />
Ðèñ. 10.12<br />
Äâîâèì³ðíèé (n-âèì³ðíèé) âåêòîð p=<br />
( p , p )<br />
0<br />
r<br />
1 2<br />
r<br />
( p = (p 1 , p 2 , ...,<br />
p n )) óòâîðþº â îêîë³ ð³âíîâàæíî¿ òî÷êè (òî÷êà â ÿê³é ïðîïîçèö³ÿ<br />
òîâàðó ñï³âïàäຠç éîãî ïîïèòîì) âåêòîðíå ïîëå.<br />
³í æå âèçíà÷ຠ³ ñêàëÿðíå ïîëå — ïîëå âàðòîñò³ òîâàðó.<br />
10.5.2. Ãðà䳺íò ôóíêö³¿<br />
Íåõàé ôóíêö³ÿ z = f(x, y) äèôåðåíö³éîâíà â òî÷ö³<br />
M 0 (x 0 , y 0 ). Òîä³ çà òåîðåìîþ 10.4.3, ó ö³é òî÷ö³ ³ñíóº ïîõ³äíà<br />
ö³º¿ ôóíêö³¿ çà áóäü-ÿêèì íàïðÿìîì. ² çðàçó æå âèíèêàº<br />
âàæëèâå ïèòàííÿ ïðî çíàõîäæåííÿ íàïðÿìó, çà ÿêèì öÿ<br />
ïîõ³äíà áóäå íàéá³ëüøîþ.<br />
Ùîá â³äïîâ³ñòè íà òàêå çàïèòàííÿ, ðîçãëÿíåìî äâà âåêòîðè:<br />
îäèíè÷íèé âåêòîð e = cos α⋅ i + sin α⋅ j , ïðî ÿêèé âèùå<br />
r r r<br />
áóëî çãàäàíî â ï. 10.4, ³ âåêòîð<br />
r<br />
r<br />
x= x0 = z′ x( x0, y0) ⋅ i + z′<br />
y( x0, y0)<br />
⋅j,<br />
gradz (10.5.1)<br />
y = y<br />
372 373
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