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❈❤✳✺✿ ❆rr❛② ❝♦♠♣✉t✐♥❣ ❛♥❞ ❝✉r✈❡ ♣❧♦tt✐♥❣ ❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥ ✶ , ✷ ❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r② ✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s ✷ ❆✉❣ ✷✶✱ ✷✵✶✻

●♦❛❧✿ ❧❡❛r♥ t♦ ✈✐s✉❛❧✐③❡ ❢✉♥❝t✐♦♥s

❲❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛ ♥❡✇ ♦❜❥❡❝t✿ ❛rr❛② ❈✉r✈❡s y = f ( x ) ❛r❡ ✈✐s✉❛❧✐③❡❞ ❜② ❞r❛✇✐♥❣ str❛✐❣❤t ❧✐♥❡s ❜❡t✇❡❡♥ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ▼❡❡❞ t♦ st♦r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ✐♥ ❧✐sts ♦r ❛rr❛②s ① ❛♥❞ ② ❆rr❛②s ≈ ❧✐sts✱ ❜✉t ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t ❚♦ ❝♦♠♣✉t❡ t❤❡ ② ❝♦♦r❞✐♥❛t❡s ✭✐♥ ❛♥ ❛rr❛②✮ ✇❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛rr❛② ❝♦♠♣✉t❛t✐♦♥s ♦r ✈❡❝t♦r✐③❛t✐♦♥ ❆rr❛② ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✉s❡❢✉❧ ❢♦r ♠✉❝❤ ♠♦r❡ t❤❛♥ ♣❧♦tt✐♥❣ ❝✉r✈❡s✦

❲❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛ ♥❡✇ ♦❜❥❡❝t✿ ❛rr❛② ❈✉r✈❡s y = f ( x ) ❛r❡ ✈✐s✉❛❧✐③❡❞ ❜② ❞r❛✇✐♥❣ str❛✐❣❤t ❧✐♥❡s ❜❡t✇❡❡♥ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ▼❡❡❞ t♦ st♦r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ✐♥ ❧✐sts ♦r ❛rr❛②s ① ❛♥❞ ② ❆rr❛②s ≈ ❧✐sts✱ ❜✉t ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t ❚♦ ❝♦♠♣✉t❡ t❤❡ ② ❝♦♦r❞✐♥❛t❡s ✭✐♥ ❛♥ ❛rr❛②✮ ✇❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛rr❛② ❝♦♠♣✉t❛t✐♦♥s ♦r ✈❡❝t♦r✐③❛t✐♦♥ ❆rr❛② ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✉s❡❢✉❧ ❢♦r ♠✉❝❤ ♠♦r❡ t❤❛♥ ♣❧♦tt✐♥❣ ❝✉r✈❡s✦

❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ✈❡❝t♦rs ❱❡❝t♦rs ❛r❡ ❦♥♦✇♥ ❢r♦♠ ❤✐❣❤ s❝❤♦♦❧ ♠❛t❤❡♠❛t✐❝s✱ ❡✳❣✳✱ ♣♦✐♥t ( x , y ) ✐♥ t❤❡ ♣❧❛♥❡✱ ♣♦✐♥t ( x , y , z ) ✐♥ s♣❛❝❡ ■♥ ❣❡♥❡r❛❧✱ ❛ ✈❡❝t♦r v ✐s ❛♥ n ✲t✉♣❧❡ ♦❢ ♥✉♠❜❡rs✿ v = ( v ✵ , . . . , v n − ✶ ) ❱❡❝t♦rs ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❧✐sts✿ v i ✐s st♦r❡❞ ❛s ✈❬✐❪ ✱ ❜✉t ✇❡ s❤❛❧❧ ✉s❡ ❛rr❛②s ✐♥st❡❛❞ ❱❡❝t♦rs ❛♥❞ ❛rr❛②s ❛r❡ ❦❡② ❝♦♥❝❡♣ts ✐♥ t❤✐s ❝❤❛♣t❡r✳ ■t t❛❦❡s s❡♣❛r❛t❡ ♠❛t❤ ❝♦✉rs❡s t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✈❡❝t♦rs ❛♥❞ ❛rr❛②s r❡❛❧❧② ❛r❡✱ ❜✉t ✐♥ t❤✐s ❝♦✉rs❡ ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ s✉❜s❡t ♦❢ t❤❡ ❝♦♠♣❧❡t❡ st♦r②✳ ❆ ❧❡❛r♥✐♥❣ str❛t❡❣② ♠❛② ❜❡ t♦ ❥✉st st❛rt ✉s✐♥❣ ✈❡❝t♦rs✴❛rr❛②s ✐♥ ♣r♦❣r❛♠s ❛♥❞ ❧❛t❡r✱ ✐❢ ♥❡❝❡ss❛r②✱ ❣♦ ❜❛❝❦ t♦ t❤❡ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡t❛✐❧s ✐♥ t❤❡ ✜rst ♣❛rt ♦❢ ❈❤✳ ✺✳

❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ✈❡❝t♦rs ❱❡❝t♦rs ❛r❡ ❦♥♦✇♥ ❢r♦♠ ❤✐❣❤ s❝❤♦♦❧ ♠❛t❤❡♠❛t✐❝s✱ ❡✳❣✳✱ ♣♦✐♥t ( x , y ) ✐♥ t❤❡ ♣❧❛♥❡✱ ♣♦✐♥t ( x , y , z ) ✐♥ s♣❛❝❡ ■♥ ❣❡♥❡r❛❧✱ ❛ ✈❡❝t♦r v ✐s ❛♥ n ✲t✉♣❧❡ ♦❢ ♥✉♠❜❡rs✿ v = ( v ✵ , . . . , v n − ✶ ) ❱❡❝t♦rs ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❧✐sts✿ v i ✐s st♦r❡❞ ❛s ✈❬✐❪ ✱ ❜✉t ✇❡ s❤❛❧❧ ✉s❡ ❛rr❛②s ✐♥st❡❛❞ ❱❡❝t♦rs ❛♥❞ ❛rr❛②s ❛r❡ ❦❡② ❝♦♥❝❡♣ts ✐♥ t❤✐s ❝❤❛♣t❡r✳ ■t t❛❦❡s s❡♣❛r❛t❡ ♠❛t❤ ❝♦✉rs❡s t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✈❡❝t♦rs ❛♥❞ ❛rr❛②s r❡❛❧❧② ❛r❡✱ ❜✉t ✐♥ t❤✐s ❝♦✉rs❡ ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ s✉❜s❡t ♦❢ t❤❡ ❝♦♠♣❧❡t❡ st♦r②✳ ❆ ❧❡❛r♥✐♥❣ str❛t❡❣② ♠❛② ❜❡ t♦ ❥✉st st❛rt ✉s✐♥❣ ✈❡❝t♦rs✴❛rr❛②s ✐♥ ♣r♦❣r❛♠s ❛♥❞ ❧❛t❡r✱ ✐❢ ♥❡❝❡ss❛r②✱ ❣♦ ❜❛❝❦ t♦ t❤❡ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡t❛✐❧s ✐♥ t❤❡ ✜rst ♣❛rt ♦❢ ❈❤✳ ✺✳

❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ✈❡❝t♦rs ❱❡❝t♦rs ❛r❡ ❦♥♦✇♥ ❢r♦♠ ❤✐❣❤ s❝❤♦♦❧ ♠❛t❤❡♠❛t✐❝s✱ ❡✳❣✳✱ ♣♦✐♥t ( x , y ) ✐♥ t❤❡ ♣❧❛♥❡✱ ♣♦✐♥t ( x , y , z ) ✐♥ s♣❛❝❡ ■♥ ❣❡♥❡r❛❧✱ ❛ ✈❡❝t♦r v ✐s ❛♥ n ✲t✉♣❧❡ ♦❢ ♥✉♠❜❡rs✿ v = ( v ✵ , . . . , v n − ✶ ) ❱❡❝t♦rs ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❧✐sts✿ v i ✐s st♦r❡❞ ❛s ✈❬✐❪ ✱ ❜✉t ✇❡ s❤❛❧❧ ✉s❡ ❛rr❛②s ✐♥st❡❛❞ ❱❡❝t♦rs ❛♥❞ ❛rr❛②s ❛r❡ ❦❡② ❝♦♥❝❡♣ts ✐♥ t❤✐s ❝❤❛♣t❡r✳ ■t t❛❦❡s s❡♣❛r❛t❡ ♠❛t❤ ❝♦✉rs❡s t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✈❡❝t♦rs ❛♥❞ ❛rr❛②s r❡❛❧❧② ❛r❡✱ ❜✉t ✐♥ t❤✐s ❝♦✉rs❡ ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ s✉❜s❡t ♦❢ t❤❡ ❝♦♠♣❧❡t❡ st♦r②✳ ❆ ❧❡❛r♥✐♥❣ str❛t❡❣② ♠❛② ❜❡ t♦ ❥✉st st❛rt ✉s✐♥❣ ✈❡❝t♦rs✴❛rr❛②s ✐♥ ♣r♦❣r❛♠s ❛♥❞ ❧❛t❡r✱ ✐❢ ♥❡❝❡ss❛r②✱ ❣♦ ❜❛❝❦ t♦ t❤❡ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡t❛✐❧s ✐♥ t❤❡ ✜rst ♣❛rt ♦❢ ❈❤✳ ✺✳

❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ❛rr❛②s ❆rr❛②s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✈❡❝t♦rs ✇❤❡r❡ ✇❡ ❝❛♥ ❤❛✈❡ ♠✉❧t✐♣❧❡ ✐♥❞✐❝❡s✿ A i , j ✱ A i , j , k ❊①❛♠♣❧❡✿ t❛❜❧❡ ♦❢ ♥✉♠❜❡rs✱ ♦♥❡ ✐♥❞❡① ❢♦r t❤❡ r♦✇✱ ♦♥❡ ❢♦r t❤❡ ❝♦❧✉♠♥   A ✵ , ✵ A ✵ , n − ✶   · · · ✵ ✶✷ − ✶ ✺ ✳ ✳ ✳✳✳ ✳ ✳ − ✶ − ✶ − ✶ ✵ A =   ✳ ✳     ✶✶ ✺ ✺ − ✷ A m − ✶ , ✵ A m − ✶ , n − ✶ · · · ❚❤❡ ♥♦ ♦❢ ✐♥❞✐❝❡s ✐♥ ❛♥ ❛rr❛② ✐s t❤❡ r❛♥❦ ♦r ♥✉♠❜❡r ♦❢ ❞✐♠❡♥s✐♦♥s ❱❡❝t♦r ❂ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❛rr❛②✱ ♦r r❛♥❦ ✶ ❛rr❛② ■♥ P②t❤♦♥ ❝♦❞❡✱ ✇❡ ✉s❡ ◆✉♠❡r✐❝❛❧ P②t❤♦♥ ❛rr❛②s ✐♥st❡❛❞ ♦❢ ♥❡st❡❞ ❧✐sts t♦ r❡♣r❡s❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ❛rr❛②s ✭❜❡❝❛✉s❡ t❤✐s ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♠♦r❡ ❡✣❝✐❡♥t✮

❙t♦r✐♥❣ ✭①✱②✮ ♣♦✐♥ts ♦♥ ❛ ❝✉r✈❡ ✐♥ ❧✐sts ❈♦❧❧❡❝t ♣♦✐♥ts ♦♥ ❛ ❢✉♥❝t✐♦♥ ❝✉r✈❡ y = f ( x ) ✐♥ ❧✐sts✿ ❃❃❃ ❞❡❢ ❢✭①✮✿ ✳✳✳ r❡t✉r♥ ①✯✯✸ ✳✳✳ ❃❃❃ ♥ ❂ ✺ ★ ♥♦ ♦❢ ♣♦✐♥ts ❃❃❃ ❞① ❂ ✶✳✵✴✭♥✲✶✮ ★ ① s♣❛❝✐♥❣ ✐♥ ❬✵✱✶❪ ❃❃❃ ①❧✐st ❂ ❬✐✯❞① ❢♦r ✐ ✐♥ r❛♥❣❡✭♥✮❪ ❃❃❃ ②❧✐st ❂ ❬❢✭①✮ ❢♦r ① ✐♥ ①❧✐st❪ ❃❃❃ ♣❛✐rs ❂ ❬❬①✱ ②❪ ❢♦r ①✱ ② ✐♥ ③✐♣✭①❧✐st✱ ②❧✐st✮❪ ❚✉r♥ ❧✐sts ✐♥t♦ ◆✉♠❡r✐❝❛❧ P②t❤♦♥ ✭◆✉♠P②✮ ❛rr❛②s✿ ❃❃❃ ✐♠♣♦rt ♥✉♠♣② ❛s ♥♣ ★ ♠♦❞✉❧❡ ❢♦r ❛rr❛②s ❃❃❃ ① ❂ ♥♣✳❛rr❛②✭①❧✐st✮ ★ t✉r♥ ❧✐st ①❧✐st ✐♥t♦ ❛rr❛② ❃❃❃ ② ❂ ♥♣✳❛rr❛②✭②❧✐st✮