■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ♦❢ ♣♦❧②♥♦♠✐❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s ❛♥❞ ❝♦rr❡❝t✐♦♥ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❏♦r❞② P❛❧❛❢♦① ✭❆ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❏❛❝❦② ❈r❡ss♦♥✮ ❏♦✉r♥é❡s ◆❛t✐♦♥❛❧❡s ❞❡ ❈❛❧❝✉❧ ❋♦r♠❡❧ ✷✵✶✼ ❈■❘▼ ✶✻✲✷✵ ❏❛♥✉❛r② ✷✵✶✼ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✶ ✴ ✸✷
■♥tr♦❞✉❝t✐♦♥ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■♥tr♦❞✉❝t✐♦♥ ✶ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ✷ ●❡♥❡r❛❧ ♥♦t❛t✐♦♥s ❖✉r r❡s✉❧ts ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ■❧❧✉str❛t✐♦♥s ♦❢ ♦✉r t❤❡♦r❡♠s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ✸ Pr❡♣❛r❡❞ ❢♦r♠ ♦❢ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ ▼♦✉❧❞ ❊①♣❛♥s✐♦♥ ❈♦rr❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ Pr♦♦❢ ♦❢ ♦✉r ❚❤❡♦r❡♠s ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✷ ✴ ✸✷
■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ■♥tr♦❞✉❝t✐♦♥ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✸ ✴ ✸✷
■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ❲❡ ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣❧❡① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ r❡❛❧ ♣❧❛♥❛r ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ ❛ ❝❡♥t❡r ✐♥ ✵ ❞❡♥♦t❡❞ ❜② X lin = i ( x ∂ x − y ∂ y ) ✇✐t❤ x , y ∈ C ✇✐t❤ y = ¯ x ✳ ❋✐❣✉r❡ ✕ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t ✵ ✐s ❛ ❝❡♥t❡r✳ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✹ ✴ ✸✷
❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄ ■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = X lin + P ( x , y ) ∂ x + Q ( x , y ) ∂ y ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷
❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄ ■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = X lin + P ( x , y ) ∂ x + Q ( x , y ) ∂ y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷
❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ ❛♥❞ ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄ ■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = X lin + P ( x , y ) ∂ x + Q ( x , y ) ∂ y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷
■♥tr♦❞✉❝t✐♦♥ ■s♦❝❤r♦♥♦✉s ❝❡♥t❡rs ❛♥❞ ❏❛rq✉❡✲❱✐❧❧❛❞❡❧♣r❛t✬s ❝♦♥❥❡❝t✉r❡ Pr♦❣r❡ss ❛❜♦✉t t❤❡ ❝♦♥❥❡❝t✉r❡ ❖✉r ❛♣♣r♦❛❝❤ ✿ t❤❡ ▼♦✉❧❞ ❈❛❧❝✉❧✉s Pr♦♦❢s ♦❢ t❤❡ t❤❡♦r❡♠s ❲❤✐❝❤ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡r✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ♣❡rt✉r❜❛t✐♦♥ ♦❢ t❤✐s ✜❡❧❞ ❄ X = X lin + P ( x , y ) ∂ x + Q ( x , y ) ∂ y ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ♣r♦♣❡rt② t♦ ❜❡ ❛ ❝❡♥t❡r ❄ ❆ ❝❡♥t❡r ✐s ✐s♦❝❤r♦♥♦✉s ✐❢ ❛❧❧ t❤❡ ♦r❜✐ts ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐s♦❝❤r♦♥♦✉s ❝❡♥t❡r ❲❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ P ❛♥❞ Q ❛r❡ ♥❡❝❡ss❛r② t♦ ♣r❡s❡r✈❡ t❤❡ ✐s♦❝❤r♦♥✐❝✐t② ❄ ❏♦r❞② P❛❧❛❢♦① ✲ ❏◆❈❋ ✷✵✶✼ ✺ ✴ ✸✷
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