Read e-book online Elements of Noncommutative Geometry PDF

By Gracia-Bondia J.M., Varilly J.C., Figueroa H.

ISBN-10: 1421451492

ISBN-13: 9781421451497

ISBN-10: 1731752113

ISBN-13: 9781731752116

ISBN-10: 1751811913

ISBN-13: 9781751811916

ISBN-10: 2402483113

ISBN-13: 9782402483117

ISBN-10: 2512672824

ISBN-13: 9782512672821

ISBN-10: 2603714104

ISBN-13: 9782603714102

The topic of this article is an algebraic and operatorial transforming of geometry, which lines its roots to quantum physics; Connes has proven that noncommutative geometry retains all crucial positive aspects of the metric geometry of manifolds. Many singular areas that emerge from advances in arithmetic or are utilized by physicists to appreciate the flora and fauna are thereby introduced into the world of geometry. "This publication is an advent to the language and methods of noncommutative geometry at a degree compatible for graduate scholars, and in addition presents enough element to be helpful to physicists and mathematicians wishing to go into this quickly growing to be box. it will probably additionally function a reference textual content on numerous subject matters which are appropriate to noncommutative geometry.

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T♦ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡ ❡q✉❛t✐♦♥s✳ Hermite interpolation ■♥ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✱ ❛♥❞ s♦❧✈❡ s✐♠✉❧t❛♥❡♦✉s ❡q✉❛t✐♦♥s ❢♦r ❜♦t❤ ♣♦s✐t✐♦♥ ❛♥❞ t❛♥❣❡♥t ✈❛❧✉❡ ❛t ❡❛❝❤ ♦❢ t❤❡ ♣♦✐♥ts ❜❡✐♥❣ ✐♥t❡r♣♦❧❛t❡❞✳ ❚❤✉s t✇✐❝❡ ❛s ♠❛♥② ❝♦❡✣❝✐❡♥ts ❛r❡ r❡q✉✐r❡❞ ❛s ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ❝❛s❡✳ ❆ ♣❛r✲ t✐❝✉❧❛r❧② ✐♠♣♦rt❛♥t ❝❛s❡ ✐s ❝♦♥str✉❝t✐♥❣ ❛ ❝✉r✈❡ ❜❡t✇❡❡♥ t✇♦ ❡♥❞ ■♥t❡r♣♦❧❛t✐♦♥ ✸✺ ✸✭✐✮✖▲❛❣r❛♥❣❡ ❛♥❞ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✉s❡❞ t♦ ❝♦♥✲ str✉❝t ❛ ♣❛r❛♠❡tr✐❝ ❝✉❜✐❝ ❝✉r✈❡ s❡❣♠❡♥t✳ ♣♦✐♥ts✱ ✇✐t❤ ❦♥♦✇♥ t❛♥❣❡♥t ✈❛❧✉❡s ❛t ❡❛❝❤✳ ❚❤❛t r❡q✉✐r❡s ❛ ❝✉r✈❡ ✇✐t❤ ❢♦✉r ❝♦❡✣❝✐❡♥ts ✐♥ ❡❛❝❤ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❛r❡ ❝✉❜✐❝s ❀ t❤❡r❡ ✐s ❛❧s♦ ❛♥ ❡q✉✐✈❛❧❡♥t ♣❛t❝❤ ✇❤✐❝❤ r✉♥s ❜❡t✇❡❡♥ ❢♦✉r ❝♦r♥❡r ♣♦✐♥ts✱ ❛♥❞ ❤❛s ✶✻ ❝♦❡✣❝✐❡♥ts✳ ❈✉❜✐❝s ❛r❡ ❢r❡q✉❡♥t s✐❣❤t✐♥❣s ✐♥ ❝♦♠♣✉t✐♥❣ ✇✐t❤ ❣❡♦♠❡tr②✿ s❡❡ ■❧❧✉str❛t✐♦♥ ✸✭✐✮✳ The problem of parameterization ■♥t❡r♣♦❧❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❞❡❝✐❞✐♥❣ ✇❤❛t t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✲ ✉❡s ❛t ❡❛❝❤ ♣♦✐♥t ✇✐❧❧ ❜❡✖t❤❡ ✐ss✉❡ ♦❢ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✖✐s ❝r✉❝✐❛❧✳ ✭❚❤❛t ✐s ❛ ♣r♦❜❧❡♠ t❤❛t ❞♦❡s ♥♦t ♦❝❝✉r ✇✐t❤ ❡①♣❧✐❝✐t✱ s✐♥❣❧❡✲✈❛❧✉❡❞✱ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✿ ❛♥❞ s♦ ✇❡ ❝❛♥ s❡❡ ✇❤② t❤❡s❡ ❛r❡ ♣r❡❢❡rr❡❞ ❢♦r ❞r❛✇✐♥❣ ❣r❛♣❤s ❛♥❞ s♦ ♦♥✳ ❆♥❞ t❡❝❤♥✐q✉❡s ❢r♦♠ ❵❣r❛♣❤✐♥❣✬ ❛♣♣❧✐❝❛✲ t✐♦♥s ✉s✉❛❧❧② ❡①♣❧♦✐t t❤✐s ❧✐♠✐t❛t✐♦♥✱ ✇❤✐❝❤ ✐s ✇❤② ✇❡ s❤♦✉❧❞ ❜❡ ✇❛r② ♦❢ tr②✐♥❣ t♦ tr❛♥s♣❧❛♥t t❤❡♠ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s✳✮ ❙♦✱ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ✐♥t❡r♣♦❧❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s t❤❛t✱ ✇❤✐❧❡ t❤❡ ♣♦s✐t✐♦♥s ❛♥❞ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥s ♠❛② ❜❡ ♣r♦✈✐❞❡❞✱ ✇❡ ❤❛✈❡ t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s t❤❛t t❤❡ ❝✉r✈❡ ❵s❤♦✉❧❞✬ ❤❛✈❡ ✇❤❡♥ ✐t ♣❛ss❡s ❡❛❝❤ ♣♦✐♥t✱ ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ❛s ✇❡❧❧ ❛s ❞✐r❡❝t✐♦♥ ♦❢ ❞❡r✐✈❛✲ t✐✈❡s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ t❤❡ s✐♠♣❧❡st ❝❤♦✐❝❡ ✐s t♦ s♣❛❝❡ ♣❛r❛♠❡tr✐❝ ✈❛❧✉❡s ❡q✉❛❧❧② ❜❡t✇❡❡♥ ♣♦✐♥t ❞❛t❛✳ ❚❤✐s ✇♦r❦s ✐❢ t❤❡ ♣♦✐♥ts ❛r❡ t❤❡♠s❡❧✈❡s q✉✐t❡ ❡✈❡♥❧② s♣❛❝❡❞❀ ♦t❤❡r✇✐s❡ s♦♠❡✲ t❤✐♥❣ ❜❡tt❡r ✐s ♥❡❡❞❡❞✳ ❙✐♥❝❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ✐s r❡❧❛t❡❞ t♦ ❝✉r✈❡ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✻ ❧❡♥❣t❤✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✇❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ✇✐❧❧ ❜❡ ❜❡t✇❡❡♥ ❡❛❝❤ ❞❛t❛ ♣♦✐♥t❀ ❜✉t t❤❛t ✐s ♣✉tt✐♥❣ t❤❡ ❝❛rt ❜❡❢♦r❡ t❤❡ ❤♦rs❡✱ ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡♥✬t ❣♦t t❤❡ ❝✉r✈❡ ②❡t✳ ❖♥❡ ❝♦✉❧❞ ✐♠♣❧❡♠❡♥t ❛ t❡❝❤♥✐q✉❡ ♦❢ s✉❝❝❡ss✐✈❡ r❡✜♥❡♠❡♥t✖s❡t ✉♣ ♦♥❡ ❝✉r✈❡✱ ❣❡t t❤❡ ❝✉r✈❡ ❧❡♥❣t❤s ❢r♦♠ ✐t✱ ❛♥❞ t❤✉s ♦❜t❛✐♥ ♥❡✇ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ❞❛t❛ ♣♦✐♥ts✱ ❛♥❞ r❡♣❡❛t t❤❡ ❡①❡r❝✐s❡✖❜✉t t❤✐s r✐❣♠❛r♦❧❡ ✐s ♥♦t ✉s✉❛❧❧② ❛t✲ t❡♠♣t❡❞❀ ✐t ✇♦✉❧❞ ♣r♦❜❛❜❧② ❜❡ ❞✐✣❝✉❧t ❡✈❡♥ t♦ ♣r♦✈❡ t❤❛t ✐t ✇♦✉❧❞ ❝♦♥✈❡r❣❡✳ ❚❤❡ ✉s✉❛❧ s♦❧✉t✐♦♥ ✐s ❝❤♦r❞✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✱ ✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ♣♦✐♥ts ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ str❛✐❣❤t✲❧✐♥❡ s❡❣♠❡♥ts ❝♦♥♥❡❝t✐♥❣ t❤❡♠✳ ❚❤✐s ✐s ❛ ❣♦♦❞ ✇♦r❦❤♦rs❡✱ ❣✐✈✐♥❣ tr♦✉❜❧❡ ♦♥❧② ✇❤❡♥ t❤❡r❡ ❛r❡ ❛❜r✉♣t ❵❝♦r♥❡rs✬ ✐♠♣❧✐❡❞ ❜② t❤❡ ❞❛t❛✱ ❛♥❞ ❝❤❛♥❣❡s ♦❢ s♣❛❝✐♥❣✳ ❋✉rt❤❡r r❡✜♥❡♠❡♥ts ✐♥✈♦❧✈❡ t❛❦✐♥❣ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ s✉❝❝❡ss✐✈❡ s♣❛♥s ✐♥t♦ ❛❝❝♦✉♥t ✭s❡❡ ❋❛r✐♥✬s ❜♦♦❦ ❈✉r✈❡s ❛♥❞ ❙✉r❢❛❝❡s ❢♦r ❈♦♠♣✉t❡r ❆✐❞❡❞ ●❡♦♠❡tr✐❝ ❉❡s✐❣♥ ❢♦r ♠♦r❡ ❞❡t❛✐❧✮✳ ❲✐t❤ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ s✐♠✐❧❛r ♣r♦❜❧❡♠s ♦❝❝✉r❀ ❛♥❞ ✐t ♠✉st ❜❡ r❡♠❡♠❜❡r❡❞ t❤❛t ♠❛❣♥✐t✉❞❡s ♦❢ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦r♠ dx dt ❡t❝✳✱ ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❛❝t✉❛❧ s✐③❡ ♦❢ t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ✉♥✐ts ♦❢ ❧❡♥❣t❤ ❜❡✐♥❣ ✉s❡❞✳ ❚❤✉s✱ ✐❢ ✇❡ s❝❛❧❡ ❛ ❝✉r✈❡ ❜② s❝❛❧✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ ✐ts ❍❡r♠✐t❡ ❝♦❡✣❝✐❡♥ts✱ ✇❡ ♠✉st s❝❛❧❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ❡①♣❧✐❝✐t❧②✳ ❚❤❛t✬s ❡❛s② ❡♥♦✉❣❤ ❢♦r ❛ s✐♠♣❧❡ s❝❛❧✐♥❣✱ ❜✉t ✇❤❛t ❛❜♦✉t ❛ s❤❡❛r tr❛♥s❢♦r♠❄ ❆❧❧ t❤❡s❡ r❡♠❛r❦s ❤❛✈❡ ❜❡❡♥ ❛❞❞r❡ss❡❞ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡r✲ ♣♦❧❛t✐♦♥✱ ❜✉t ❛❧s♦ ❛♣♣❧② t♦ ❝✉r✈❡ ✜tt✐♥❣✳ ❆❣❛✐♥✱ t❤✐s ✐s ❛ ♣r♦❝❡ss t❤❛t ✇♦r❦s ✇❡❧❧ ✇✐t❤ ❡①♣❧✐❝✐t ❣❡♦♠❡tr②✱ ❛♥❞ ❢❛✐r❧② ✇❡❧❧ ✇✐t❤ ✐♠♣❧✐❝✲ ✐ts ✭❡①❝❡♣t t❤❛t ♥♦r♠❛❧✐③❛t✐♦♥ ❝❛✉s❡s ❛ ♣r♦❜❧❡♠✮✳ ❲✐t❤ ♣❛r❛♠❡tr✐❝ ❣❡♦♠❡tr②✱ ✇❡ ❛❣❛✐♥ ❤❛✈❡ t♦ ❞❡❝✐❞❡ ✐♥ ❛❞✈❛♥❝❡ ✇❤❛t ♣❛r❛♠❡t❡r ✈❛❧✉❡ ❡❛❝❤ ♣♦✐♥t ✇✐❧❧ ❝♦rr❡s♣♦♥❞ t♦✳ ❇✉t ✐❢ t❤❡ ♣♦✐♥ts ❛r❡ ❛t ❛❧❧ ❞❡♥s❡✱ t❤✐s ✐s ❞✐✣❝✉❧t✿ ❝❤♦r❞✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ✐s ❝❡rt❛✐♥❧② ✉s❡❧❡ss✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤✐s s❡❝t✐♦♥ ✇✐t❤ ❈ ❝♦❞❡ ❢♦r ▲❛❣r❛♥❣❡ ❛♥❞ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❚❤❡ ✜rst ♣r♦❝❡❞✉r❡ ✇♦r❦s ♦✉t t❤❡ ▲❛❣r❛♥❣✐❛♥ ✐♥t❡r✲ ♣♦❧❛t✐♥❣ ❝✉❜✐❝ ♣❛r❛♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧ t❤r♦✉❣❤ ❢♦✉r ♣♦✐♥ts ✐♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ♣♦✐♥ts ✇✐❧❧ ❜❡ s✉♣♣❧✐❡❞ ✐♥ ♣①✱ ♣②✱ ❛♥❞ ♣③✳ ❚❤❡ 0✱ ❛♥❞ t❤❡ ♣❛r❛♠✲ ♣❛r❛♠❡t❡r ♦♥ t❤❡ ❝✉r✈❡ ❛t t❤❡ ✜rst ♣♦✐♥t ✇✐❧❧ ❜❡ 1❀ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❜❡ ♣♦❧②①✱ ♣♦❧②②✱ ❛♥❞ ♣♦❧②③❀ ♣♦❧②①❬✸❪ ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ❡t❡r ❛t t❤❡ ❧❛st ♣♦✐♥t r❡t✉r♥❡❞ ✐♥ ■♥t❡r♣♦❧❛t✐♦♥ t3 ✐♥ x ✸✼ ❛♥❞ s♦ ♦♥✳ ❚❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ♠✐❞❞❧❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ ✇✐❧❧ ❜❡ r❡t✉r♥❡❞ ✐♥ t✶ ❛♥❞ t✷✳ ★✐♥❝❧✉❞❡ ❁♠❛t❤✳❤❃ ★✐♥❝❧✉❞❡ ❁st❞✐♦✳❤❃ ✴✯ ❆❜s♦❧✉t❡ ✈❛❧✉❡ ♠❛❝r♦ ✯✴ ★❞❡❢✐♥❡ ❢❛❜s✭❛✮ ✭✭✭❛✮ ❁ ✵✳✵✮ ❄ ✭✲✭❛✮✮ ✿ ✭❛✮✮ ✴✯ ❆❧♠♦st ✵ ✲ ❛❞❥✉st ❢♦r ②♦✉r ❛♣♣❧✐❝❛t✐♦♥ ✯✴ ★❞❡❢✐♥❡ ❆❈❈❨ ✭✶✳✵❡✲✻✮ ✐♥t ❧❛❣r❛♥❣❡✭♣①✱♣②✱♣③✱t✶✱t✷✱♣♦❧②①✱♣♦❧②②✱♣♦❧②③✮ ❢❧♦❛t ♣①❬✹❪✱♣♦❧②①❬✹❪❀ ❢❧♦❛t ♣②❬✹❪✱♣♦❧②②❬✹❪❀ ❢❧♦❛t ♣③❬✹❪✱♣♦❧②③❬✹❪❀ ❢❧♦❛t ✯t✶✱✯t✷❀ ④ ❢❧♦❛t ①❞✱②❞✱③❞✱❞❢❧❀ ✐♥t ✐❀ ✴✯ ✯✴ ❙✉♠ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts t♦ ✉s❡ t♦ s❝❛❧❡ t✶ ❛♥❞ t✷✳ ◆♦t❡ t❤❡ ❡①tr❡♠❡❧② t✐r❡s♦♠❡ ❝❛st✐♥❣ t❤❛t ♥❡❡❞s t♦ ❜❡ ❞♦♥❡ ❜❡❝❛✉s❡ ❛❧❧ ♦❢ t❤❡ st❛♥❞❛r❞ ❈ ♠❛t❤s ❧✐❜r❛r② ✐s ✐♥ ❞♦✉❜❧❡s✳ ❞❢❧ ❂ ✵✳✵❀ ❢♦r✭✐ ❂ ✶❀ ✐ ❁ ✹❀ ✐✰✰✮ ④ ①❞ ❂ ♣①❬✐❪ ✲ ♣①❬✐✲✶❪❀ ②❞ ❂ ♣②❬✐❪ ✲ ♣②❬✐✲✶❪❀ ③❞ ❂ ♣③❬✐❪ ✲ ♣③❬✐✲✶❪❀ ❞❢❧ ❂ ❞❢❧ ✰ ✭❢❧♦❛t✮sqrt✭✭❞♦✉❜❧❡✮ ✭①❞✯①❞ ✰ ②❞✯②❞ ✰ ③❞✯③❞✮✮❀ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✽ ⑥ ✐❢ ✭✐ ❂❂ ✶✮ ✯t✶ ❂ ❞❢❧❀ ✐❢ ✭✐ ❂❂ ✷✮ ✯t✷ ❂ ❞❢❧❀ ✐❢ ✭❞❢❧ ❁ ❆❈❈❨✮ ④ ❢♣r✐♥t❢✭st❞❡rr✱ ✧▲❛❣r❛♥❣❡✿ ❝✉r✈❡ t♦♦ s❤♦rt✿ ✪❢❭♥✧✱❞❢❧✮❀ r❡t✉r♥✭✶✮❀ ⑥ ✯t✶ ❂ ✯t✶✴❞❢❧❀ ✯t✷ ❂ ✯t✷✴❞❢❧❀ ✴✯ ✯✴ ❈❛❧❧ t❤❡ ♣r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡❢❢✐❝✐❡♥ts ✐♥ ❡❛❝❤ ❝♦♦r❞✐♥❛t❡✳ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣①✱♣♦❧②①✱t✶✱t✷✮✮ r❡t✉r♥✭✷✮❀ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣②✱♣♦❧②②✱t✶✱t✷✮✮ r❡t✉r♥✭✸✮❀ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣③✱♣♦❧②③✱t✶✱t✷✮✮ r❡t✉r♥✭✹✮❀ r❡t✉r♥✭✵✮❀ ⑥ ✴✯ ❧❛❣r❛♥❣❡ ✯✴ ✴✯ ✯✴ r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡❢❢✐❝✐❡♥ts ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ▲❛❣r❛♥❣✐❛♥ ❝✉❜✐❝ t❤r♦✉❣❤ ❢♦✉r ♣♦✐♥ts✳ ❚❤❡ ❝♦❞❡ r❡❢❧❡❝ts t❤❡ ❛❧❣❡❜r❛✳ ✐♥t ❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣✱♣♦❧②✱t✶✱t✷✮ ❢❧♦❛t ♣❬✹❪✱♣♦❧②❬✹❪❀ ❢❧♦❛t ✯t✶✱✯t✷❀ ④ ❢❧♦❛t ❞✶✱❞✷✱❞✸✱t✶s✱t✷s✱t✶❝✱t✷❝✱❞❡♥♦♠✱tt❀ ■♥t❡r♣♦❧❛t✐♦♥ ✸✾ ❞✶ ❂ ♣❬✶❪ ✲ ♣❬✵❪❀ ❞✷ ❂ ♣❬✷❪ ✲ ♣❬✵❪❀ ❞✸ ❂ ♣❬✸❪ ✲ ♣❬✵❪❀ t✶s ❂ ✭✯t✶✮✯✭✯t✶✮❀ t✷s ❂ ✭✯t✷✮✯✭✯t✷✮❀ t✶❝ ❂ t✶s✯✭✯t✶✮❀ t✷❝ ❂ t✷s✯✭✯t✷✮❀ ❞❡♥♦♠ ❂ ✭t✷s ✲ ✭✯t✷✮✮✯t✶❝❀ ✐❢ ✭❢❛❜s✭❞❡♥♦♠✮ ❁ ❆❈❈❨✮ ④ ❢♣r✐♥t❢✭st❞❡rr✱ ✧▲❛❣r❛♥❣❡❴❝♦❡❢❢s✿ ✐♥❝r❡♠❡♥ts t♦♦ s❤♦rt✿ ✪❢❭♥✧✱ ❞❡♥♦♠✮❀ r❡t✉r♥✭✶✮❀ ⑥ tt ❂ ✭✲t✷❝ ✰ ✭✯t✷✮✮✯t✶s ✰ ✭t✷❝ ✲ t✷s✮✯✭✯t✶✮❀ ♣♦❧②❬✸❪ ❂ ✭❞✸✯✭✯t✷✮ ✲ ❞✷✮✯t✶s ✰ ✭✲❞✸✯t✷s ✰ ❞✷✮✯✭✯t✶✮ ✰ ❞✶✯t✷s ✲ ❞✶✯✭✯t✷✮✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✷❪ ❂ ✭✲❞✸✯✭✯t✷✮ ✰ ❞✷✮✯t✶❝ ✰ ✭❞✸✯t✷❝ ✲ ❞✷✮✯✭✯t✶✮ ✰ ❞✶✯t✷❝ ✰ ❞✶✯✭✯t✷✮✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✶❪ ❂ ✭❞✸✯t✷s ✲ ❞✷✮✯t✶❝ ✰ ✭✲❞✸✯t✷❝ ✰ ❞✷✮✯t✶s ✰ ❞✶✯t✷❝ ✲ ❞✶✯t✷s✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✵❪ ❂ ♣❬✵❪❀ r❡t✉r♥✭✵✮❀ ⑥ ✴✯ ❧❛❣r❛♥❣❡❴❝♦❡❢❢s ✯✴ ❚❤❡ s❡❝♦♥❞ ♣r♦❝❡❞✉r❡ ❝♦♠♣✉t❡s t❤❡ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛♥t ✐♥ t❤r❡❡ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✹✵ ❞✐♠❡♥s✐♦♥s t❤r♦✉❣❤ t✇♦ ♣♦✐♥ts ✇✐t❤ t✇♦ ❣r❛❞✐❡♥t ✈❡❝t♦rs ❛t t❤❡ ❡♥❞s✳ ❚❤❡ ♣♦✐♥ts ❛r❡ ♣✵ ❛♥❞ ♣✶✱ ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ❛r❡ ❣✵ ❛♥❞ ❣✶✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❛r❡ r❡t✉r♥❡❞ ✐♥ ♣♦❧②①✱ ♣♦❧②②✱ ❛♥❞ ♣♦❧②③✳ ❚❤❡ ❛❧❣❡❜r❛ ✐♥ t❤✐s ❝❛s❡ ✐s ♠✉❝❤ s✐♠♣❧❡r t❤❛♥ t❤❛t ❢♦r ▲❛❣r❛♥❣✐❛♥ ✐♥t❡r♣♦❧❛t✐♦♥✳ ✈♦✐❞ ❤❡r♠✐t❡✭♣✵✱♣✶✱❣✵✱❣✶✱♣♦❧②①✱♣♦❧②②✱♣♦❧②③✮ ❢❧♦❛t ♣✵❬✸❪✱♣✶❬✸❪✱❣✵❬✸❪✱❣✶❬✸❪❀ ❢❧♦❛t ♣♦❧②①❬✹❪✱♣♦❧②②❬✹❪✱♣♦❧②③❬✹❪❀ ④ ✈♦✐❞ ❤❡r♠✐t❡❴❝♦❡❢❢s✭✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✵❪✱♣✶❬✵❪✱❣✵❬✵❪✱❣✶❬✵❪✱♣♦❧②①✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✶❪✱♣✶❬✶❪✱❣✵❬✶❪✱❣✶❬✶❪✱♣♦❧②②✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✷❪✱♣✶❬✷❪✱❣✵❬✷❪✱❣✶❬✷❪✱♣♦❧②③✮❀ ⑥ ✴✯ ❤❡r♠✐t❡ ✯✴ ✈♦✐❞ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵✱♣✶✱❣✵✱❣✶✱♣♦❧②✮ ❢❧♦❛t ♣✵✱♣✶✱❣✵✱❣✶❀ ❢❧♦❛t ♣♦❧②❬✹❪❀ ④ ❢❧♦❛t ❞✱❣❀ ❞ ❂ ♣✶ ✲ ♣✵ ✲ ❣✵❀ ❣ ❂ ❣✶ ✲ ❣✵❀ ♣♦❧②❬✵❪ ♣♦❧②❬✶❪ ♣♦❧②❬✷❪ ♣♦❧②❬✸❪ ❂ ❂ ❂ ❂ ♣✵❀ ❣✵❀ ✸✳✵✯❞ ✲ ❣❀ ✲✷✳✵✯❞ ✰ ❣❀ ⑥ ✴✯ ❤❡r♠✐t❡❴❝♦❡❢❢s ✯✴ Surface patches ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✶ Q = F(t, u) 0 ≤ t ≤ 1, 0 ≤ u ≤ 1✳ ✸✭✐✐✮✖❆ ♣❛r❛♠❡tr✐❝ ♣❛t❝❤ ✐♥t❡r✈❛❧ ❞❡✜♥❡❞ ♦✈❡r t❤❡ ❙✉r❢❛❝❡ ♣❛t❝❤❡s ❛r❡ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s ♦❢ t❤❡ ❢♦r♠ x = f1 (t, u) y = f2 (t, u) z = f3 (t, u) ✭✇❤✐❝❤ ✇❡ ❝❛♥ ❛❧s♦ ✇r✐t❡ ✇✐t❤ ✈❡❝t♦r ❝♦❡✣❝✐❡♥ts ✱ ❛s Q = F(t, u) ❛ ♣❛♣❡r✲s❛✈✐♥❣ ♠❡❛s✉r❡ t❤❛t ✇✐❧❧ ❜❡ ✐♥❝r❡❛s✐♥❣❧② ✉s❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✮✳ ❆ ♣❛t❝❤ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦✈❡r ❛♥② ♣❛r❛♠❡tr✐❝ ♣♦rt✐♦♥ ♦❢ t❤❡ (t, u) ♣❛r❛♠❡t❡r s♣❛❝❡ ✱ ❜✉t ✐s ❡❛s✐❡st t♦ ❞❡❛❧ ✇✐t❤ ♦✈❡r t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✐♥t❡r✈❛❧✿ 0≤ t ≤1 0 ≤ u ≤ 1.

Dt ❚❤❡ ❝✉r✈❛t✉r❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r✲ ♠✉❧❛✱ ✇❤✐❝❤ ✐♥✈♦❧✈❡s ❛ ✈❡❝t♦r ♣r♦❞✉❝t ❜❡t✇❡❡♥ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿ dQ × d2 Q .

T✱ ❢♦r ❛♥② ✜①❡❞ ✈❛❧✉❡ ♦❢ t ♦r u✱ ✇❡ ❣❡t ♦✉t ❛ ❝✉❜✐❝ ✐s♦✲♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ t❤❡ ♦t❤❡r ♣❛r❛♠❡t❡r✳ ❆s ✇❡❧❧ ❛s ❦♥♦✇✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❝✉r✈❡s✱ ✇❡ ♥❡❡❞ t♦ ♠❛t❝❤ t❛♥❣❡♥ts✖❛♥❞ ♠❛②❜❡ ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s✱ r❛❞✐✉s ♦❢ ❝✉r✈❛✲ t✉r❡ ❡t❝✳✖ ❛❝r♦ss t❤❡ ❜♦✉♥❞❛r✐❡s✳ t❛♥❣❡♥t ❛❝r♦ss t❤❡ u=0 ❞✐✛❡r❡♥t✐❛t❡ ✇✐t❤ r❡s♣❡❝t t♦ ∂x = ∂u + + + ❚❤❡♥ s❡t u=0 ▲❡t✬s ✜♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❡❞❣❡ ♦❢ ❛ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t ♣❛t❝❤✳ ❋✐rst u✿ 3a0 t3 u2 + 2a1 t3 u + a2 t3 3a4 t2 u2 + 2a5 t2 u + a6 t2 3a8 tu2 + 2a9 tu + a10 t 3a12 u2 + 2a13 u + a14 .

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