❈❛❧❝✉❧✉s ■■■ ✭♣❛rt ✶✮✿ ❱❡❝t♦rs ❛♥❞ ✸✲❉✐♠❡♥s✐♦♥❛❧ ●❡♦♠❡tr② ✭❜② ❊✈❛♥ ❉✉♠♠✐t✱ ✷✵✷✶✱ ✈✳ ✹✳✵✵✮
               ❈♦♥t❡♥ts
               ✶ ❱❡❝t♦rs ❛♥❞ ✸✲❉✐♠❡♥s✐♦♥❛❧ ●❡♦♠❡tr②                                                                                            ✶
                   ✶✳✶   ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ❛♥❞ ✸✲❙♣❛❝❡         ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳    ✶
                         ✶✳✶✳✶   ●r❛♣❤✐♥❣ ❋✉♥❝t✐♦♥s ♦❢ ✷ ❱❛r✐❛❜❧❡s✿ ❙✉r❢❛❝❡s ❛♥❞ ▲❡✈❡❧ ❙❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳              ✷
                         ✶✳✶✳✷   ●r❛♣❤✐♥❣ ❋✉♥❝t✐♦♥s ♦❢ ✸ ❱❛r✐❛❜❧❡s✿ ▲❡✈❡❧ ❙✉r❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳             ✺
                   ✶✳✷   ❱❡❝t♦rs✱ ❉♦t ❛♥❞ ❈r♦ss Pr♦❞✉❝ts✱ ▲✐♥❡s ❛♥❞ P❧❛♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳             ✻
                         ✶✳✷✳✶   ❱❡❝t♦rs ❛♥❞ ❱❡❝t♦r ❖♣❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳           ✻
                         ✶✳✷✳✷   ❚❤❡ ❉♦t Pr♦❞✉❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳         ✽
                         ✶✳✷✳✸   ❚❤❡ ❈r♦ss Pr♦❞✉❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳        ✶✶
                         ✶✳✷✳✹   ▲✐♥❡s ❛♥❞ P❧❛♥❡s ✐♥ ✸✲❙♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳        ✶✷
                   ✶✳✸   ❱❡❝t♦r✲❱❛❧✉❡❞ ❋✉♥❝t✐♦♥s✱ ❈✉r✈❡s ❛♥❞ ▼♦t✐♦♥ ✐♥ ✸✲❙♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳              ✶✺
                         ✶✳✸✳✶   ❱❡❝t♦r✲❱❛❧✉❡❞ ❋✉♥❝t✐♦♥s ❛♥❞ ❈✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳           ✶✺
                         ✶✳✸✳✷   ▼♦t✐♦♥ ✐♥ ✸✲❙♣❛❝❡✿ ❱❡❧♦❝✐t② ❛♥❞ ❆❝❝❡❧❡r❛t✐♦♥❀ ❆r❝❧❡♥❣t❤❀ ❯♥✐t ❚❛♥❣❡♥t✱ ◆♦r♠❛❧✱ ❛♥❞ ❇✐♥♦r✲
                                 ♠❛❧ ❱❡❝t♦rs❀ ❈✉r✈❛t✉r❡ ❛♥❞ ❚♦rs✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳           ✶✼
               ✶     ❱❡❝t♦rs ❛♥❞ ✸✲❉✐♠❡♥s✐♦♥❛❧ ●❡♦♠❡tr②
               ❋r♦♠ ♦♥❡✲✈❛r✐❛❜❧❡ ❝❛❧❝✉❧✉s✱ ✇❡ ❛r❡ ✈❡r② ✉s❡❞ t♦ ✷✲❞✐♠❡♥s✐♦♥❛❧ ✏♣❧❛♥❛r✑ ❣❡♦♠❡tr②✿ ❣✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f(x)✱ ✇❡ ❦♥♦✇
               ❤♦✇ t♦ ♣r♦❞✉❝❡ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝✉r✈❡ y = f(x) ❛♥❞ ✉s❡ ❝❛❧❝✉❧✉s t♦ ✐❞❡♥t✐❢② s♦♠❡ ♦❢ ✐ts ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s s✉❝❤
               ❛s ❧♦❝❛❧ ♠✐♥✐♠❛ ❛♥❞ ♠❛①✐♠❛✳ ❲❡ ❛❧s♦ ❦♥♦✇ ❤♦✇ t♦ ❣r❛♣❤ ♣❛r❛♠❡tr✐❝❛❧❧②✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s t❤❡ ❝②❝❧♦✐❞
               x = t+sin(t)✱ y = 1+cos(t)✳ ❆♥❞ ✇❡ ❡✈❡♥ ❦♥♦✇✱ ♠♦r❡ ♦r ❧❡ss✱ ❤♦✇ t♦ ❣r❛♣❤ ✐♠♣❧✐❝✐t❧②✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥s✱ ❧✐❦❡ t❤❡
               ❝✐r❝❧❡ x2 + y2 = 1✳
               ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✐♥tr♦❞✉❝❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✭♦❢t❡♥ ❝❛❧❧❡❞ ✸✲s♣❛❝❡ ❢♦r s❤♦rt✮ ❛♥❞ st✉❞② t❤❡ ❣❡♦♠❡tr② ♦❢ ❧✐♥❡s
               ❛♥❞ ♣❧❛♥❡s✱ ❛s ✇❡❧❧ ❛s ♠♦r❡ ❣❡♥❡r❛❧ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ ✉s✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❆❧♦♥❣ t❤❡ ✇❛②✱ ✇❡ ✇✐❧❧
               ✐♥tr♦❞✉❝❡ ❛♥❞ st✉❞② ✈❡❝t♦rs✱ ✇❤✐❝❤ ✇✐❧❧ ❤❡❧♣ t♦ ❝❧❛r✐❢② ❛ ❣r❡❛t ❞❡❛❧ ♦❢ t❤❡ ❝♦♥❝❡♣ts✳
               ✶✳✶     ❋✉♥❝t✐♦♥s ♦❢ ❙❡✈❡r❛❧ ❱❛r✐❛❜❧❡s ❛♥❞ ✸✲❙♣❛❝❡
                   • ❆ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✐s✱ ❛s t❤❡ ♥❛♠❡ ✐♥❞✐❝❛t❡s✱ ❛ ❢✉♥❝t✐♦♥ t❤❛t t❛❦❡s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ ♦✉t♣✉ts
                      ❛ ✈❛❧✉❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✐♥♣✉ts✳
                         ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❝t✐♦♥ f(x,y) = x+y t❛❦❡s ✐♥ t✇♦ ✈❛❧✉❡s x ❛♥❞ y ❛♥❞ ♦✉t♣✉ts t❤❡✐r s✉♠ x+y✳
                         ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❝t✐♦♥ g(x,y,z) = xyz2 t❛❦❡s ✐♥ t❤r❡❡ ✈❛❧✉❡s x✱ y✱ ❛♥❞ z✱ ❛♥❞ ♦✉t♣✉ts t❤❡ ♣r♦❞✉❝t
                           xyz2✳
                         ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❝t✐♦♥ d(l,w) = √l2 +w2 ❣✐✈❡s t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❡❝t❛♥❣❧❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢
                           t❤❡ r❡❝t❛♥❣❧❡✬s ❧❡♥❣t❤ l ❛♥❞ ✐ts ✇✐❞t❤ w✳
                         ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❝t✐♦♥ V(r,h) = 1πr2h ❣✐✈❡s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ✭r✐❣❤t ❝✐r❝✉❧❛r✮ ❝♦♥❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐ts
                                                                   3
                           ❜❛s❡ r❛❞✐✉s r ❛♥❞ ✐ts ❤❡✐❣❤t h✳
                                                                               ✶
                 • ❆s ✇✐t❤ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❤❛s ❛ ❞♦♠❛✐♥ ❛♥❞ ❛ r❛♥❣❡✿ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡
                   s❡t ♦❢ ✐♥♣✉t ✈❛❧✉❡s ❛♥❞ t❤❡ r❛♥❣❡ ✐s t❤❡ s❡t ♦❢ ♦✉t♣✉t ✈❛❧✉❡s✳
                      ◦ ❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s f(x,y)✱ t❤❡ ❞♦♠❛✐♥ ✐s ♥♦✇ ❛ s✉❜s❡t ♦❢ t❤❡ ✷✲❞✐♠❡♥s✐♦♥❛❧ ♣❧❛♥❡ r❛t❤❡r
                        t❤❛♥ ❛ s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✳ ❇❡❝❛✉s❡ ♦❢ t❤✐s✱ ❞♦♠❛✐♥s ♦❢ ❢✉♥❝t✐♦♥s ♦❢ ♠♦r❡ t❤❛♥ ♦♥❡ ✈❛r✐❛❜❧❡ ❝❛♥ ❜❡
                        r❛t❤❡r ❝♦♠♣❧✐❝❛t❡❞✳
                      ◦ ■♥ ❣❡♥❡r❛❧✱ ✉♥❧❡ss s♣❡❝✐✜❡❞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ s❡t ♦❢ ✐♥♣✉ts ❢♦r ✇❤✐❝❤ t❤❡
                        ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♠❛❦❡s s❡♥s❡✳ ❲❡ ❣❡♥❡r❛❧❧② ❛❞♦♣t t❤❡ ❝♦♥✈❡♥t✐♦♥s t❤❛t sq✉❛r❡ r♦♦ts ♦❢ ♥❡❣❛t✐✈❡
                        r❡❛❧ ♥✉♠❜❡rs ❛r❡ ♥♦t ❛❧❧♦✇❡❞✱ ♥♦r ✐s ❞✐✈✐s✐♦♥ ❜② ③❡r♦✳
                                                  √     √
                      ◦ ❊①❛♠♣❧❡✿ ❋♦r f(x,y) =       x +   y✱ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ ✜rst q✉❛❞r❛♥t ♦❢ t❤❡ xy✲♣❧❛♥❡✱ ❞❡✜♥❡❞ ❜② t❤❡
                        ✐♥❡q✉❛❧✐t✐❡s x ≥ 0 ❛♥❞ y ≥ 0✳
                                                  p 2     2
                      ◦ ❊①❛♠♣❧❡✿ ❋♦r f(x,y) =       x +y −1✱ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ xy✲♣❧❛♥❡ ✇❤✐❝❤ s❛t✐s❢②
                        x2+y2 ≥1✿ t❤✐s ❞❡s❝r✐❜❡s ❛❧❧ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ♣❧❛♥❡ ❡①❝❡♣t ❢♦r t❤♦s❡ ❧②✐♥❣ str✐❝t❧② ✐♥s✐❞❡ t❤❡ ✉♥✐t ❝✐r❝❧❡✳
                      ◦ ❊①❛♠♣❧❡✿ ❋♦r f(x,y) =      1  ✱ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ xy✲♣❧❛♥❡ ✇❤✐❝❤ s❛t✐s❢② x − y 6= 0✿
                                                 x−y
                        t❤✐s ❞❡s❝r✐❜❡s ❛❧❧ ♣♦✐♥ts ✐♥ t❤❡ ♣❧❛♥❡ ❡①❝❡♣t t❤♦s❡ ♦♥ t❤❡ ❧✐♥❡ y = x✳
                 • P♦✐♥ts ✐♥ ✸✲s♣❛❝❡ ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ tr✐♣❧❡t ♦❢ ♥✉♠❜❡rs (x,y,z)✳ ❚❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡ z r❡♣r❡s❡♥ts ❤❡✐❣❤t
                   ❛❜♦✈❡ t❤❡ xy✲♣❧❛♥❡✳
                      ◦ ❋r♦♠ ❛♥ ❛❧❣❡❜r❛✐❝ st❛♥❞♣♦✐♥t✱ ✸✲s♣❛❝❡ ❜❡❤❛✈❡s q✉✐t❡ s✐♠✐❧❛r❧② t♦ ✷✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳
                      ◦ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ✸✲s♣❛❝❡✱ ✇❡ ❝❛♥ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts✳ ❇② ❛ s✉✐t❛❜❧❡ ♣❛✐r ♦❢
                        ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts (x ,y ,z )
                                                   p                                                                   1   1  1
                        ❛♥❞ (x ,y ,z ) ✐s ❣✐✈❡♥ ❜②   (x −x )2+(y −y )2+(z −z )2✳
                               2  2  2                  1    2      1    2       1    2
             ✶✳✶✳✶   ●r❛♣❤✐♥❣ ❋✉♥❝t✐♦♥s ♦❢ ✷ ❱❛r✐❛❜❧❡s✿ ❙✉r❢❛❝❡s ❛♥❞ ▲❡✈❡❧ ❙❡ts
                 • ❚❤❡r❡ ❛r❡ t✇♦ ♣r✐♠❛r② ✇❛②s t♦ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥ f(x,y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳
                 • ❚❤❡ ✜rst ✇❛② ✐s t♦ ♣❧♦t t❤❡ ♣♦✐♥ts (x,y,z) ✐♥ ✸✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ s❛t✐s❢②✐♥❣ z = f(x,y)✳
                      ◦ ❆t t❤❡ ♣♦✐♥t (x,y) ✐♥ t❤❡ ♣❧❛♥❡✱ t❤❡ ❣r❛♣❤ ❤❛s t❤❡ ❤❡✐❣❤t z = f(x,y)❀ s♦ ✇❡ s❡❡ t❤❛t ❛s (x,y) ✈❛r✐❡s
                        t❤r♦✉❣❤ t❤❡ ♣❧❛♥❡✱ t❤❡ ❢✉♥❝t✐♦♥ z = f(x,y) ✇✐❧❧ tr❛❝❡ ♦✉t ❛ s✉r❢❛❝❡✱ ❝❛❧❧❡❞ t❤❡ ❣r❛♣❤ ♦❢ f(x,y)✳
                 • ❙♦♠❡ ❡①❛♠♣❧❡ ♦❢ s✐♠♣❧❡ ❣r❛♣❤s z = f(x,y) ❛r❡ ❣✐✈❡♥ ❜❡❧♦✇✳
                      ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = 0 ✐s t❤❡ xy✲♣❧❛♥❡✳
                      ◦ ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = x+y ✐s ❛ t✐❧t❡❞ ♣❧❛♥❡✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ax+by+cz = d
                        ❢♦r ❝♦♥st❛♥ts a,b,c,d ✭✇✐t❤ ♥♦t ❛❧❧ ♦❢ a,b,c ③❡r♦✮ ✇✐❧❧ ❣✐✈❡ ❛ ♣❧❛♥❡✱ ❛s ✇❡ ✇✐❧❧ ❞✐s❝✉ss ❧❛t❡r✳
                 • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = x2 +y2 ✐s ❛ ♣❛r❛❜♦❧♦✐❞ ✭✐✳❡✳✱ ❛ ♣❛r❛❜♦❧✐❝ ❞✐s❤✮✳
                 • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = px2 +y2 ✐s ❛ r✐❣❤t ❝✐r❝✉❧❛r ❝♦♥❡ ♦♣❡♥✐♥❣ ✉♣✇❛r❞✱ ✇✐t❤ ✈❡rt❡① ❛t t❤❡ ♦r✐❣✐♥✳
                                                                      ✷
                                                       2    2
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ ♦❢ z = x −y ✐s ❛ ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞✱ ❛❧s♦ ❝❛❧❧❡❞ ❛ s❛❞❞❧❡ s✉r❢❛❝❡✳
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = x2 ✐s ❛ ♣❛r❛❜♦❧✐❝ ❝②❧✐♥❞❡r✳
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = x3−3xy2 ✐s s♦♠❡t✐♠❡s ❝❛❧❧❡❞ t❤❡ ✏♠♦♥❦❡② s❛❞❞❧❡✑✱ ❛s ✐t ❤❛s t❤r❡❡ ❞❡♣r❡ss✐♦♥s r❛t❤❡r
                     t❤❛♥ t❤❡ t✇♦ ❢♦r t❤❡ r❡❣✉❧❛r s❛❞❞❧❡ ✭♦♥❡ ❢♦r ❡❛❝❤ ❧❡❣✱ ❛♥❞ ♦♥❡ ❢♦r t❤❡ t❛✐❧✮✳
                                                      √ 2    2       p           
                                                   3− x +y /12             2     2
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❣r❛♣❤ z = e                     cos    x +y       ♣r♦❞✉❝❡s ❛ s✉r❢❛❝❡ t❤❛t ❧♦♦❦s ❧✐❦❡ r✐♣♣❧❡s ✐♥ ❛ ♣♦♦❧ ♦❢
                     ✇❛t❡r✳
                   • ❚❤❡ s❡❝♦♥❞ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥ f(x,y) ✐s t♦ ♣❧♦t t❤❡ ♣♦✐♥ts (x,y) ✐♥ t❤❡ ♣❧❛♥❡ ♦♥ t❤❡ ❧❡✈❡❧ s❡ts
                     f(x,y) = c ❢♦r ♣❛rt✐❝✉❧❛r ✈❛❧✉❡s ♦❢ c✱ ❛s ✐♠♣❧✐❝✐t ❝✉r✈❡s✳
                         ◦ ❋♦r ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ f(x,y) ❛♥❞ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦❢ c✱ t❤❡ ♣♦✐♥ts (x,y) s❛t✐s❢②✐♥❣ f(x,y) = c ❛r❡ ❝❛❧❧❡❞
                           ❛ ❧❡✈❡❧ s❡t ♦❢ f✳
                         ◦ ❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s t❤❡s❡ s❡ts ✇✐❧❧ ❣❡♥❡r❛❧❧② ❜❡ ❝✉r✈❡s✱ s♦ t❤❡② ❛r❡ ❛❧s♦ s♦♠❡t✐♠❡s ❝❛❧❧❡❞
                           ❧❡✈❡❧ ❝✉r✈❡s✳
                         ◦ ▲❡✈❡❧ s❡ts ❛r❡ ♦❜t❛✐♥❡❞ ❜② ✐♥t❡rs❡❝t✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ z = c ✇✐t❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s✉r❢❛❝❡ z = f(x,y)✳
                         ◦ ■❢ ✇❡ ❣r❛♣❤ ♠❛♥② ♦❢ t❤❡s❡ ❧❡✈❡❧ ❝✉r✈❡s t♦❣❡t❤❡r ♦♥ t❤❡ s❛♠❡ ❛①❡s✱ ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ✏t♦♣♦❣r❛♣❤✐❝❛❧ ♠❛♣✑
                           ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f(x,y)✳
                                                                                2    2
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❧❡✈❡❧ s❡ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f(x,y) = x +y ❛r❡ ❝✐r❝❧❡s ✐♥ t❤❡ ♣❧❛♥❡✳
                                                               2    2                                           √
                         ◦ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ ❧❡✈❡❧ s❡t x + y = c ✭❢♦r c > 0✮ ✐s ❛ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s          c ❝❡♥t❡r❡❞ ❛t (0,0)✳
                         ◦ ❋♦r c = 0 t❤❡ ❧❡✈❡❧ s❡t ✐s ❥✉st t❤❡ s✐♥❣❧❡ ♣♦✐♥t (0,0)✱ ❛♥❞ ❢♦r c < 0 t❤❡ ❧❡✈❡❧ s❡ts ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛♥② ♣♦✐♥ts
                           ❛t ❛❧❧✳
                         ◦ ❇❡❧♦✇ ♦♥ t❤❡ ❧❡❢t ❛r❡ ♣❧♦tt❡❞ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ✐♥ t❤❡ ♣❧❛♥❡ ❢♦r c = 1,2,3,...,9✱ ✇❤✐❧❡ ♦♥ t❤❡ r✐❣❤t t❤❡
                           ❧❡✈❡❧ s❡ts ❤❛✈❡ ❜❡❡♥ ❞r❛✇♥ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s✉r❢❛❝❡ z = f(x,y) ✐ts❡❧❢✱ t♦ ✐❧❧✉str❛t❡ ❤♦✇ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s
                           ❛r✐s❡ ❜② ✐♥t❡rs❡❝t✐♥❣ t❤❡ s✉r❢❛❝❡ z = f(x,y) ✇✐t❤ t❤❡ ✈❛r✐♦✉s ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡s✳
                                                                              ✸
                                                                               2    2
                   • ❊①❛♠♣❧❡✿ ❚❤❡ ❧❡✈❡❧ s❡ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f(x,y) = x −y ❛r❡ ❤②♣❡r❜♦❧❛s✳
                        ◦ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ ❧❡✈❡❧ s❡t x2 − y2 = c ✭❢♦r c 6= 0✮ ❛r❡ ❤②♣❡r❜♦❧❛s ✭❡❛❝❤ ♦♥❡ ❝♦♥s✐sts ♦❢ t✇♦ ♣✐❡❝❡s✮✱
                           ✇❤✐❧❡ ❢♦r c = 0 t❤❡ ❧❡✈❡❧ s❡t ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t✇♦ ❧✐♥❡s y = x ❛♥❞ y = −x✳
                        ◦ ❇❡❧♦✇ ♦♥ t❤❡ ❧❡❢t ❛r❡ ♣❧♦tt❡❞ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ✐♥ t❤❡ ♣❧❛♥❡ ❢♦r c = −5,−4,−3,...,4,5✱ ✇❤✐❧❡ ♦♥ t❤❡ r✐❣❤t
                           t❤❡ ❧❡✈❡❧ s❡ts ❤❛✈❡ ❜❡❡♥ ❞r❛✇♥ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s✉r❢❛❝❡ z = f(x,y) ✐ts❡❧❢✳
                        ◦ ❆s✇❡♥♦t❡❞❡❛r❧✐❡r✱ t❤❡ ❣r❛♣❤ ♦❢ z = x2−y2 ✐s ❝❛❧❧❡❞ ❛ ❤②♣❡r❜♦❧✐❝ ♣❛r❛❜♦❧♦✐❞✿ ✐t ❤❛s t❤✐s ♥❛♠❡ ❜❡❝❛✉s❡ ✐ts
                           ❤♦r✐③♦♥t❛❧ ❝r♦ss✲s❡❝t✐♦♥s ✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❧❡✈❡❧ s❡ts✮ ❛r❡ ❤②♣❡r❜♦❧❛s ✇❤✐❧❡ ✐ts ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥s
                           ❛r❡ ♣❛r❛❜♦❧❛s✳
                                                                                                               2   2
                                                                                                   2    2 2 −x −y
                   • ❊①❛♠♣❧❡✿ ❇❡❧♦✇❛r❡♣❧♦tt❡❞t❤❡❧❡✈❡❧❝✉r✈❡s❢♦rt❤❡❢✉♥❝t✐♦♥f(x,y) = (x −y ) e                         ❢♦r c = 0.04,0.14,...0.54✱
                     ❛❧♦♥❣ ✇✐t❤ t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ♣❧♦t ♦❢ z = f(x,y)✿
                                                                             ✹