calculo

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪
❈➪▲❈❯▲❖ ■■ ✲ P❘❖❏❊❚❖ ◆❊❲❚❖◆
❆❯▲❆ ✵✽
❆ss✉♥t♦✿❋✉♥çõ❡s ❞❡ ✈ár✐❛s ✈❛r✐á✈❡✐s r❡❛✐s ❛ ✈❛❧♦r❡s r❡❛✐s✱ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠✱ ❝✉r✈❛s ❞❡ ♥í✈❡❧✱
❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s r❡❛✐s ❛ ✈❛❧♦r❡s r❡❛✐s
P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ❢✉♥çã♦ ❞❡ ✈ár✐❛s ✈❛r✐á✈❡✐s✱ ❞♦♠í♥✐♦✱ ✐♠❛❣❡♠✱ ❝✉r✈❛s ❞❡ ♥í✈❡❧✱ ❣rá✜❝♦
❋✉♥çõ❡s ❞❡ ✈ár✐❛s ✈❛r✐á✈❡✐s r❡❛✐s ❛ ✈❛❧♦r❡s r❡❛✐s ✭❋✉♥çõ❡s ❞❡ Rn ❡♠ R✮
❯♠❛ ❢✉♥çã♦ ❞❡ n ✈❛r✐á✈❡✐s r❡❛✐s ❛ ✈❛❧♦r❡s r❡❛✐s é ✉♠❛ r❡❣r❛ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♥❴✉♣❧❛ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦
❞♦ Rn ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧✳
❙❡ B ❞❡♥♦t❛ ♦ s✉❜❝♦♥❥✉♥t♦ ❡ f ❛ r❡❣r❛✱ ❛ ❢✉♥çã♦ é ❞❡♥♦t❛❞❛ ♣♦r f : B ⊂ Rn −→ R

♦✉

f : B ⊂ Rn

−→

(x1 , x2 , ..., xn ) −→

R f (x1 , x2 , ..., xn )

♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ ♣♦r f (x1 , x2 , ..., xn )

◗✉❛♥❞♦ n = 2 ❡s❝r❡✈❡♠♦s f (x, y) ♦✉ z = f (x, y)

◗✉❛♥❞♦ n = 3 f (x, y, z)

▼✉✐t❛s ❣r❛♥❞❡③❛s sã♦ ❡s❝r✐t❛s ♣♦r ❢✉♥çõ❡s ❞❡ ✈ár✐❛s ✈❛r✐á✈❡✐s r❡❛✐s ❛ ✈❛❧♦r❡s ❡♠ r❡❛✐s✳ P♦r ❡①❡♠♣❧♦✱
✶✳ ❆ t❡♠♣❡r❛t✉r❛ T ❞❡ ✉♠❛ ❝❤❛♣❛ ♠❡tá❧✐❝❛ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❛s ❝♦♦r❞❡♥❛❞❛s x ❡ y ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❛ ❝❤❛♣❛✳

✷✳ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ✉♠ só❧✐❞♦ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❛s ❝♦♦r❞❡♥❛❞❛s x, y, ❡ z ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞♦ só❧✐❞♦✳

✸✳ ❖ ✈♦❧✉♠❡ ❝♦♥t✐❞♦ ❡♠ ✉♠ r❡s❡r✈❛tór✐♦✱ q✉❡ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦✱ q✉❡ ❡stá s❡♥❞♦ ❡♥❝❤✐❞♦
❞❡ á❣✉❛✱ ❞❡♣❡♥❞❡ ❞❛ ❛❧t✉r❛ h ❞♦ ✧❝♦♥❡ ❞❡ á❣✉❛✧❡ ❞♦ r❛✐♦ ❞❛ ❜❛s❡ r ❞❡ t❛❧ ❝♦♥❡✳

✹✳ ❖ ❧✉❝r♦ ♦❜t✐❞♦ ♣♦r ✉♠❛ ❡♠♣r❡s❛ ❝♦♠ ❛ ✈❡♥❞❛ ❞❡ ✉♠ ♣r♦❞✉t♦✱ ♣♦r ❡❧❛ ♣r♦❞✉③✐❞♦✱ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❡

✷

x1

=

❝✉st♦ ❝♦♠ ♠❛tér✐❛ ♣r✐♠❛

x2

=

❝✉st♦ ❝♦♠ ♣❡ss♦❛❧

x3

=

❝✉st♦ ❝♦♠ ✐♠♣♦st♦s

x4

=

❝✉st♦ ❝♦♠ ♣r♦♣❛❣❛♥❞❛

x5

=

✈❛❧♦r ✈❡♥❛❧ ❞♦ ♣r♦❞✉t♦

l(x1 , x2 , x3 , x4 , x5 )

=

❧✉❝r♦

❆s ❢✉♥çõ❡s ❝♦♠ ❛s q✉❛✐s tr❛❜❛❧❤❛r❡♠♦s s❡rã♦ ❡①♣r❡ss❛s ♣♦r ❢ór♠✉❧❛s✱ t❛✐s ❝♦♠♦

f (x, y)

=

x2 − 2xy 3

= ex cos y x−y f (x, y, z) = z f (x1 , x2 , x3 , x4 ) = x21 + x22 + x23 + x24 f (x, y)

❈♦♥s✐❞❡r❡♠♦s ♠❛✐s ❡st❡ ❡①❡♠♣❧♦ f (x, y) =

x2 x−y ❚❡♠♦s q✉❡ f (2, 3) =

22
4
=
= −4
2−3
−1

P♦❞❡♠♦s ✐♥t❡r♣r❡t❛r ❡ss❛ ❢✉♥çã♦ ❝♦♠♦ s❡♥❞♦ ✉♠❛ ✧♠áq✉✐♥❛✧✳