Tauri Eu=Ei*u+Ei¥7 Hi-Fi

y-- flu ) ; U=gCx) ; x=xH) Y

dyf duUoff -- adf.dfx.DE'tIIIx

DXLatt

-

case 't w=fCx,y ) food x -- xct ), y

-

- yet) ur dewat

W

:#Tats

.

fdaf-iff.daftzey.IT/ chain ruleX Y vcsidonorruonooisillls

⇒ da-

dt di

t

case 2 w -- f- (x.y ) Tauri x - Xiu ,u ),y=yC4N ) at Itf , Zhou

ETE, Eu=Ei*u+Ei¥7x y

*Eau Hi-Fi + Ey ' :L,

u v u v-

ïììĊę ôŦÜÖŤßĆîĀúć÷êĆüĒðøĒúąĂîčóĆîíŤ÷ŠĂ÷

ĂîčóĆîíŤ÷ŠĂ÷×ĂÜôŦÜÖŤßĆîðøąÖĂïĒúąôŦÜÖŤßĆîēé÷ðøĉ÷ć÷ 1BSUJBMEFSJWBUJWFT PG DPNQPTJUF BOE JNQMJDJU GVODUJPOT

ĂîčóĆîíŤ÷ŠĂ÷×ĂÜôŦÜÖŤßĆîðøąÖĂï

ĔîĀĆü×šĂîĊĚđøćÝąÖúŠćüëċÜĂîčóĆîíŤ÷ŠĂ÷×ĂÜôŦÜÖŤßĆîðøąÖĂïÖŠĂî àċęÜÖĘÙČĂÖćøýċÖþćÖäúĎÖēàŠÿĈĀøĆïôŦÜÖŤßĆî

Āúć÷êĆüĒðøîĆęîđĂÜ đîČęĂÜÝćÖôŦÜÖŤßĆîĀúć÷êĆüĒðøöĊĂîčóĆîíŤĕéšĀúć÷Ēïï ÖäúĎÖēàŠÿĈĀøĆïôŦÜÖŤßĆîĀúć÷

êĆüĒðøÝċÜöĊĕéšĀúć÷ĒïïđßŠîÖĆî ĔîìĊęîĊĚđøćÝąóĉÝćøèćÖäúĎÖēàŠÿĈĀøĆïôŦÜÖŤßĆîÿĂÜêĆüĒðø ēé÷ĒïŠÜÖćø

óĉÝćøèćđðŨîÖøèĊêŠćÜė éĆÜîĊĚ

ÖøèĊìĊę ĔĀš w = f(x, y) đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš×ĂÜ x Ēúą y ēé÷ìĊę x = g(t) Ēúą y = h(t)

đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš×ĂÜ t ÝąĕéšüŠć

dw

dt=

∂w

∂x· dxdt

+∂w

∂y· dydt

ÿćöćøëîĈĕðđ×Ċ÷îđðŨîĒñîõĎöĉĕéšéĆÜîĊĚ

êĆüĂ÷ŠćÜ ÖĈĀîéĔĀš w = sinx cos y ēé÷ìĊę x = sin t Ēúą y = cos t ÝÜĀć dz

dt

⇐ we ¥=Fx¥ttEydat.ArX Y =(cosxcosy) (cost)t( sinxlfsiny)f-sint )d¥/dI = cosxcosycosttsinxsinysint #tdt


ÖøèĊìĊę ĔĀš w = f(x, y) đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš×ĂÜ x Ēúą y ēé÷ìĊę x = g(s, t) Ēúą y = h(s, t)

đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš×ĂÜ s Ēúą t ÝąĕéšüŠć

∂z

∂s=

∂z

∂x· ∂x∂s

+∂z

∂y· ∂y∂s

∂z

∂t=

∂z

∂x· ∂x∂t

+∂z

∂y· ∂y∂t

ÿćöćøëîĈĕðđ×Ċ÷îđðŨîĒñîõĎöĉĕéšéĆÜîĊĚ

êĆüĂ÷ŠćÜ ÖĈĀîéĔĀš z = arctan(2x+ y) ēé÷ìĊę x = s2t Ēúą y = s ln t ÝÜĀć ∂z

∂sĒúą ∂z

∂t

êĆüĂ÷ŠćÜ ÝÜđ×Ċ÷îÖäúĎÖēàŠÿĈĀøĆïôŦÜÖŤßĆî w = f(x, y, z, t) ēé÷ìĊę x = x(u, v) y = y(u, v) z = (u, v) Ēúą t = t(u, v)

3¥53 ¥ -

- II. Est # Is-

x y3¥# Effort =¥¥tIl2st)t4×¥¥Cent )s t s t # =3¥Fft II. Ft

-

- 4*1*151+1 HE '

w

m II -- Ex '¥u+Fi7utFEutIf3Ix y z t

an ax 37--39 ' Ey + FistU V U V U V U V


êĆüĂ÷ŠćÜ ÖĈĀîéĔĀš M = xey−z2 ēé÷ìĊę x = 2uv Ēúą y = u− v Ēúą z = u+ v ÝÜĀć ∂M

∂uđöČęĂ u = 3, v = −1

êĆüĂ÷ŠćÜ ÝÜđ×Ċ÷îÖäúĎÖēàŠÿĈĀøĆïôŦÜÖŤßĆî w = f(x) ēé÷ìĊę x = g(r, s)

ĂîčóĆîíŤ÷ŠĂ÷×ĂÜôŦÜÖŤßĆîēé÷ðøĉ÷ć÷

ĔĀš y = f(x) đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš ÿøšćÜôŦÜÖŤßĆî F ēé÷ÖĈĀîéĔĀš F (x, y) = F (x, f(x)) = 0

ÝąđĀĘîüŠćôŦÜÖŤßĆî F (x, y) đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤĕéš Ēúą÷ĆÜđðŨîôŦÜÖŤßĆîēé÷ðøĉ÷ć÷ĂĊÖéšü÷ ēé÷ÖäúĎÖēàŠ

ÝąĕéšüŠćdy

dx= −∂F

∂x

/∂F

∂y= −Fx

Fy

o- - -

"II --3¥ .EU -17731+3731←FZ 2M = (e't

-"

Ha, + (xe.tt/(n1tfxeY-ZTfzzHil1uAuuuvYe6?uDHtb , in - statue"¥1113,-DI -2-6+24 = -8424=16#

ME 1×(31-1)=-6,413,47--4,713,41--2-1-

a¥=dFi¥rIF = dawg . #

Implicit function y=x2 (y=fCx) explicit function )--

rheum cos (x -y ) -_xeY (20oz's y=fCx ) )and : £×[cos Cx -y)) = dz×CxeY]

- sin (x -ylddxcx- y) = XeYdaI×teY- sink-yiu -¥] =xe"day, +

EY

-sin Cx-y) + sin (x-4M¥, =xeYddI×yeYxeYd¥ -sink-y) day = - sin Ix- y) -EY

Ida¥=e¥iT #s-

Iet z=fCx,y ) = cos (x - yl - xeY=o ; y=fCx )-7=76,41

Ex 'Ey un IE -- ¥, -12¥ - dat, ← an dat,*:* ;÷÷÷÷÷÷÷ )-

EI 3.5.6 n'nun cos ( x-y) =xe"

our dayoff, fcx ,y) = Cos (X - y) - Xe

'tso fy=fCx ) )

:da¥= - ¥,

= - -sisnnYhe=Tx÷÷#


êĆüĂ÷ŠćÜ ÖĈĀîéĔĀš cos(x− y) = xey ÝÜĀć dy

dx

ĔîìĈîĂÜđéĊ÷üÖĆî ëšćĔĀš z = f(x, y) đðŨîôŦÜÖŤßĆîìĊęĀćĂîčóĆîíŤ ĕéš ÿøšćÜôŦÜÖŤßĆî F ēé÷ÖĈĀîéĔĀš

F (x, y, z) = F (x, y, f(x)) = 0 ēé÷ÖäúĎÖēàŠÝąĕéšüŠć

∂z

∂x= −∂F

∂x

/∂F

∂z= −Fx

Fz

∂z

∂y= −∂F

∂y

/∂F

∂z= −Fy

Fz

êĆüĂ÷ŠćÜ ÝÜĀć ∂z

∂xĒúą ∂z

∂y×ĂÜôŦÜÖŤßĆîēé÷ðøĉ÷ć÷ìĊęÖĈĀîéĔĀšêŠĂĕðîĊĚ

x− z = arctan(yz)

xyz = cos(x+ y + z)

-

F(x,4,Z ) = F (x

, y , fix ,yl ) x'+y42-2 = 1

1071kt x = I 7Voinqaonwfgn.gg

÷:±¥÷i¥¥¥÷÷i:"o -- Ex -' EFFIo¥=T→

qq.ee#yeEzrEy:.fEy---FfzTFlx,yp-)=X-2--arctanly 't ) ; Z -- 2- (x, y l

-

'

. Ex = - FEZ =- .# C- = 1447€¥,zq

= -1¥ 1+4+42-12It (YZ)2

'Ei - 'Ez -- i:¥IEE= -m¥⇒z

#FIX

,y ,z) -- Xyz - cos Cxtytt ) ; 2- = 2- (x ,y )

-

'

. 3¥ =- ¥1 = - YZtsincxty-#Xy + sin Cxtytz )

Fy = - FIZ = - XZtsinCXty- #Xy t sin Cxtytz)

Differential nomads :moral- Local linear approximation

*:i¥÷÷÷÷÷÷÷÷÷÷:* ÷:*:÷÷÷÷÷÷÷sf-

( fan .- ×. .. % . . . .,

r"

2- Differential Vardon'

Nooorishis

-÷¥÷÷t÷÷÷÷:÷:t÷÷÷÷:i÷÷i:i:÷:÷:i.

÷:::÷÷:7


ñúêŠćÜđßĉÜĂîčóĆîíŤøüöĒúąÖćøðøą÷čÖêŤ 5PUBM EJGGFSFOUJBMT BOEBQQMJDBUJPOT

ĔĀš z = f(x, y) ñúêŠćÜđßĉÜĂîčóĆîíŤøüö×ĂÜ z đ×Ċ÷îĒìîéšü÷ dz îĉ÷ćöēé÷

dz = fx(x, y)dx+ fy(x, y)dy =∂z

∂xdx+

∂z

∂ydy

đöČęĂ dx, dy ÙČĂ ñúêŠćÜđßĉÜĂîčóĆîíŤ×ĂÜ x Ēúą y êćöúĈéĆï

øĎðìĊę ÙüćöĀöć÷ìćÜđø×ćÙèĉê×ĂÜñúêŠćÜđßĉÜĂîčóĆîíŤøüö dz

Ùúšć÷ÖĆïôŦÜÖŤßĆîĀîċęÜêĆüĒðø øĎð ĒÿéÜĔĀšđĀĘîüŠć dz ÙČĂÖćøđðúĊę÷îĒðúÜÙüćöÿĎÜ×ĂÜøąîćïÿĆöñĆÿ

đöČęĂÝčé (x, y) đðúĊę÷îÝćÖ (a, b) ĕðđðŨî (a+!x, b+!y) Ĕî×èąìĊę !z ÙČĂÖćøđðúĊę÷îĒðúÜÙüćöÿĎÜ

×ĂÜóČĚîñĉü z = f(x, y)

êĆüĂ÷ŠćÜ ÖĈĀîéĔĀš z = f(x, y) = x2 − xy + 3y2 ÝÜđðøĊ÷ïđìĊ÷ïÙŠć×ĂÜ dz Ēúą !z đöČęĂ(x, y) đðúĊę÷îÝćÖ đðŨî

•

-11-IZ

dz+I

•

It -_ fxcxiyldxtfycx ,y)dy bakufu At-_ flxotsx ,Yotsyl -fl xoxo)DX -- DX = - 0.04 DZ =f( 2.96, -0.95) -fl3, - it dy = by = to. 05 = 14.2817-15

=- 0.7189

¥::". ¥. ¥:c::::/ -t.dz = (7) C- 0.04) tf-9)(0.05 )= - 0.28 - O . 45

= -0.73


êĆüĂ÷ŠćÜ óĉÝćøèćĒÙðàĎúàċęÜðøąÖĂïéšü÷ìøÜÖøąïĂÖĒúąÙøċęÜìøÜÖúöéĆÜøĎð ëšćĔĀš r Ēúą h ÙČĂøĆýöĊĒúąÙüćö÷ćü×ĂÜĒÙðàĎú ĒúąĔĀš A ĒìîóČĚîìĊęñĉü×ĂÜĒÙðàĎú ÝąĕéšüŠć A = 4πr2 + 2πrh ëšćüĆéÙüćö÷ćü×ĂÜĒÙðàĎúĕéš àö ĒúąüĆéøĆýöĊ×ĂÜĒÙðàĎúĕéš àö ēé÷êŠćÜÖĘöĊÙŠćñĉéóúćéĔîÖćøüĆéĕöŠđÖĉî àö ÝÜĔßšñúêŠćÜđßĉÜĂîčóĆîíŤøüöđóČęĂðøąöćèÙŠćñĉéóúćéĔîÖćøüĆéÙŠć A ìĊęđðŨîĕðĕéšöćÖìĊęÿčé

êĆüĂ÷ŠćÜ ÙčèĀîšćÖćÖìčđøĊ÷îÝąìĈÖćøðøąöćèÙŠćÙüćöÿĎÜ×ĂÜêšîìčđøĊ÷îĀöĂîìĂÜ ēé÷ĔĀš xđðŨîøą÷ąìĊę÷ČîĀŠćÜÝćÖ êšîìčđøĊ÷î Ēúą h đðŨîÙüćöÿĎÜ×ĂÜêšîìčđøĊ÷îĀöĂîìĂÜüĆéÝćÖøąéĆïÿć÷êć×ĂÜÙčèĀîšćÖćÖìčđøĊ÷îÝîëċÜ÷Ăé éĆÜøĎð Ĕî×èąìĊęüĆéöčöđÜ÷ìĊęÙčèĀîšćÖćÖìčđøĊ÷îöĂÜđĀĘî÷Ăéĕéš 45

ÿćöćøëüĆéøą÷ąìĊę÷ČîĀŠćÜÝćÖêšîìčđøĊ÷î ĀöĂîìĂÜĕéšđðŨîøą÷ą đöêø ÿööêĉüŠćÙŠćñĉéóúćéÝćÖÖćøüĆéöčöđÜ÷đðŨî 1 ĒúąÙŠćñĉéóúćéÝćÖÖćøüĆé øą÷ąìćÜđðŨî đöêø ÝÜðøąöćèÙŠćÙüćöñĉéóúćé×ĂÜÙŠćÙüćöÿĎÜ×ĂÜêšîĕöšÝćÖøąéĆïÿć÷êć h


êĆüĂ÷ŠćÜ êĆüêšćîìćîøüö R đÖĉéÝćÖÖćøêŠĂĒïï×îćî×ĂÜêĆüêšćîìćî R1, R2 Ēúą R3 ēé÷öĊÙŠćÙüćöêšćîìćîøüöđðŨîĕðêćöÿöÖćø

1

R=

1

R1+

1

R2+

1

R3

ëšćÖĈĀîéĔĀš R1 = 25Ω, R2 = 40Ω Ēúą R3 = 50Ω ēé÷ìĊęêĆüêšćîìćîĒêŠúąêĆüöĊÙŠćñĉéóúćéìĊęđðŨîĕðĕéšĔîÖćøüĆéÙČĂ ÝÜĀćÙŠćñĉéóúćéìĊęđðŨîĕðĕéšĔîÖćøüĆéÙüćöêšćîìćîøüö R

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Tauri Eu=Ei*u+Ei¥7 Hi-Fi