gradient ,y)=µy¥g - math.ucdavis.edumgaerlan/teaching/wq_2018/017c_a03/mat... · 10.53 gradients...
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10.53 Gradients and Directional Derivatives2/1
Jtx ,y )
oftx,y)=µy¥g] Gradient is a vector
Directional Derivative
Dunflx . ,y . ) = oftx . ,y . ) .u^ Hunky^ "
EH nation a. ¥,¥%
22 flay )=e*Y ( 0,0 ) [ I, ]
point vector'¥
th ,y)=keIYy] oslo .FMhiktsttmtrz a=¥n=⇐fIffYYr]
oflaoju - ftp.fyhzgfi.fftl.fz-2#scalar
v. 6. Maxima and Minima
5- ( x,y ) ( x. ,y° )
offx . ,yD=O=8=[0o]=( 0,0 )
DHess -xx,y)=[ftp.f# =ytheinconclusive T sapdognlky
D= fxxfyy - H×y)2 yoyoT= fxxtfyy local localmin Max
10.6.
5. f( X. g) = -2×2 + y'
-6g
Htxykfzyh . ]=[ ;] ⇒�1� .
4x⇒⇒x=o@ zy -6=0 ⇒ y=3( 0,3 )
Hessfcx ,y)=ffIfHess f( 0,3 )=[ to{ ] D= C-4) (2) - (0×0)=-8-0
-saddlepoin-al.CO#
1-1
J ( x ,y)=xy - Zyz
osixytf,?4y]=[ :]�1� Y=0 @ × - 4y=o
( 0,0 )
g. 49=0x=o
{ 91,] D=o.t4 ) - a) ( D= -1<0 -→ saddlepoint.ae#
8.Hx.ykyxe-YIfgY9.txtdefTDHg5-tgtgsFx-YeTFy-lyjxeTtfxEYjy-1xeT-xeis.y-CxyYeT.Ce-yyxy-XEY-xye-ylxeiExje.d-fIxFIyoenooo@ye9-o20xeT.x
yet
yEo↳E#Xeo -
x. 0eY=o
tDfh)g(x)=oXil -0=0
-51×7=0 or g(x)=oX=0
Eg .( x - 1) ( x -4=0
E:*
:H::]
¥x=YeT ¥y=(yjxeT+(xEY)'y=1xe '- xeity
5×x=£x(¥x)=§×(yey)=o=×EY -
xye'T
fyy=Ey(¥y)=Fy( XEY - xyEY)= - XEY - ( xeihxye 's )
5yx=¥(¥y)=F×(xe.hn/yED=EY-yeTfxy=fyxf××( 0,4=0 kixfyy - #y)2=D
fyylo ,o)= - oeotoeao .oeo)=O 0.0 - (1) 2= -1<0
t×y( 0,0=50-0 .eo=1saddle point
a. th ,y)= xcosy
o*⇒=h%nIft :L
:BnT:�1� Cos y=O @ - × sin y =O
y=kntzDiI,nEk-
xfN=¥#( at Is ,
( QEI,
( 0,3¥ ), ... . all saddle points
Enginedoo
'iE¥'
0u - 1<0