Linear algebra and its applications

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Chapter 6 Othogonality and beast squares

How to find Pm projectionsplane

I 2 IMsg

Goal Equilibrium solin for heat equationshigher dimensional heat equations

ID f CUT Ko Ux x t QIf f C Ko coolantUt K Uxx t Q Qdoes not dyeon time

Intuition appoach equilibrium solutions as f w

If 0,1 170 evens out the initial leapand insulated boundaries

Equilibrium solutions means Ut o

Ex Ut kUxxut o UH O

uI

o u axt b

µI

O LDirichlet

Dc Unit ToUh t T H b To a

T

L

Insulated boundary Uno t o llxfc.tk

a o stillhlediaitialcond.gsµ

Hitt lol U dx then LE to find bIt o

for insulated boundaries

Why 1µ Has fol Ut dx

folk Uxxdxk Ux 0

Initial condition UH o fixas t too U Lx f aix _b

equilibrium14107 1 fix d solutionHI 7 54 bdx b Lb t fol fix dx

Equilibrium solutions with heat source

Ut Ux t heatsourse Q

4 10,1 1 3 heat gainless at two endsUx Litt he BoWX o fix IcEquilibrium solution Uix lim U Ix f

C to

Nxx 2 Ux 2X 1C

u x't G X T Crthen Ux 107 C

Ux IL 2L t C L 4 327Htt J U ix E It

II teaH6 fix at Hc 7 f X 5 4 dx

a Ifffindxts IL

so equilibrium solution 4 7 xt 3xttfotf.ydx.fi zHigher dimensional heat equation

EUI J cpucx.y.toor

E t faput ror

Heat gainfloss through 2h MatataFourier's law Of ko finale Il n ace

Ko sonF

tarot nE a fkf.OU.tn tf Qar r

div Koon t f Qn

put divlk.su tQthermal properties

f f C Ko constants 0 0Ut k ou