WEBVTT
Kind: captions
Language: en
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Okay So let’s continue with the fifth lecture
So as you have seen that this two property
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for function of one variable completely determines
as written in other board the derivative F
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prime X not that it is it approximates the
increment F X not plus H minus FX not and
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it approximate linearly This leads us to a
definition of differentiability of function
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of several variables and that’s the correct
definition Now I will write the correct definition
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Remember we have discarded the other two first
one completely discarded second one with the
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reservation so we just see now what is the
correct definition here it is
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Here is my setup again and U as I said unless
otherwise specified open connected connected
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is not that important but sometime it is open
connected set and I have X not in U F is differentiable
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at X not if what will happen now there exist
a linear approximation of F for the increment
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F X not plus H minus F X not if what will
happen now their exist a linear approximation
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of F for the increment F X not plus H minus
F X not that is if there exist a linear operator
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let us denote is by DF at X not from RL to
RM such that I follow the exact notion of
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for single variable
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I write this as F X not plus H minus FX not
the approximation is D FX not H or as you
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have written before same thing I am writing
D FX not at H where goes to 0 as H goes to
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0 Remember H is in RN right H goes to 0 means
norm of H goes to 0 So this is the exact analog
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of this property 1 and 2 So DF X not as a
linear functional so X on this vector H and
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approximate FX not plus H up to an which goes
to 0 norm H and goes to 0 as H goes to 0 So
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this is the correct definition you will see
right now for derivative of function of several
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variables and it exactly matches a function
upon variable
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And remember we wanted our differential function
to be continuous and we immediately have it
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Same setup
like that F is differentiable at X not implies
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F is continuous at X not How It is too easy
from the definition Let me write it again
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in this form correct Now I take mod on both
sides and equal to mod comes out correct
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Now you put H goes to 0 H goes to 0 norm H
goes to 0 so this quantity goes to 0 as well
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and this norm H goes to 0 and E X not H goes
to 0 this norm goes to 0 so what do we get
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continuity because now if I have XN converges
to X not XN X not both are in the domain U
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then you just put the H to be XN minus X not
apply this you will see F of XN converges
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to F of X not
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In this equality from first line to second
line I have used something here so what we
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have used here
okay what we have used here is let me explain
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on the other board So definition is clear
look at the definition again Look at this
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proof module of this any which I have written
which I am going to explain now So see DF
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X not is a linear operator from RN to RM linear
operator means all of you know it takes X
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plus alpha Y to so A of X plus alpha Y AX
plus AY right
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If from your linear algebra you know linear
operators along with the linear operators
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comes the concept of operate so what we have
used here is this Suppose A suppose RN to
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RM is a linear operator correct Then we have
this notion of norm of A which is supremum
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of norm X so X in RN norm X equal to 1 AX
which is same as by dividing X in RN just
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RN X not equal to 0 AX by norm X
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So norm of A is a supremum of this quantity
so that will imply for all X in RN is the
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supremum bigger than anything AX is less than
equal to norm A mod X That is what we have
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used here DF X not is a linear operator acting
on H so norm of DF X not H is less than equal
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to norm of DF X not into norm of H and there
is a little fact that you verify from linear
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algebra that every A RN to RM as norm bounded
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This is a fixed quantity so we have it So
this continuity problem is taken care of Okay
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so this notion of derivative is really nice
for us and actually what if you what I said
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was one possible definition was to fix one
direction and then claim that fix one direction
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look at this limit along directions and then
claim that for every direction it exist and
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every differentiable we have to discard it
for some reason
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But that is sometime actually we can recover
that derivative from this definition so what
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I mean here that we will call that second
definition as directional derivative and let
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me show you how to recover the directional
derivative from derivative
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So again
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so suppose U is a fixed direction U in vector
that gives a direction What was our definition
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second definition Limit H goes to 0 F of X
not plus HU minus F X not divided by H okay
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Let us give it a name let us call DU FX not
and call it directional derivative of F at
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X not in the direction U Suppose DF X not
exists that is F is differentiable in the
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origin of the definition we just made Then
what will happen from here I know from the
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definition HU minus F X not will be DF X not
right acting at HU plus norm of HU EX not
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HU correct
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This is equal to this is linear operator H
is a HD F X not U plus H into mod of U norm
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of U but norm of U is 1 so EX not HU okay
So I get this So I divide by H goes away correct
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Now you put X goes to 0 what happens H goes
to 0 implies norm of HU which is mod H this
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does to 0 so EX not HU this goes to 0 according
to definition of derivative and this quantity
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is bounded it is plus or minus 1 is bounded
by plus 1 or minus 1 according to H the sign
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of derivative so what will happen finally
we get
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Okay let me write on this board itself The
entire analysis leads to DUF at X not this
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is equal to DF X not acting at U and we will
use it to actually calculate the derivative
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of function Remember this what is there in
the box I will show you in today’s class
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how to actually compute this fellow DF X not
with examples may be next class but we have
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to use this idea
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So now let us come to this point How to compute
DF X not given an F We have to recall a bit
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of linear algebra here
How to realize a linear operator This question
00:15:45.520 --> 00:15:58.080
is related to how to realize a linear operator
Suppose A is a linear operator from RN to
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RM A linear operator to determine a linear
operator actually depends or it is completely
00:16:09.660 --> 00:16:13.950
determined how do you fix basis for RN and
RM
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So I will show you or else all of you know
it from your linear algebra course but still
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let me do it it is very important for us So
let E1N E2N ENN this is the basis for RN You
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can take Canonical basis 011 100 sorry 100
011 so all of you know the Canonical basis
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but it doesn’t matter You take any basis
I will give some example in assignment And
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E1N E2N and ENN this is the basis for your
targets space RM
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Okay let us take in any X in RN this fellow
being a basis I can write X as summation J
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equal to 1 to N XJ EJ N correct Very good
What is AXAX is linear operator so on the
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sum it goes inside it goes J equal to 1 to
N XJ A EJ N This is by linearity Now AEJ is
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where A is from RN to RM so AEJ is a fellow
in vector in RM So now I have a basis for
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RN So I can write in terms of the basis or
RN so for each J I can write EAJ N as summation
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I equal to 1 to M sum AIJ J is fixed I will
vary from 1 to M EIM correct
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I have written each of this fellow in terms
of these basis So what happens to AX okay
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so this you see you can write suppose I fixed
my basis EIM equal to 000 at 1 place and 0
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that is Canonical basis at I place You see
you can write it as a vector summation J equal
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to 1 to N AI1 XJA sorry A1 XJ J equal to 1
to N A to J XJ and so on summation J equal
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to 1 to N AMJ XJ Simple exercise you can do
but you immediately recognize what is this
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this is if you look at this matrix AIJ which
is matrix AIJ I equal to 1 to M J equal to
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1 to N acting on the vector X1 X2 XN
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So what is says that if X in the Canonical
basis X1 X2 XN and AX equal to this fellow
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and this this matrix acting on this and this
is called matrix of A Let us write this as
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matrix A So there is a way you calculate matrix
of a linear operator Let me show you one example
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and end today’s lecture
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So 1 example let us take this operator A from
R2 to R2 A of X1 X2 X is a vector in R2 given
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by X1 X2 let us say X1 minus X2 X1 plus X2
basis 1001 or R2 both side or both are R2
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I say this side also this basis this side
also this basis what is matrix of A From here
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you not from this calculation you note that
first column of A is AE1 and second column
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of A is AE2 Just look at this calculation
you will get it
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So what is AE1 1011 what
is AE2 minus 1 and 1 so matrix (let me write
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here) matrix A equal to first column is 11
and second column is 1 minus 1 This is one
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example Next day we will do this exercise
of D of FX not
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Thank you!