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Kind: captions
Language: en
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Okay today in the fourth lecture we will start
with a derivative of function of several variables
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Just once again I have a function F define
from sub set U in RN to RM M and N as before
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greater than and equal to 1 We’ll talk about
derivatives unless otherwise specified you
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know it happens in for function of one variable
also So unless otherwise specified
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we assume U to be an open connected set
Because there is always a problem in defining
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derivative at boundary it becomes one sided
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Okay so before defining or what should be
the definition of derivative here let us try
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to see or recall back what happens in function
of one variable So recall back suppose I have
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a function from an open interval AB to R X
not is a point in AB and we say F is differentiable
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at
X not something like this happens that limit
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H goes to 0 correct This exist and we call
it F prime at X not
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So what is the meaning of this definition
Let’s put a star here what is the meaning
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of this definition start well that is here
You have defined here now you have a point
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X not and here is your FX not X not plus H
H goes to 0 H can go to 0 from either side
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negative side and positive side If X not If
you take negative then X not minus H here
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and if you take positive then okay let us
do it little bigger so this is the interval
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H length of the interval H this is X not plus
H and here is your FX not plus H
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Look at the numerator this is precise the
difference this
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correct I am dividing by H which is this length
So you see if I join this two points by a
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straight line if I have this angle this numerator
if I call this angle depending on this theta
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H theta H is this ratio tan of theta H And
what is idea of defining this well as H goes
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to 0 this as H goes to 0 this ratio actually
gives you this ratio goes to tan theta which
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is
the slope of tangent at FX not This is the
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geometry behind this definition
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If you do it from X minus H X not minus H
similarly I have to approximate this side
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and this will also the demands both way it
should converse to the slope of the tangent
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at FX not at X not Sorry slope of the tangent
of F at X not yes okay So simple geometrically
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geometric interpretation of derivative
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Now if I try to just generalize this notion
directly to function of several variables
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how should I think Well look at this ratio
this ration actually measure the difference
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between the value of FX not plus H and FX
not divide by the length of the interval correct
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So maybe our first instinct will be that well
let us try this
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I have a function an open set U open connected
set U in RN to RM so given X not in U I should
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say F is so this is a possible definition
okay X not in F is differentiable at X not
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while what I do I look at X not plus H minus
so this is RM value so it is a vector in RM
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minus this vector FX not and H is increment
so I divide by the length of the increment
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and here the concept of length is norm and
then try to put limit H goes to 0 we call
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that H goes to 0 in this is same as saying
norm of H goes to 0 thus follows for the definition
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of the norm right
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And if this exists then I will call it differentiable
and I will denote it by F primes Well what
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is wrong I mean fine we just lifted the idea
from one variable to several variables but
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whatever we lift if we go back when N equal
to 1 U is an interval and M equal to 1 that
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is the case of single variable this should
match to our original definition of concept
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of notion of differentiable function for one
variable And looking at this definition you
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can see that immediately there is a problem
What is the problem So what I am trying to
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do here I have come up with a possible definition
of derivative function of several variables
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Now I see it is if it matches with my original
notion when N equal to 1 M equal to 1 U equal
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to AB and X not is a point in AB look at this
function FX R to the famous 1 mod X and X
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not equal to 0 just put this definition Limit
now single variable H goes to 0 0 plus H minus
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F of 0 divide by length of H which is Limit
H goes to 0 by definition 0 plus H minus 0
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divide by H which is equal to 1 so exist
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That means according to my definition this
F will be differentiable at 0 But all of you
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knew it is not it is the first example you
do of a differentiable function that what
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happens here this is mod X graph of mod X
at 0 This is like this and as I said that
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existence of derivative means it gives the
slope of the tangent at a point but at this
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point of course you cannot do any unique tangent
There is no unique tangent exist F is not
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differentiable So what is going wrong
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So this definition I have to discard
What went wrong Well if I look back then I
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see the problem is that I have put the length
here and while actually derivative I don’t
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put a length here I just divide by H so what
is happening why mod X is not differentiable
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that all of you know if I approach 0 from
this side this limit is plus 1 and if I approach
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0 from this side then the limit is minus 1
So limit doesn’t exists such 0 therefore
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it is not differentiable so I cannot put this
mod here
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So now I reflect once again maybe is because
that I have to approach through direction
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this direction matters positive direction
or negative directions and in real line there
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is only direction this one and its negative
one so basically one So maybe whilst defining
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derivative for several variables we should
take care of directions That we should be
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able to approach for all possible directions
and what we can come up with a possible solution
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…
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Well here is my U I want the derivative at
X not I should be able to approach X not from
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all possible direction and then we will see
that if the limit exists that is what I mean
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here There is another possible definition
okay X not is in U and I take U in RN a norm
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vector that will define the direction right
Any normal vector defines a direction Direction
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is precisely given by all normal vector
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And I should define and I should look at F
of X not plus I should approach through these
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directions then F of X not divided by H (I
cannot put sign here because I have to approach
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from negative as well as positive of U) and
then put H goes to 0 H is real And if this
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exists then F is differentiable I am not saying
that this is the definition I said it is another
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possible definition which we after reflection
about the drawbacks of the previous definition
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we make come up with this definition Okay
not bad
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This will take care for N equal to 1 M equal
to M this takes care of FX equal to mod X
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it is not differentiable at 0 if I put this
definition That is the modification I have
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done not bad And I want for every possible
I should write every possible U this thing
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happens Okay but let me put another example
here
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Let us say X from simple one R2 To R let me
recall F of XY equal to XY square X square
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plus Y power 4 where X not equal to 0 and
0 if X equal to 0 You can just see that this
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function according to this definition if X
not equal to 0 Y not equal to 0 so at 00 except
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00 everywhere it will satisfy this definition
everything is very nice But let us check it
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check at X not Y not equal to 00
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I put this definition Limit H going to 0 F
of 00 plus – So it if is a U with norm of
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U1 U2 equal to 1 plus U1 (sorry let me write
it more clearly) F of 0 plus U minus F at
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0 0 means vector 00 divided by H that will
give me what my definition U1 U2 square oh
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sorry there is a H That will give me H cube
U1 U2 square divided by H1 square U1 square
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plus H24 sorry H square H4 U power 4 correct
everything divided by H Check that is correct
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And now if I put so I have a Limit H going
to 0 if I put it will go to 0 you can calculate
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this will turn out to be U1 by sorry U1 square
by U2 sorry U2 square by U1 if U1 not equal
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to 0 and 0 otherwise This is the Limit Very
good Limit exists at 00 so according to my
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this definition F is differentiable at 00
But you can easily check if F is not continuous
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at 00 why you just approach through the line
Y equal to root X you will see that at 0 it
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doesn’t match If you check F is not continuous
at 00 so then I have a serious problem
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Because I do not I want the differentiable
function to be at least continuous if I am
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able to draw tangent to a graph or tangent
to a point tangent at a point to the graph
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and the function is not continuous at all
there is a jam then what do you mean by a
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tangent this defies our geometry So this derivative
again I have to discard but possible you will
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not discard as it is not too bad so we will
discard it with a little bit reservation and
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we will come back to it
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We will call it later on directional derivative
and it will be useful for us but for time
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being this cannot be the definition of differentiable
function of several variables because if it
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is differentiable function at least we want
continuous and this function is not continuous
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but turns out to be differentiable according
to this definition
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So I have to look back my star a little bit
carefully what was star F AB to R
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Limit H going to 0 F of X not plus H minus
F of X not divided by H exists and we call
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it equal to F prime X not We observe what
is happening is actually that this F of X
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not plus H minus F of X not there is an increment
that is equal to from this I can write F prime
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of X not into H plus H into to some (21:28)
where goes to 0 as H goes to 0
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So that means to say F prime X not into H
approximate this increment F of X not minus
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F X not correct in this sense And second this
approximation is linear
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What does it mean that H going to F prime
X not into H this is a linear function from
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R to R because and R to R any linear function
I am not writing it listen to me carefully
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Any linear function from R to R
that is given by alpha in R such that L of
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H equal alpha H by multiplication
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And the last thing property 1 and 2 I have
written on that board determine F prime X
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not uniquely I will leave it as an assignment
exercise and what it mean by this is if F
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X not plus H minus F of X not equal to some
alpha H plus H into EX not 1 H and also equal
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to beta H plus H into some EX not 2 H where
E1 X not H and E2 X not H both goes to 0 as
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H goes to 0 then alpha equal to beta So if
F prime X not is uniquely determine based
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2 properties this I put an exercise and the
next class will start from here and come up
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with its right definition of differentiability
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Thank you!