Vector fields with potential on domains

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Vector fields with potential on domains
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the let me please start with a charge review the so whatever we want where and short form the situation is that given the vector field is and it has to be C 2 vector field but let me the for a summary of the and write all that down and that we call s with grad f or map has a few physicists equals F a potential home and we've seen not only last class where there was actually class before last class that if x has In a potential then and by the shots lemma x the differential is symmetric or if you think about the Jacobi matrix of self adjoint if you think about the linear map In any case this means for best this holds for all indices and all points which are missed admissible and physicists physicists here say X is conservative since uh where k is preserved and here they say uh as I explained to you rotation equals 0 or on in language X is an irritation or in Turkish the good at so once flight OK so perhaps this is actually the more concise wording and the main problem we want to address and wants us to do for today is uh can we actually true for controversy namely how problem is the does the other implication so in short form this is the problem can wait save it irrotational field 1 satisfying this it also has a potential is conservative and problem here is answer will depend on the main on domain so everything i've written down without doing this depends on the domain X is a vector field on a subset you in Rn and it would depend on um it would depend on this domain you if or not this holds yeah that's all we have to um study domains and we would do this in 2 portions the first one comes today the next 1 will be in a week from now so all this is the next section about could make have or connectivity if you like some Monday and it as a real what we need to as a reminder of and this is what our relatively open sets so and like I can and although this is always on you in Rn uh and let me faces in the terminology of metric spaces but if you don't know what is here anybody except falls high school students who doesn't know what a metric space is depending on the class he attended analysis to OK everybody everybody knows or doesn't dare to show it here she doesn't um OK so if M d say is a metric space and we consider a subset of all may go in and say you we consider subset then and of course then all DE is also a metric space how just choosing the same D as here and restricting to the set all manner and and now we want to uh consider open sets and 1 of the open sets of omega well most of the intersections of opposites of M with maker that's what I want to and remind you of so open sets sold for omega opens the says the open sets to write this out and short form I just given by the intersection of or may go out with you know where you in M it is OK I it is I was very precise at right end comedy yeah but it's the same he you remember you remember that open notion of open sets will always depend on the metrics we impose OK so uh so we need to intersect all may with open sets and encased omega itself is not open this uh leads to some surprising results like uh correct intervals which seem close object open so let me give a few examples 1st perhaps uh here is it's all may go fast say is the intervals from 0 to 1 and this is contained of course in which is an a then whatever the open and what open sets then for instance in into sub interval like this is open and since uh this statement looks and somewhat we had we often say here they're relatively open sets so now ellipses heavily on open right this is the common notion so often we say here actually relatively all the sets of uh where do I write this out um steps here here these are the relatively and all sets are relative just to distinguish open sets in my subspace from open sets in he I would say open sets and if both spaces play a role here I say relatively open sets of OK and another example OK is if the omega itself or menu open then uh nothing nothing surprising happens then and if you intersect to open sets result is open so that all open subsets of all of the major you and contained in omega km is also so this is open say open and then it is also relatively open or open the relatively open meaning open in omega OK that change week I covered last term when an when introducing metric spaces and for the notion of connected use of connected sets we need we need the notion of a planet of the open sets so me right next definitions of knowledge In a problem I'm sorry yes know so usually my sign convention right is that whenever I riots contained it means tool that it doesn't exclude it so means this a subset of all equal yet including the equality case does that answer your question yeah convention is is you know already right here this and then you have also the in nonzero and so forth since in analysis what's reason for adopting this Convention analysis uh mostly we need we do need to be don't need proper subsets but it we don't care if it's equal or not so that's why I and tend
to use that and that's why in general people in analysis uses quote came not we the however if it's the same space nothing is added in the definition OK so the next definition here it is I know which we need is that if I have a metric space that m or n d be a metric space or not In fact could be a topological space uh since I only need the notion of open sets and then and a set is called connected so all omega subset n In connectedto this is unnamed and off long and hard work and so is connected if the following holds a by definition if it cannot be written as the union of 2 non MT open sets which are disjoint so is a positively or negatively to phrase this written down of across nodes the negative way so why not right in the positive way each disjoined peach disjoint union uh also Reagan so omega is equal to a union D and for disjoint union 1 often uses the sign so you said if you like when hey so for each such disjoint union where a and B of open what with a b both open we have that either a is empty will be is and and the other set is the full space or make it so there's so long revolution who is the a is empty or B is empty and if this is the case then be of his is omega in a in the other case is a is obviously on OK so there's no way to decompose may into open disjoint sets that's the definition and before I come to uh examples let me add another definition a subset of the set omega naught in omega the is called a component of a connected component of connected components 1 the sometimes component and if Hong period but also I should say also perhaps of all major if an only if 1st of all I omega naught is connected to end and a 2nd or um should and it's uh sorry so that that notice it's missing even know as thing I want is empty and I didn't write this out in the notes they share with there may be a cell and non empty on OK subset of society act because OK so I don't want to call the empty set is connected component OK in in which case well if omega no end all and its complement omega naught omega without omega naught open to all mn and omega naught is connected will be more likely I can write open and closed since complement there is also look uh of results this also kind is by definition is closer and these are complements of 1 another and from again orders open then this is closed yeah and vise versa so actually it's no matter the writers other not as cell and the 2nd condition and and important condition is connected omega naught itself is connected to and are OK so let me give a few examples um on all examples so the 1st example is an interval in our in our is by definition is a connected subset so connected to subset loops of crime is by definition at the end of the and you can check that this uh that this agrees with the end of the definition of interval I gave in analysis what it contains all intermediate in Internet interval contains all intermediate points since live in order not this there this is easy at and to give and non and a trivial example say choose or this days the ambient space omega are being the closed interval minus 0 point so I'm talking about the the interval from 0 to 1 level point and a half is missing right then the who carry then this space here this set for instance in our this is a non connected uh connected a set in our wire is missed while these 2 intervals at each relatively open uh and so they have a disjoint union yeah so this is not connected it's not connected digit so what you have to do in order to show that it's not connected all have to scientists is a disjoint union with maybe open and closed on answer uh so I choose a nonempty I choose a the interval from 0 to what's going to a house and b the interval from a from a have to 1 yeah these are 2 relatively open subsets if instance and they arises intersectional flight and all the open interval so minus a half to plus I have say I would say that the center right so they are relatively open lenitively open and non empty so they satisfy the conditions just to see how the condition works yeah and I mean to draw a picture here of course my pictures with picture has the mind will be such an example we have 2 subsets at which are not connected and and this 1 is the 1 open set a and just a speed here this Bisanz covering the general case and let me now prove and what I stated here and he's for bounded interval the case or is the same so true that a a B is this subset of that that is connected to a Crohn's nected it God as his goal
just to see how we can uh work with the definition well so assume we can devise a different devise a contradiction from the fact that a b is the union of 2 open a disjoint non-empty sets unit cell are supposed suppose an said a B maybe equals the disjoint union of a and B the suppose that both are the non empty at the of the well OK and open on OK and derive a contradiction why does not work well um take taking a so a is a non empty so I can take this loop and also a when defined as a number TA indeed part it located now an what can TV I claim it must be uh the terminal point b yet so suppose that a suppose this were not the case if t is less than the yeah I want to rule out this case uh I suppose at less than the wealth and since a is open we know and as open we know that she is not in a since otherwise if t is the name and they and then a neighborhood of T is also contained a in and in a so tk so cannot be the so however if is uh not in any way I want to write so not in here since meant as how does it looks like here's my interval from a to B and you have assumed universe somewhere here is a perhaps the some halts this is the what if T is a in a well then some Absalom neighborhood was also in a when she's not the supremum OK so it is not in a so that means since this is a disjoint union T is being well but then uh since keys all
moved uh have so if t is indeed than the neighborhood of B is also I also contained here and this is a contradiction here so that he cannot be the supremum cannot be subsector there is a lower bound for the set and so um again the other writers to cannot be which is not a upper bound you have to use them what do you see well men uh assumed a so it's less antique OK can sell this will not be concurrent case so at t equals speed long-term so the super so a reaches all the way through now actually actually still still um if t equals the events is in sense otherwise be if it's in the then again the small neighborhood will be in be and so in so this will be would not be the supremum OK so T is a may about openness of speech and so uh well this is a contradiction since that run from the same arguments with this set be in place of a and derive and here you slip and and that means you arrive to the same conclusion for a floor just right so how or um so here I could actually D. N. A. to make more like condition and now same arguments so arguments for these in place of a gives me that B is an the user and the hand this is a contradiction yeah so cannot be cannot be a disjoint union of this interval into tool sets which are non empty perhaps you can do it in a quicker way but it's it's the what came up the today is approved the all you like and I on and the proof of the other intervals so is similar there when right infinity in case of billions cells which is somewhat simpler the OK so intervals are connected and and the thing and so be also we could save by example to that this set interval minus appointed in the middle is to connect the components please this non its in our the here we are 1 of the 2 the the thank you his book of so I am I need what I need a further motion actually in analysis um it's not a good wording but it's common to say but the domain so useful definition it is an analysis that he will on the is the following as we call it from now on 1st a domain the domain how and in English it's ambiguous and German and German and so called could beat and it's uh and so we can distinguish it from the different its own specialized but here we cannot distinguish it from the general notion of the domain so english is no good here but this is this is the common notion so a domain and analysis the use in our and say is an old man connected subgraphs it is all in connected connected set move the hand and all follows functions will be defined on domains that's important so that's why I the the notion in English came about OK so that I need actually another notion of connectivity which is also com and less abstract now mean this definition I gave it seems on the 1st side not so easy to handle whatever give you know and is more geometric sparse connectedness so you know if I'm in a sense I want to see consider the property that 2 points a P and Q can be joined the bypass and if the set is not connected some service stuff here and I take a point here is no path from p to this point yeah so uh this is the notion of Path connectedness which does not quite agree with connectedness so omega in over and the it's past committed I'm unfortunately need both notions the Prof connected dicts as amended all with the there when a ropes and if so how can you can guess what the definition should be Inc for all p q rule in the space there should be a part of who cares for all q in all major very exists curve C and now I want this curve to be far from easy it exposition I want this curve piece wise C 1 so that curves and piecewise C 1 saved from she going from 0 to 1 and 0 1 2 well maker yeah so it may have some uh vertices of chaos and under French abilities such that the initial point maps to P the terminal point maps to q
which but uh and CEO of security focused PC see the single includes so if they're very obvious motion and in fact um I mean what I write here can make a difference here if I from if if I'm a topologist that I right continuous colors here and perhaps I'm asking for more and rights moves because there but in the end it will not matter and age at least it will not matter if omegas OpenMS since I can all this this Smove edges yeah if I haven't this this is this is all my have a small neighborhood here and without telling you how exactly this goes You can smooth itself so there's no problem and so this is just to make everything consistent here uh with my center the OK so know what is how are these 2 notions related path-connected isn't connectedness well this is a stronger notion so proposition in that in says that omega is passed connected all then on then accession might be I should I it like this in our OK then omegas connected world so have connected path connecting is a strong emotion and um perhaps I'm I would give a proof the were so for this 1 so how do I do this and so on the continent so what I'm doing the to show up as I have 2 connectedness I want the if every path connected as I will connectedness so I assume I have a uh composition a disjoint union of all mankind to a and B open and non empty and uh I want to show that 1 of these that this gives a contradiction and so suppose suppose Promega is a disjoint union of a and B where a b is open and maybe column and nonzero and on empty not and then what I do is I let take 2 points so here's my on here's my space omega R and here as a of the and without telling you how it exactly looks at this may subdivide omega into a and B OK now I take a point p in a Q of these side take points p and q and I can buy connected and bypass connectedness or I can find a path connecting the 2 when some of the theories that power and that part of C C from P to pure from P to kill will and now since see is piecewise differentible in particular it's all I need here is it's continuous the case of continuous the you have to see as continues the then well that's what does it mean about the uh and the preimages of a and B so let's consider now a parameterization say 0 1 to this curve here see right now I want to look at the pre images about pets as Laurent at the pre images of a and B in which of being subdivide this interval well that this this here is a decomposition also into the 2 disjoint open non empty uh sets I claim 1 is this will if for continuous mapping the inverse image of an open set is open again 1 of the 3 characters nations of continuity we covered last term their so in this image is open so I here I have to 2 subsets which are open and the preimages also since a and B are disjoint unions of omega I claim that also see in those of the a and c and there's 2 the good also disjoint unions of the perspective into the similar to what is this well that disjoint since I mean if there is a common point if they share appointment this point goes to a and 2 b well this is a disjoint union that's impossible why why do they get them when do I get the entire interval as a union will since all points are Ivan A. and B. since the union of a and B is all made us all look all point Ivan an empty so this must be the entire interval from 0 to 1 so it's in fact a disjoint union so by the connectivity of closed intervals which I approved of there 1 of the into 1 of the sets is empty when cell well the inverse of the ball see inverse of the is empty well on the other hand I chose points in there so that's a contradiction the yeah so I cannot play the so that that shows that in fact after that check to I have a omega is or is connected since then 1 of the sets must be the empty here here I'm mean it yeah I should have written on suppose on the contrary that p is an a and she was in the air yeah so say on the contrary so this is not this is OK this is OK at this is a contradiction I I assumed the non empty OK so that's the proof so a half connected so in particular is connected it's a strong
notion that it is in fact a figure 4 questions all is yeah Elvis being this which is why I put the relatively and parenthesis hopefully so I did that yeah the I always mean relative yield to say that OK so if this is an implication and I claim that the converse is not true what would be an example um to look through the different past connected to warm up take at about the I all the hand here that the room here which was and this work can what yeah world and I whom Qin so you know some further examples and the you know OK so for connected sets some that's easy so perhaps I give you an example of a set which is connected but not path-connected connected but that's the interesting case not path could make this kind of connected some we the example I have is an example in our 2 it and that it's as follows take the unit circle solvers set is the unit circle union something and now I take a spiral approximating the unit circle from the inside what any curve approximating it will do it in so I take something spiraling out getting closer and closer no I reached the limit of drawing quality semantic the union was a spiral saying and some um for instance that could be given by T is mapped to OTT over t plus 1 each of the i th which he's uh in 0 to infinity you cell the radius here goes from for fatigue with 0 this has made is there and 40 tending to infinity this approach is 1 that is always less than 1 and this is just a standard circle parameterization so it's exactly as at all as drawn it it's there it goes round and round and the radius is 1 is strictly monotone ously a increasing to 1 but it never reaches 1 so it's actually a problem uh for I don't know keep forgetting for problem session of homework to prove that this is the case but what's the idea well connectedness is about open sets and open sets will reach beyond the unit circle a however pass connectedness so that that will show you this kind this kind of consideration will show you that this set is connected it's not path-connected since you can never you can never get to the circle as 1 actually now it's it's infinite along the way here if I'm uses language OK and so I miss some of examples with graphs which
oscillate and whatever and all the others all the other examples are not so interesting since they're both connected and connected but perhaps it's so good to write a solid so here that is to say we mostly origin from Martin well then so what do we have so it's past connected it's past connected for and at least 2 and not path-connected NIPS not path-connected for any clothes what so why so what is this set for n equals 1 well that's the real line with emissions of the origin so why is it not path-connected well tell me why is it not path-connected the the was the argument to use you each ha of world superior OK take 2 points minus 1 and 1 say on the real line connect them with a possible why it impossible here yes it takes actually campus by the intermediate value theorem as I go from here to there and I will reach 0 as a value yeah the continuous so in fact it's not postulated but they are without serious past connected for larger than to there is OK means that you can connect 2 points 4 and I don't know exactly you do this for instance you do it was polar coordinates like this whatever yeah you must do something about the fact that if you take a straight line connecting the points you might hit this point so you made of also argue by case distinction whatever you like OK and let me before we come to the break let me just right out 1 1 Theorem of authority could proposed the next proposition which tells me mad OK it's it isn't there is a distinction path connected sets and connected sets however you can forget about it since for open sets with no distinction so therefore you in our and In this open then then the it's the notions agreed you is connected this only if you sparse connected the had it now yeah so no that that this is not an open set in rn near this this counterexample in or that was a very sort of their method that if we had with the say something about open sets banned the 2 notions agrees so if X is a disjoint union into new open on empty sets that's and if I have no such union that's equivalent to being able to connect every pair of points OK so let me perhaps make should make 10 minutes break at this point OK so perhaps you I will then I will skip the proof of this although we were actually the use the results or what yeah so 1 of the implications t anyway I wrote it out before I improved it past connected implies connected it's have direction connected implies path-connected way to do this is I just that-a got the idea to look at the set of points that point P in you and look at the points which can which can be reached by past and most which cannot be reached by past and those are OK you will see that that uh these are in general tool open sets and to open and they are non empty in case it's non-connected so that is the and and so that will give you a sheep proof uh for the Congress statement and I won't go into detail OK so and let me rather go on and and prove something about the problems of potential so so that's the next sessions I come back to our original problem niches great existence problem for the potential of its is strong enough to not all of the an entity it's the 1st the 1st the part of the solution and so what do we want to do we want to use curve integrals path integrals for line integrals I want to say uh to integrate out the function here so given a point x yeah and a domain you uh and a point wise we want to and given the vector field x we want to integrate out the respective field bypass and it will and we care about the problem is so not the path integral depends on how we go between these 2 points and for following will say what is um equivalent self In general the result will depend on how we go from X to Y that's and here are some characterizations of the case that it does not depend on the path so we use cuff integrals the set such a leg this section if you're in theorem OK so I suppose user domain that's oppose you know this the name so it's open and connected to being hard and I have a vector field and so also know it's sufficient that it's a continuous vector feared also from you to our and the end then the following is equivalent to man but the following is occurring the 1 from but it's looking so I won't tell you when it's true but I will tell you what's equivalent standard mathematics tactics so there exists a potential for potential functions for X so the potential S and C 1 of you meaning that put the gradient of this is x throughout OK so this is what I want to have some that this is equivalent to the uh the fact that the curve and the line integral doesn't depend on the
particular curve chosen path chosen so for and the true I heard say seen on to C 1 um say iron piecewise differential let's just parameterize them by the same interval b 2 you were carry with cocaine was common common initial and terminal client where's so you know what of an 8 equals C 1 of any equals say X and seems and be equals C 1 of the most why say uh what following holds the profit of the Greek integral of scene not of X path integral is equal to the C 1 I have to develop NOW wine into I should say the sir yet so it doesn't matter if I ran from seeing what was C 1 in order to get from X to Y and the 3rd equivalent properties that some loops closed curves I get 0 has a and world for each new life at harm see you in a piecewise C 1 but I'm 1 its parameterized by a great tool you the now we have the integral of that inside we have integration of the New World XTX expenditures but the saw the strategy here is 1st I show you today I show you that the cycle and neck and a week from now I will show you a condition under which the these refrains actually become true and it will need the further study of properties of domains and the gender it has locate so the why is this a proof and so on easy bit the easy bit is from 1 to 2 and also easiest to entry are equivalent hot is from 2 to 1 wrong period saw how does it go now no it's really a that it becomes a a crucial character nation so what do we want to show we want to show that as if I has a potential for any 2 pass I get the same uh into will also I should I want to show that this I always use the a difference as of Y minus F of X and Y and fall color for each curve so each curves see saved from X to Y yet if this holds for each curve and then and then add in particular and these must agree since then it's in both cases is f of y minus F of X OK so let's let's just do the calculation all AC so the line integral effects in case I have a potential it is well integrated you integrate the gradient had right and how they can integrated gradient while by this emission by definition this is the integral also uh say a to B yeah so sees parameterized from a to B C of exons seals these like uh from a to B also well the vector field ends at the curve times the tangent vector of notes that's just by definition of the line integral so
that now look look carefully what we have here the gradient of s and the uh of F of the of TE well this is this at times c prime of well this is just a consequence of the chain rule this is the very same thing and this is the derivative of f ast as the prime of T B and T from a to being here at why is this so remember that we has occurred as a greedy it maps by to you maps by F 2 hour so the chain rule the chain rule gives me that if I want to compute the derivative of a composition now this is granted have times see prime as with not less mn OK so I did s after the prime and now it remains to use the fundamental theorem of calculus so long so this is chain world and then this is fundamental CERN most continents the while this is in fact f of upper limit minus s at this function at the minus this function it then a which is f of y minus OK so if you have a potential if you have a potential then and of course uh gets uh that would pass interval is just for potential at the at the at the initial and the difference of the potentials at the terminal points that you should know this calculation and for this reason it was actually in the uh the problem session problems that may be also since I was asked then at the In the break let me make 1 remark regarding problem session and so this this that each finishes 1 to 2 and 2 2 4 2 4 2 2 1 I come in a minute but the 1st say something I forgot to say about the problem sessions which is we have a problem of problem sessions which is that some of them month before class and some of us after class and so uh we take it all we do take notice of this effect and it means that in so that sometimes you find as something which is defined in class uh you find this definition explicitly uh on the on the problem she'd for those of you who go to propositions before class so the whole with no difference in form of those problems there done and the problem sessions uh we keep an eye on that it's possible to solve them both cases nodes that makes sense to do that but it it may not it may look different if you depending on in which of the 2 cases you up and it's is not a good situation I'm so but it's so hard to change in something there so it's endorsable OK itself there was the easy of a hot direction is you Combaz if just the line integral does not depend on the particular path chosen why do I have a potential well and the how do we do this the effects of point p and you so let's start once again we fix a point p and now we know that our said this past connected so as to give so what do we need to do we need to define a potential a potential s we want to do it by just by line integration and the result will not depend on the particular of particular correct so everything is good up to now so for each point set X we can just take any curve C and define a if Ivory line integral of the vector field X along c and this is what we want to do who cares so with and safety said so let me confused stop can said to the potential potential to be F of X equals X the best near and here what is C where C is any any profits going from P to x for see piece wise uh piecewise to French from say a b to you this with that he I have still 1 whatever I do will the but this was a common focus on the and fine sorry and with that correct and points see also a equals the c of being the x and in particular this is well defined this is verified by assumption to no matter what kind of C I take I get the same value back now so at by 2 is above the 45 the the OK and also I should say says I'm in domain you domain he was passed connected the so that means you is open and connected
so that in particular it's past connected by the sphere and uh i did not prove in before the before the break there is a set of position 12 and if it's postulated it means that always can find such a so the exists yeah so this is perhaps set of OK so this is all fine I were in particular I reach for each x i can find such a per year their so C exists for all x in you Robert so what's the problem well the only problem is that F to show that f potential so great as is greatest as is what should be the name its X that's what I need to show here are some so uh to be shown to be shown is credits equal stakes in more we are we actually want to show it is that we show a the eye of f of the eyes partial is the air 5 component projects for all x in you and for all OK that we need to show how do we do this well there's a nice a way of doing this as follows of any problems sky noisy and you're busy have something completely different know what you have a question OK now comes the key idea here how do you how do I show this will show this is the difference quotient along that path in the direction the i so here's my point x his my point P is saying and my the main yours and what I want to do is I want to use that particular path that runs from the EI direction and then compute this equation as a difference quotient that's the idea there use use the difference difference quotient and I for an appropriate pass and where do this as I say it well for small interval I'm was in you and then I can connect this and then run here and no matter which path I choose I get the same result as this year if I go like here to this point I get the same result so why not choosing this particular path here where I can easily compute the difference quotient so uh I need to write out all of this here and now theorem in order in order to have to surrender so what I do is I let's see 1 from minus epsilon 2 epsilon Ireland tool when the the curve see 1 of t equals x plus t believe what that's a parameterization of this to and I say that and so noted see 1 of 0 old this fixed c prime in yeah I threw up who OK the the good the so now since was open to this means that this part of the stays within you if I choose EP Simon small enough what on search method uh C. Wong takes values and you see the you to claim and now am by the know this is no I need to connect this path to peace in order to uh has my definition of the if it right this will help me was the difference quotient at a point x in order to verify this here and now and in order to use my definition I must join this past uh to the point p OK so I do this as follows use open open and connected so use past connected so that's the easy direction past connected groups so that means there exists a path see from 0 up to say 1 minus apes silent and to you piecewise uh um the piecewise differential the enormity of 0 in quotes please seen all of 1 minus epsilon because the initial point here the yeah so all of this is X minus IP silently I yeah this is the terminal point of C not here will be seen what here will be C 1 and in a minute I'll take the connected to union anything the union of these curves and any problems if any idea what I'm doing right now it doesn't sound as so so this is this I believe is is nice the most proof in the in which what and it's it's 1 of the 2 key calculations which follows so no I take the something the union the some of signal and and see what which means I 1st ran from C and then from C 1 OK and now why do they do this whole thing while in order to differentiate f at x uh in the eyes coordinate direction yeah and this will be no be easy since I have ever and I have my integrals now defined to exactly solve this point and can take uh the difference quotient so this is a piece wise and and piecewise um eventually curve say on the interval from 0 to 1 + ep signed on to you and so I can compute the difference quotient non-nuclear little various social groups shown charlatans curricula Schmidt have been explored the end of the the world on the the you the the so the you should have that right the OK so the now for so assuming assuming that T is small enough that I can do the following calculation I'm interested in the I've partial of f at x what is this well it's been meant the students to 0 0 of the left of a what by definition of and fun not limit there so this is just x and f of X plus TTI minus the strengths and that for unit OK so this is OK by definition of F of X i have to plug in and I have to plug in the past it's is still limited to Polish um now this I can actually has been limited of f of C. of the last year 1 test the minus the f of sea of 1 the I since i've chosen over T since i've chosen my points and yeah I need and next since I take now 2 points 1 here and 1 with this interval here and given by 4 curve see that these 2 points and now I can calculate not like in my definition these are both line integrals so I apply position based on my definition of a potential this is the moment of T tending to 0 all the let's write that 1 over T 1st integral from OK I I don't right now I don't mind out uh gradient vessel so but I plug in my very definition which is the integral from 0 to 1 plus the also it of of clearly of times there's the prime of told he told mine this so that was Mrs. here the definition of this of this covetable by using this some curve by using the some curve C so this means actually that up to minus 1 minus T. I choose my curve not from 1 minus T 2 1 plus the the I use uh 1 minus is accelerations in 1 minus epsilon through 1 custody I use the of curve and there's something missing sorry and he I take the same interval but it only extends to 1 from 0 to 1 sent the the the and right so OK and now all of this is OK since this extends about since from 0 to 2 1 0 1 minus epsilon and this extends over the same curves I get that this is equal to the interval I he tends to 0 or 1 over T uh Demetrius integral from uh um from 1 to 1 class the yeah this is not necessary larger than 1 but 4 minute you can believe that and the off um and of your of T time see prime and also I should write or the choice 1 case since intervals from 0 to 1 would In case is less than 1 T is less than 0 of this book as being an interval running now what is minus it's a minus the interval with flipped and borders now this is just my direction EI so my integrand here is there was the whole point of it if X times the eyes is just the i'th component there was wired were run by a uh want to run in the EI direction in order to have the text I hear them so the whole thing is i that have separated out once again limited t tends to 0 uh 1 over at the interval everything which remains is XI also was strong the X I the or 2 the from and 1 to 1 class to you so now I have only this tiny bit uh object to integrate and through so I can I claim this is in fact takes on a fixed the this limit wisely so well the this is a short piece where integrated and this occurs in the neighborhood of x so there should be true formally formally this a Spivey um mean value theorem of integration yeah so this is in fact the length of the interval t times the function at the intermediate at some intermediate value see also say but in the limit this tends to tends to 1 and so uh and so in fact this tense have to X and it's in fact it's I of X yeah so so what we did here is the the and you see by 2 by 2 is it's a tool does matter which cover I take I can take a curve which runs in the eye direction then I can compute of my differential um as the difference quotient and the difference quotient for these integrals can be computed explicitly at least climate here and this is XI effects or OK so this is good enough in order to show to this is exactly what I needed to verify DI of sex is excite effects so little complicated but it has to be that and now all you remove the only remaining steps and is to show that tool and 3 I equivalent and that's not hard no and it's got this key and so I will do this the yeah parents all the world what he you was the the the he of the but conclusions so remaining step is to show that to entry prevalent the the the but the well that's not why is this well since loops and 2 curves are closely connected if I have see knowledge and see 1 to running like this uh between the same end points I get the loop by taking 1st seen knowledge and then running backwards from C 1 so if I take the Nord and c 1 minus I get a loop the this is the new emphasis to to crops the the OK that's the entire idea here and you will see since actually the integral is additive in some uh we have proclaimed selects to versus and so if so in order to see 1 has the same endpoints I have to use say in points then and yeah events CDP equals C mon plus C 1 minus it's not which the and vise versa if I have a loop and conversely and good the if c is the in Fresno powdery gets so my care how do I get to 2 curves well I take just any intermediate point and save the 1st part of the loop is my 1st curve and running through the 2nd part backwards is my significance so if c is a new then but then and for each uh t knowledge in the interval events defined the I have the uh had a right to subdivision gives um the which can decompose decompose the into as some sleep with C naught plus the C 1 minus where C 1 and see more in C 1 have same endpoints we the the OK if I had said when I can simply right the part of the line integral as a sum of integral xt overseer of years is equal to or seen thanks active players C 1 minus exteriors and this is uh by additivity of into sums now that we have this early on and then this is uh the interval the Indigo from the backwards curve gives a minus and assigns a minus sign to the integral so I get to see north xt s minus C 1 it's and now what any need to show if the integral overlooks is 0 then the interval for any 2 curves with the same endpoints agrees well then this is 0 so actually all we need to do is to say now is is left-hand side is 0 then the likelihood right inside 0 there is the left hand side equals 0 and think of England to right hand side equal 0 and then the left hand side because 0 corresponds to and the right hand side corresponds you could 0 corresponds to 2 and I'm done with the entire proof so we are beds you 2 5 process for instance and terminal time so what have we achieved now we know where we have to solve the problem but we know equivalent problems that next time we will show when all this becomes true and we need to a study domains in a more in detail and then we're done with this section thanks for listening