Irrotational fields on simply connected domains

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Irrotational fields on simply connected domains
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OK welcome to the 5th class on complex and out of this and this time I've put them the review on the slide has so since the I hope to come to the calculation of the end of this class all sold most of this or the upper half of this was actually at the same time very review uh last time so if to remind you once again uh on we know that a um vector field which is opposed to potential so that perspective it must be and must have an a Jacobian which itself the joined or symmetric and or in short form a conservative field is irrotational and the problem we deal with is can we at what happens if the is it true but we can reverse the science of this that this becomes an equivalence uh well uh this will depend on the domain and infect uh for what's called a domain and analysis namely for what's connected in a connected and open set so last class we characterize conservative views so the left we uh showed equivalences for the left hand side here namely at the existence of the potential or that the vector field as considered conservative that is equivalent to the effect that perhaps this is easier to say that for each loop for each loop closed curve the vector feud integrates up to 0 or so that if I take a general occurs with 2 different endpoints that it only depends on the end points what the line integral of the vector field gives me back so these are equivalences for the left hand side of some of its implications and today or what we have to do is we will need 1 more property of domains that's that will be the 1st half of a class and and the 2nd class we will characterize the right inside the the uh irrotational fields um with some properties which actually show us what the uh backwards implication it's OK so I started which of properties of domains which makes up uh section 2 . 6 simply connected of the connected to but domains have we we I could a simply connected sets and cell what is the problem here the name basically we want to the colloquial language I would say we want to detect holes so we want to distinguish domains you which has evolved from domains you which do not have all but how can we do this way to do this is to give a good definition come up with a good definition so there's lots of technical stuff but basically it's stand to in order to distinguish this from the it's booking itself let me start who by telling you how this is done with how can we distinguish these 2 cases um we as we look at curves from 2 points in the domain and we're interested so here 2 curves and he also to 1st and the difference here between these 2 sets of curves and these here is that these can continuously be deformed into 1 another while these cannot since visible so this is the property which I will formulate is minus that's the 1st point of my definition and actually I want to assume use open the set so late you would be open you in Rn the open and now let me give property 1 of K which is the most complicated 1 so I take tool piecewise differentible curves selves the to see 1 be piecewise differential Belisarius from AP to you here so please think of this as the knowledge this is the 1 uh and now I say what it means to be continuously deformable from 1 but this curve is continuously deformable to that could so that means there is a so-called homotopy uh OK actually I want to say from thanks to y which means this point will be X and this point will be 1 the notes Rasmus Prehn pq OK so then I say a homotopy the the the homotopy summation the a homotopy him German a from to see 1 i is a matter is a map OK which introduces an additional auxiliary parameter so that we can continuously deform these curves into 1 another so what kind of man is used as map h from 0 1 close ABE say the domain also uh my curves yeah this is a common domain of both curves to make it simpler you will in written as s T maps school age of face steep H. for homotopy um and uh I want to fix the endpoints here it's and why so that means uh I want to say and I want to assume that the following holds H 0 of T uh up to OK phlegmy 1st say H 0 H 0 of states this image 1 is they have a curve so the this out 1st the in age 1 of t it's C 1 will grant for a and also I want to fix the importance
of so each H S here and now we have h s for each parameter between 0 and 1 defined each H S will be a curve here here so this here is a particular kind of age let's say this is a age 0 see knowledge and this will be H wall proof see what while and also I want to say that the end points for each H S curve from here to there I want to fix the endpoints to X and y so that means that h of as a is x and age of B is y for all s the and who OK the so this is a key the definition I should give and now in order to make my life a little easier I will assume a stronger condition namely that this is not a continuous map well it's not ready yet written here um perhaps it should be there was any map and analysis is continuous um so think she I want to say that it's differential and so I have a more overstatement Malvern and how can we assume that all the following and how can they at age it is In H is differential is actually it's C 1 In and has a continuous s derivative so this when I go from these curves is actually a tangent vector pointing into the estimations of the S curves would be something like this is and and so this these curves have a tangent continuous tangent that's this condition end of the moreover end uh what else do I want that get t derivatives exist th the key and and share 1 2nd derivatives uh which on the right and the st t but these exist and are a key piece wise um piecewise continuous in T carry cell AH this um so this I want to be um harder right there's it may be a bit sloppy this piece wise continuous continues to gain the what I mean yeah I mean OK actually I started with piecewise continuous curves so my initial curves may look like this year and so my homotopy may have the same the kings here for a given value t once they have this would be to naught T 1 key tool and all I want to do is every I want to require that these things as a function of T I actually differentible uh I actually the well when restricted to certainty intervals they are a these give continuous functions and this is true for all its yes I'm a bit sloppy here with my definition and let me say at this point that this condition here is the it's basically this is a not necessary to assume it's only for convenience in fact if you have a continuous homotopy when you can smooth it by the can smoother by certain procedures and you will end up with uh these 2 French ability conditions yeah so so this is in a sense this is yeah I say this is only uh I could say this is superfluous who is the assumed for assume for convenience the and you would see where I needed later today OK and and so if you look at the definition of a homotopy you will usually find universe OK and so now that becomes more important things the 2 curves see knowledge and C 1 as before are called homotopic but of we were told and provide there exists as a homotopy between the and C 1 it homotopy from see in order to see what so think about my 2 pictures here here C 1 and C gnawed on what topic and here there not yeah here the whole is a problem to get continuously from here to there the in of can now and I want to um consider the particular case of the loop um then I say it's a loop it's so what's a little bits of point with a constant the constant initial point so if the loop is is anxiously homotopy homotopic to this initial point I call it contractible won't cell if 1 is the loop base et um underwriters uh say AIX equals y men and so homotopic homotopic to see knowledge constant to x and that's the Petek like has 1 of these curves is a point then um I call this loop I call this u contractible there and C 1 is called contractible the some Ciba but if the ends I also also another in another and notion is that we cause such loops and now homotopic contractible or only poem what topic and the what to say OK ends no finally I come to give a label to attribute the label to domains in order to distinguish this demand from that a menu uh I call it simply
connected if each loop is contractible so uh and over them the well I don't need that you this called simply connected as and since it this is such a long thing and will also my SC for that simply connected inferences are made answer seems aluminum a hearing and OK if each group is a in you is contractible and each the loop C 1 or whatever in uh you'll it's contracted so this is actually good enough I could take in new round this uh . so just forget about why and consider this no this is a loop which cannot be contracted what if this occurs it can be contracted here and it's in fact equivalent equivalent to perhaps a problem from problem session uh there at each pair of curves uh is homotopic in this case the focus as it were not have the fuel remarks regarding the notions notes for Marcus remark that you remarks on is OK 1 thing is that the homotopy uh is in fact homotopy also curves is affected in occurrence relation home until it home it's OK uh to finds an prevalence relation yet so if you yeah on curves with fixed endpoints so if you have if you have and what to curse which on 1 topic and a 3rd curve which is topic to the 2nd 1 then you can compose it for instance and see that this is uh a transitive relationships all OK the so this will be the problem sigh guest segments um you are you is simply connected if and only if OK 1 thing I can make more special that I wrote each loop is contractible no matter where it starts no matter what x is in fact I can um I can um assume OK uh I should write about the delay the and domains you will sorry that paying you it's simply connected and if and only if all loops with a fixed x are contractible this is more special body Copeland load carrying 100 interest exist so uh 1 of 4 and X in you all loops care with an at x Our contractibility yeah I don't need no I don't need to consider arbitrary points X but 1 1 particular point is enough there will be a problem the reason for this to be true or is if I have to you and say 2 points x and y if I'm connected which I am in a domain then there is a path from x to y so that means that I can be consider can come to loops ed banks by composing a path from X to Y going around a loop and going back near this the idea here and vise versa so in fact loops in connected in connected sets it does matter there is some at which point I actually um require the property that it's a contemptible on OK also include they don't write this out is that any 2 pair of curves with different endpoints is con contractible this holds an arbitrary set you doesn't have to be connected and on every right sort Senseval use probably there you is you it's simply connected if and only if each pair 10 also curves with the same endpoints you always same endpoints what the there I homotopic my uh how OK there's a lot more to say and but um let me as I not generalize slide here but give you some examples examples um 1 interesting examples OK 1 thing perhaps to know is that the look at the entire space Rn then it's simply connected it doesn't have any holes is simply connected why is this well I think to have take the loop the take a loop say based OK by this property here uh I could for instance look at this particular point 0 at the origin OK take any loop whenever it does more OK and now how would you contractors loop won't just like uh uh use home of varieties to right and Canada obvious that probably not in the but this is too complicated occur if I'm sorry here so what I use is is I simply uh mn and multiply the loop by this you know I can wait for solid um age of S T so given essay given the kind of given new C I could see this will see how define h of this T to be uh say 1 minus this times c of T cell add S equals 0 this is
all of this is the very loop and it s equals 1 this the uh vanish so what I do here is i shrink radially I shrinker radially with 1 minus what and yet and it's a similar idea will work to show that to any 2 curves in our in our home topic
so foreseen or to see 1 with the same endpoints I would provide the age of AST as linear interpolation 1 minus this scene because it's time see what that would also work care so as to give an example which is not simply connected to an example with the whole uh the the most the as the most obvious idea is to consider as 1 in the said you all equals on tool without the origin so we have a hole at the origin and we look at the loop at the group unit circle will I claim this is not contractible actually this is also easy to show it's it's very obvious right thing of male here here and a uh and a loop of rope it will definitely eventually it will have uh been there was uh hinder it to shrink to point so that means that it's physically it's pretty clear mathematically it's it's a little more complicated and in fact I cannot prove it right away that it will follow from what we do so uh this so you is you is not simply connected has a whole here but it will this is not so easy to prove so proof later she will follow from calculating some line integrals which are nonzero but it's hard to show this directly carry and here is yet another example if you look at the shelter be of zeros the the without be little r of 0 minutes to make opening I do feel like this so what are these will love these are spherical shells so these are 2 years from which removal of the sphere and in dimension 2 will be that this will be an annulus 4 are greater than and now this set this here was hard to that that don't draw too much uh this set it depends now on the different on the dimension if some of this set is simply connected the so that simply connected 4 and at least 3 so if you have any loop here and the loop here I don't know of you are able to mention this but take any loop here with in here there's 1 way to shrink at this little sphere little hole is no problem on the other hand if you have a loop in the annulus then you can cannot shrink this 2 . it's not contractible him so I'm not simply connected for n equals 2 and event not connected friend equals 1 if you want to have the but I won't write it down yet cell what we do is we detect uh holds of low dimension and fact New of caring and and this is a much more general concepts so the being a mathematician so please put the camera now this last time we did work so if you topologist yeah how do you distinguish you might argue from well this is the best so I have a football yeah it's solid without both Sorry it's the best I had my office how do you distinguish this well here this market has occurred mainly in the handle going once around it's a loop and this you cannot be contacted was in the body like this and you can be contacted so these are the In covalent 6 yeah you can actually come and make the world and equivalence relation and uh by just the number of the number of holes this way we have this is this is done and topology and I will not do anymore and that's that and actually that if you look at the Wikipedia page on the homotopy you find a nice homotopy of the entire my courses so there lots of more general ideas which I cannot gives here however what I want to do is and I will give you some I would give you some special I will discuss some special domains which are which are commonly used in non convex analysis that's all communities of the 1st of
the the in it and been on the right and the thank but the In OK so and what I want to discusses on particular sets for classes of sets namely a convex sets you probably know and so in this sense uh say why in our and probably even wondered non-empty um is convex as you know if for any 2 points in the set for any pair of points x 1 the straight line IX Y is contained in the set so so for all x y in why we have that x y straight line segment actually is contained in y OK I guess you know that you know that but a particular notion useful for complex analysis is a so-called star-shaped set or star domains so why star-shaped stemmed each um that if well let's make a picture of rest at yet we get closer to Christmas so um what a star-shaped mean well there is a particular point called the center z such that any other point has a straight line connection to this very point so you can from any point you can get on a straight line and to the center the z and so over exists at the point that and y called the center but a such it such that for all x in y said is contained a 1 that's the so called Starship demand and what's critical about these 2 part a kind of kinds of domains uh what both interconnected they cannot have hold since the are radially I mean we set radiographs so uh they cannot have holes that will be the next theorem but let me also kiss if you make a few remarks uh so what I want to say 1st if 1 is star-shaped and once this if conversely if it's convex then its star-shaped 1 shape it is 1 of the points and what he is fixed by here does not here so this here in this set which is star-shaped but not convex OK but on the other hand convex means with any point you can join in if you can take any point is center for this to finish then and uh my 2nd remark in 1 is star-shaped then why is connected can and he is a nice example which is important for complex analysis of a star-shaped set a so for example phase so costly playing you should stay even that slip plane the the call it s equals C C c was found to be the interval from minus infinity 2 0 this is my notation for he's the so I removed I removed the numbers on the real makes is apt to 0 so this are removed and remaining search is it makes up for it OK this is usually called slit domain OK was
playing and this is um certainly not convex since these 2 points and don't have a a straight line with the set S but it's star-shaped with respect to well with respect to what want to any to what would be my what do I need to to take so center the yeah yet OK Rotosound Turkey had source or any point can be connected to these points to this point here and and take instead of 1 and here I could also take any point with positive any point on the real axis with a positive which is positive yet so this is sort of not what is the 8th what you expect for a domain which is called star but it's starship the that what can use the and now I have the proposition saying that the red starship domains um so long simply connected staff shape domains on a simply connected the they don't have false and the proof is simple so what do you need to show also simple connectivity uh I need to show that any loop and loop can be contracted to a point In fact by the property tool which is still out there uh for this 1 point I can make the take in particular the center what so what they need to show that any kind of any new based at the center can be contracted while this is no surprise since uh I can make you use the same method I used for uh I use foreign I can simply simply met these points on the straight line that into so if CE let's see be rule that exceed in the loop uh based faced with endpoints and at the center the and then I all I need to do the in order to to define the homotopy is uh I take a 1 minus this times U of T plus at this time the center so interpolate i'd interpolates linearly between the center and micros for any t for any t I follow a straight line here and this will shrink because 2 point of its this is very obvious and if you when you look at at the literature references therefore this class a many do not deal with a homotopy but they restrict complex analysis to this kind of uh sets star-shaped domains they can they can be done but I I uh I think this is the more honest approach really OK so this is the 1st part of a class so after a short break when we come back to the line integral OK I would like to continue please get seated so I would like to come back to the original problem which is that which rotation of fields uh do will have um I conservatives have a potential so here is the most important example to understand the full the theory I believe OK so this is an irrotational vector field name is very effective you've written out here what is this vector field where this is my dignity re 90 degree rotation and this means that OK this is divided by the length squared but since this is the length of the whole thing has length 1 over r yeah is 1 over length there is the 1 over the length of a point you at here is the length of X so you see this you on the left-hand side of all it's not entirely accurate a it's longer than need a uh upon the point it's longer in the in the middle then the air at the boundaries so it's it's has length 1 over a distance from the origin OK so this is very easy to uh vector fields and it a in the and in the homework you study did yeah this was the homework you yesterday I guess right correct hopefully um OK so what is and particularly about the respective you well as you computed I guess uh they all these line integrals um for giving them for given curves you can forget about the ROC and just take side or and just take the function 5 minutes 5 1 minus
5 0 it's only the angle variable which is no surprise that if you if you integrate this up yeah then uh it's only the angle which matters the OK and also and you figured out uh but this I mean this looks very rotational but it's a rotational vector view buys this well since the length of the vector field is larger ambient in the middle then radio outside so it's exactly the correct scaling to make it the rotational it has no rotation no matter what you think a rotational speed looks like this has no rotation and so In fact on the domains they call this perhaps you will on you'll broad x it is 0 it's the rotational so our problem is is this vector fields conservative doesn't have the potential the answer is no on you why is this uh well it's since OK don't take this literally then the clique potential there's a problem of potential as you can see if you do it on smaller sets of this little man it's OK well
this is just a problem that the end of variable the angle variable of a plane is if you like a multi valued function you can continue this this graph Iago's it would lined up and went up exactly yeah so if you integrate opposite view and go once around dictionaries such curfew non closed curve if you go under threat courtesy and integrate out the vector fields as well then the difference is 2 pi you won't do do not end up at the same point so locally you have a potential globally you down here if you walk on a small occurs in any any closed any closed curve uh groups within set then it's OK and shaking out on the top occurs of the particles you get uh um integrals there just of this vector futures that due to the fact that this is files 1 point minus 5 at the same point now this is for the 3rd you if I look at this a subset this contained in u which is to select plan and which is actually uh the basis of the graphics here CARE the negative the negative uh really makes is is not included in this is exactly vary running up the stairs let you fall down by 2 pi and the this is the graphics for the slip planes then actually there is a potential and so and infect more general than here I could write actually on yes there is a potential and here it's written in a complicated way Oct 10 also Y over X remember that if some of the complex plane you and I have my point z here with imaginary part of X and Y and real-time X then the angle here is the argument of of what it's the argument of z and instead of the tangent tangent of this angle is the Y over X so 10 of arguement um is the Y over X or the argument is equal to Y over X right so we have that's right to the about the tangent of art z equals Y over X or equals factor Y over X which is what was written out here but actually on S or the better way to write out this potential is use the odds the argument or the end of the the polar angle function if you like right N so uh
if you uh why do I point is I well it matters if the domain is simply connected or not right you will is not simply connected not simply connected which I haven't proven to you but um good it's obvious but this little plant is connected so what we see out here is that the the we have a an irrotational steals an irrotational see here on a general demand on simply connected 1 that the potential only exists on the simply connected subdomains and actually doesn't matter that I take exactly this slip plane I could I mean just make an experiment and replace S by instead you OK what could SPE uh for instance I could take is said which looks like an no loops yeah like this end on this subset of you still a potential would exist why is this so well the polar angle function here would be saying uh 0 here minus pi there and if I keep on going then it would be pi + pi vector and it would be too high where for instance never imagined mentioned this to continue the polar angle function uh around the
spiral uh then you can go up so for any simpler commended connected domain it works and this is the and this I will now the goal is now to prove process on the so this is the next subsection so units military a self-adjointness of the Jacobian of by vector fields as while I call it as an integrability condition can why the at and I have to justify who have have to justify what this means later on OK so and the we would like she proof of following Syrian as and this is formulated as an it as a character characterization of irrotational fields for stuff up Solomon X be praised seeing actually it's the 1 is sufficient here C 1 vector few you take to shoot up on uh and you in Rn the the so I don't assume I know that simply connected can be general then and I want to characterize conservative vector field and um who Edna conservative but imitation of active users then X is irrotational or meaning that the Jacobian uh a of x is equal to its transpose the In persist as the condition have used you find here at TI exchanged and this means that United States DJ comparing that's kernel and to the come through the fact that each integrated over each uh null-homotopic loop we get to all were each each null-homotopic the each contractible look the but reached contractible loops see in new the line integral vanishes the line integral vanishes uh of what we have the integral of x the interval of C over xt as I mentioned monkeys and uh why is this this is is important for us well if you always simply connected them this condition here applies to any group so this is it is true for you simply connected and I will write this out fully next class uh and rather uh show you and and talk about implications for simply connected domains uh but let me now prove this fact to you so what I do here is only 2 remind you if I have this is stated on that that x is so this statement here applies to domains with holes you'll the but it applies only to curves which are 2 loops which are um uh contractible tend not to groups which are not contractible yeah so OK and that is so there is no a complicated approved but that well it's complicated in the census computational but it's not complicated in the sense that is nothing but uh what I believe there's nothing difficult going on in fact what we want to do is we want to integrate up this condition the overall multiple and we will see that this implies uh implies this welcome so 1 direction here is obvious namely if this is true when the source while it is his so the well the this is so since we had that have a characterization of this condition this condition means there is a um is a potential and we can use that so the the I let me do this as follows it's not entirely directly P and you and consider OK he is you know I consider a point P and what I want to do is I consider that ball a the the rst
it's quite noisy actually she isn't too large what I do it's too hard OK to most people believe it's too hard OK you will see annex a year before Walker will set next class and then we get back to complex analysis the complex analysis setting and it will be simpler but this is sort of to the the hot stepped where we have to go we have to go through OK so we take a point p and z be considered a small ball B our in you OK then this ball is simply connected for yet clearly can be contracted any loop in P can be contracted by the linear contraction I showed you the OK and then this means that OK and so I want OK so now how much I can consider now topic loops in B the I restricted statement not to uh I restrict the statement to be up given it tells me that now homotopy groups and be out OK any loop in our infect any loop in our has integral 0 well then I know that a if any loop has intervals ourselves that's all writers out 1st interval the old lower I C sup for loops loops see and you know we have integral over C and xt s vanishes this is just my assumption here with restricted to the case that use br OK if that holds magnetosphere embed in under this condition there's a potential OK this potential only within this set local potential OK and this was fearing 13 which I give you showed you in the previous section this today so that means there exists a potential the potential energy from the OROS P 2 up what if I have a potential in the all of them I know by the shorts slimmer reasoning that it's a rotational saw there was also theory number 8 I hope proposition idea so this should propositions and at MIT was um and that j x equals j X transpose the short form OK so the let me say this holds folk p itself all actually it holds for any point in within the body of people you know for 1 the but as medicines peace opportunities OK so that means this direction from here to there is trivial the hot the hard case is to get from the differential condition to an integral condition since we have to integrate and something up and fact graph to integrate this very condition up to the rise of it here and this uh what I do next OK so this is the hot direction integrated up you OK so what do we have we have a but we have 2 conditions on um we need to look at a now homotopy dilute see in you and show that this integral vanishes what does it mean to be null-homotopic well you can contracted with the homotopy cells for the given blue seeing consider consider the homotopy age from 0 1 cross a view um to you the ST as usual and mapped tool edge of S T of and the edge of 0 to is a might previous condition so at times it parameter as equals 0 it's my given loop and a parameter 1 it's my given point and point of the loop so how does it look like perhaps I draw in the picture um so here's my point PCA is my given groups see and now what I consider is a homotopy which sure contracts seem who why to my given point P cell perhaps it's a good idea here to uh right out also the domain of homotopy cell in is uh perhaps contrary to the alphabet let me do it like this the cocaine so
uh the loop the see all 0 Curtis et a parameter value uh um which primitive value doesn't occur at parameter values 0 I is here at parameter values in OK so this here maps to my loop see the endpoints of the loop P. the not only a for loop but also of each uh of each loop in the homotopy of each contract the the final is constant to the point p yeah this is the contracted and so this is actually how it maps and wanted maps like where M use colors uh at a given value of S a given area of S I would have some but I would be somewhat intermediate times more on this side here so this is this this is what I have it on carries ends of in the the at the end we will book and now all of them for In the following I will actually OK I have all piecewise differential but business let me just assume that the curve C is differential with itself in whatever I say and write out uh intervals if it's only piecewise then you have to write some of the intervals rather than interval yeah but let me let me assume we undersea 1 sitting so I can avoid the sums and make the the right or to it easier OK so but where's the following claim the how the claim is that the integral over of the loop all of the into the line integral of x over loop can be written out as the it's a double integral and I don't even want to know I do need moment the summary all the Jacobian of the in each um th the D T and now the avatar derivative the HTS minus and now I put J x and on the other side so it's minus j x of age and but it's now the HTS not DHT t and th t t here and forget the DST dt so it what the tedious whatever I hope so this is my claim I can I can write the groups the integral of the uh um the the integral over the and I North Sea in terms of the Jacobian of X so why is this and why is this good well what happens with j if you put it on the other side so I Note that here j j of the Jacobian sits on 2 different sides of the scalar product product what happens if you put it on the other side you need to transpose right so if you have a condition that the transpose equals a matrix the inverse here vanishes absolutely and so that means in this I know that if j x equals j X transpose which is my irrotational condition I wrote right kernel equals 0 then then while the integrand vanishes since that means that let's just write it out in in more the well more familiar from this is the same as this when fall W and not there this is the very definition of the of a transpose A into with respect to the scalar product of self-adjointness with respect to the given the scalar product OK so if this holds now this is the and this is w this is w and this is the but that means that the integral vanishes integrand and the integrand managers pointwise not as an integral that point delta the and so that means and that's exactly what I want to show if the claim of if I can write X in this form then I have achieved what I wanted if the integrand vanishes that means that integral CD xt it's it was 0 it's design OK so this is good enough to show that the yeah so the stare at this and realize that of the right hand side you find our condition this condition is integrated as a double integral and it's integrated over homotopy and then you get this uh you get the line integral over the OK so why is this true why does the flame old perhaps I need a star something right my case 1 now we need a calculator the OK so 1st thing is considered the uh loops consisted of loops teII maps tool and I call them toward s of t equals of St so what do I do I consider were when I write out of this thing here what I what I do is I say that um T. know what the is this TU taller so torn what is the given rule so 1 is my final point and this year would be of this year will be exactly some intermediate choice no and very her so 1st I want to consider these loops in the 2nd part I will like she's which isn't he or and um now I'm I'm interested in well character and consider comes the the s dependence there also the integral x D S over my call microbes choice so over each loop taller taller this 1 is drawn in yellow I can consider the line integral and again this look and how it depends on my parameter s the is 1 thing which is obvious which is that at various final parameter as equals 1 this vanishes since I'm at a
constant point and so this must manage so I call this function uh ecology so to be an short let me call the line integral along over toy as denoted by GA so is the interval over a to B of x of the curve choice of T times taught as prime Prof t dt and now I am I want to the rewrite this by differentiation er er and caring and I want to write the left-hand side of my claim so I start here I can write and let me just it's it's more intuitive to right left hand side of the stars which is an integral over a c in yes and K is equal to 1 g you know all too well this is I mean the loop integral over CC is tall knowledge and the integral over toneladas only complicated notation is just g of g naught why do they use is complicated notation since I want to write this is the difference to g of 1 right this as g of 1 minus G of knowledge when use the fact that g of that loop T. tall 1 is is a sorry here's p toll 1 should be PDS correct is P cell 6 the line integral the integral over tall 1 xt mentions it's a constant portend talk and now I use the fundamental theorem of calculus it and right this difference of G values as the integral of uh the derivatives of g right so I write this is minus G prime of S and ideas the long-term and I have to compute what this G prime of S some very compatible left and right hand side I want to introduce another interval and his I stand here ice into this is infected T integral and here by the fundamental theorem of calculus as introduced Austin s integral and so this all together is the double interval which we want OK let me computer she primal the what is it how can it's the is this so I have to differentiate this with respect to this the so the DDT of of integral of the right hand side uh it's destroyed X also told t as of to tall Prime of course C and do to you don't care from a to B and now remember what we did an analysis tool we can integrate under the uh the can differentiate under the integral sign if if this here is the difference of a function C 1 function which it is as we will see so me a there's solve this is differentiated differentiate an integral so but the and this we did in connection with the Frobenius theorem at the end of uh uh at the end of the analysis to class case I can do this many put this in here i have to differentiate a scalar product How do I know what is a scalar product well it's the some of the components so in order to differentiate this i have to differentiate by the product rule of this and that meaning that I have 2 different this liters of a common as it is an edge other the other way to differentiate it so this gives altogether a indirect differentiating the 1st term for it is going to use for this crime and the class and now i have to differentiate the 2nd term plus I copy the X off toward the soft but now I differentiate the 2nd term which means the DD as in front of the DT where this is a d d t prime so it's close its both derivatives no the the and no um but the now I have to recognize what I did the 100 do this and to so why is this good times have said added to write this out here some the but this here's a travesti by definition what is tedious vector field of age well as a composer function use the chain rule right so in order to differentiate this I have to use the Jacobian of X and she liked post differentiate uh this with respect to its so this can be done by the chain rule and similarly OK here there's no need for the channels and this is a DDT age of St anyway by definition OK so if I summerize and perhaps this can be done here what I get a OK this was the G prime so the left hand side of style all my interval over X D S Hey I can now compute as OK areas minus the interval of t prime DSP G prime is this so I have to uh I have to write 1 interval from 0 to 1 uh of cheap prime front won't care let's just do this G prime of interest years minus sign by and now I can plug in this long term and I do it in a good way hopefully so 0 2 1 a to B is and what I have to do here is I use the chain rule so I differentiate X which is that it gives the Jacobian of it's age and no obviously leave out the arguments St the thought is calculation to make it was a bit more
obvious and all of this in the French and I have to differentiate by so which this it gives the partial DHT yeah it's nothing complicated I say and what is tall primal is here is the d DDT of age X I should use my should use partial here there so it's the age the key whom carries interest there has to be integrated DTDs OK that's the 1st term and the 2nd term copy this with uh and don't forget the integral whoa over various the and write it out in while we're caves this is x of age said times uh and now this is a 2nd partial injectes clear key which transit DST over t of age of was age but integrated over DTDs is nano colorable but uh in fact this is possible ways this half well well compared to a terms I denotes I'm able to do this on the blackboard let me see so this here yes so this term here with the correct sign is this here looking so 1 of the terms we have is good we OK now what about this term here yeah this term is not a client system well what's necessary to do in order to get to this yellow term here well we need to of up and down there we have a double derivative partial ds dt DH but because this credits by t but we don't have any 3 um derivative on this side so what we need to do actually is make sure that is the 2nd derivative here did you teach d over d t that this goes In front here then we have then we would be done and this will be again by the chain rule and a J X times uh um DHT t term and this will be a tedious h time just as it is written here and this can so what we need is a partial integration and thick this is g times f prime and we want to make a to G prime F and as you know partial integration only works if uh F times a F times g at the boundary it is uh vanishes and this is the case here so let me do various Nesterov no at the end of the year and the whole of the the in the some of the rest can go along errands this cannot go on memories and and the the the being in the know it and OK so what remains to be done is a partial integration all of while if you like of the but to see that this term this yellowtail is the Yellow term up there and way to do this is now we do I mean before we consider to be a we consider the S curves here now let us do the opposite thing we consider T. curve the so if I fixed the way how does it look like well when I have curves like this here and I want to be of he these are well perhaps I need 1 the now I do the other way around no and this solids writer don't consider this what you and maps to inequality sigma t of best that which is nothing else as before h of t so I consider these curves we have around wise good will at the final points at the final points I mean along along here and there I'm in both cases I'm fix that P and this will be widely f times g of the boundary term vanishes so I started I start with that can do in the way I do this computation backwards I'm sorry so I write down a bit interval of x t as minus integral of xt s over sigma b times sigma a so I claim that this is a row-wise visceral well uh since sigma A. is constant to p and sigma sigma uh B is also constant of the constant values so certainly so here we have sigma may also be is identical to P and sigma B the 1 of once the parameter of t of x y and sigma as pets without parameter but is constant to right so certainly both integrals not only the difference but both intervals Spanish OK pay attention to that final computation in man next class we can harvest what we got located so we have to buy so this is good mouse since I can use again the fundamental theorem in order to write a solid is a different same trick as before we want we want to double integrals so this is the integral from a to B of the the key an interval of year over 6 months he uh the t I know that this is forbidden rotations sorry you know this is not allowed to uh but I don't want to use a toy here and so forth figure out how this works correctly you know it's I think it's more obvious like this so please allow me to do that so now the same thing as before I differentiate and the interval so I right inside and I get the interval made to be the interval from 0 to 1 DDT got what is this and this line integral well it's the scalar product the scalar product of x and sick marked healed s times the sigma T. derivative which now is the derivative in the broad direction uh of various units and the old dt this store and home care and now the same thing as before i've 2 terms since I differentiated scalar product so I use the product will so scalar products and I get uh D. D. T. tanks uh DDT aches perhaps I forgot as of now can no problem there x of security of France times the original signal prime t of this uh um and had STT maybe this time separately and it's plus and maybe 0 1 x of a page and now I differentiate the same term if I differentiate the 2nd term in its uh it's day um well perhaps I write it in their original form so this is the key is a um that and S. derivative uh there's another T. derivatives so this is really d this could do to you also which DST th or is duty sort and in fact I let me just do the same thing here and write a solid is tedious age the image ij units moocrank to and this 1st thing here what is this well as before as before so long calculation but it's nothing complicated happens claimed its fighters chain will take the Jacobian of sex and age and different impose differentiate this segment t which is a service s of T by DDT solids the edge detail mn OK and now do comparison in check when we done at all all came cell hero we the yellow term with the 2nd derivatives here is the yellow term was the 2nd derivatives hopefully the very same OK I was very same by Fubini's theorem I can flip various and and diverse false alarm I can flip this year we did this at the end of last class this is OK and below K. and then the yellow terms it's meaning meaning that I can replace this yellow term by this term here but this term should be this term here well by minus this term and that term should be the why the branches and uh maybe this is a bad color choice to not too late sigh the blue term here the to distinguish on it is it is this term here yeah comparative it's exactly the same thing OK so what have we done here for long intake by them that the by our claim written or XT as in this form of a difference very integrated up conditions the rotational conditions and by long calculation overview we know that over loop indigo vanishes the writers who as a homotopy and integrates is a condition over homotopy uh due to the noise level and I'm sure that you and don't appreciate the beauty of what's going on here but let me still say that I find this proof as beautiful as can be We have a condition we have a conditional upon we have a differential condition and we want to uh to get to an interval conditions is the differential conditions and want to get an integral condition what can we do what just integrate this condition up and we get it isn't that nice I mean anyway I know it's a long coagulation next class I would draw the conclusion for simply connected domains and then we will be invariance of complex analysis and it would be much easier it's