Line integrals

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Line integrals
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and the today we'll deal with the line integral so what's continuous the section on line integrals and let me quickly what have you done so far 1st class we cover it complex multiplication and some of the stuff of complex numbers 2nd class we covered the costly Ramon equations and as a consequence of the uh C. linearity of the differential um and last class I started also with um line integrals so let me quickly review of the back I so so I'm an however so specific properties on a specific problem is to find the primitive call the find a primitive of the given function from the complex plane to complex numbers set of primitives f of gives little effort in the complex number k so a from C to C by integration of meaning that f prime equals f so this is what the the real numbers completed the um the a fundamental theorem of calculus to us and actually do this a little more generality so more general problem which i want to deal with here what is all the we want to find a given the vector field we want to find a potential so there's savers and find potential so s from uh mn while it's still this on the entire space uh a potential also a given vector fields Oscarsson vectors here has a aches from say are into our and all and an open subset tho and that means we want to find and a little F such that the gradient of f coincides with the X and I a nice and relation for this problem if you not a physicist that need this uh is uh think of s say from are 2 2 are and look at the graph what is the graph of efforts it's a hilly area right with the rest of for the case n equals tool and to so here what we look for is we look so all we have is a given vector field x and corresponding to the slope it's so here only of the domain we have where of say a vector field X and this is the gradient of f is equals x this means that the EC the given as the length corresponds to the slope so I want to find a a landscape such that such minutes uh for my given x and a such that x corresponds to the slope meaning to the direction and intensity of the essence the uh office really Netscape and as you know um so from this interpretation it's pretty obvious that this cannot be done in general why not well think of the think of following following you vector field the vector field always points to the direction of the S and so if you follow the if you follow the vector field on and suppose you have such an f you will always walk uphill right you will always go in the direction of the steepest S and so is nonzero so you will always move laughter so if X has has closed curves grossed integral curves here if there is a curve uh in the planes sir which is close to n to which x is tangential for this means on 1 hand you go uphill all the time on the other hand you must end up at the same point where you started that's impossible so effect closed if the vector field X has closed loops uh this uh this task here cannot be achieved OK so last time which I gave my stated the necessary condition for this problem here to the solution namely and necessary condition and so and low for a start to be solvable the is it is solvable I all of that is that some of the vector field X as the Jacobian a differential which is symmetric so it has some limitations onerous areas so that the x in symmetric or if you like you rather continued not a matrix but in a linear mapping is itself the giant which means that whatever the entries of the Jacobian well views of a partial derivatives of f of x so that means that the j also XII or this is my shorthand for d by dx j equals well well yeah the same thing with permuted indices and this must hold for each point x the cells for all x in OK say for AIX going from you in Rn the 2 power in making a more general so I requirements for all x in you for all points and for all indices hi j between 1 and between 1 and 10 the and let me call this
condition yeah let me call this condition practiced seminar for symmetric him so this is a necessary condition and sensor process so easy let me let me recall prove once again it's just if x if x is the gradient of f rich it means that the ice component the ice component of X is the i th component of a gradient of that is the eyes partial of so that means we need to verify so if f at this so that if it exists then then we have DJ and i now D I S and then I can OK I should actually now that a right of the proof I remember that hypotheses I should actually require that X is a C 2 vector fields since I want to apply the lemma of products of to say that this is the same as DID J. F what this the i X that this was a huge 6 OK so this is true of the lemma of the shots lemma holds for c 2 functions I I guess I should better write this out although I've formulated this is a theorem that list is like C 2 of you 2 and where you is in our who sorry OK so this clearly is a necessary condition actually we did this in the uh and analysis to course last year and also in physics it's common to write this the notation for this is uh use or rotation and so physics the standard way of writing this out is that the the uh how well let me stick here to the German notation that the color uh vanishes for all for all arguments x in you and what is what is the color What is the rotation and well it depends on dimension where for n equals 2 as you know uh we also covered his last term for rotation of X goes from form is a scalar function namely it's simply the difference of these 2 will be only significant difference the 1 which around is as I keep forgetting this is the 1 this is correct so that like out right Costa x equals uh t 1 x 2 minus the 2 X 1 end for any constraint uh the rotation goes from you'll which is now in uh so this is our tool and now it's in of 3 to and now the target is also vector hassles are free and now or products then after 1 to write out all components it's d 2 extreme minus Steve freely connects to now we have all differences with uh indices uh which do not coincide so and D 1 x 2 minus 2 gates what so OK end of so this is called are in english that's kernel and uh it's actually usually abbreviated call of X or if you physicists then you write the and not uh times X so the investors and because of this is the uh this is the rotation rotation and if you like it's rotation rate of the vector field and and the sum so this also shows you what the rotation and several dimensions should be doesn't come up and physics really that cell so tell me how many components do I need enough for now to write out the same things I 1 to give this condition for all indices i j uh uh as scalar equations how many equations to a need and an for n equals 4 PC then how many choices of nonequal indices are there for n equals 4 show 6 yeah that's correct know so it's it's actually for general and it would be what is it it's like the lower diagonal of the matrix is n times n minus 1 over 2 conditions it and I don't know all the ones you like to call these a not too bad this is what you need a the all k and l and the Hamming by chance for any that's the uh these 2 numbers uh coincide OK so this this um this is the kernel and uh and um it's up to you if you want the index notation or you can actually also view it as as the symmetry of the Jacobian matrix and then this these this means that the lower that sorry that the lower diagonal of the matrix coincides with the upper diagonal and there's no uh Abbott triangular matrix and there's no condition on the diagonal here so these these are the same that's exactly the rotation condition the OK then just gives a simple example which I'm sure you know uh which is uh if I have 3 standard rotation vector fields in our tool uh which is minus Y X so what is this near 0 0 head on the X. AIX's it points up or down and the y axis same thing to the left to and here I like this so it actually it's looks like a rotation and in fact if you compute uh efforts commutator D 1 X minus the 1 x 2 let's just do it so I D 1 so the kernel x is the 1 of x minus the 2 of the uh um of minus y which is the 1st component so this is this is 1 minus minus 1 equals 2 non-zero is expected OK so this for instance is a field uh with uh nonzero rotation and adverse
thanks a lot and it has a rotation grade at any point not only at the origin In so at any given point here you feel like it's more ring uh it's moving more to your right then to left OK the so there's so far thank you sir about about this theorem which I uh stated as if you were in last time changed and now let me come to the line integral of the definition of line integrals i which actually current and at the very end of analysis tool but uh as I was told from students for the exam I didn't expect anything to uh to turn up in the exam so I guess you don't expect anything of that to turn up here in class so let's recall that the material which is necessary here so our basic question now is um the in a gaseous state that the fall of all basic question is not so we have a necessary condition for vector future every potential is it sufficient years so this purpose that should go and find him the for the last year that cited out in between I'm sorry that yeah problem for us what is it a condition that the Jacobian itself to join is symmetric is that necessary and necessary but necessary for stop to have a solution so and this will take quite a while in fact uh it will depend on you know that's the problem here surprisingly is not we don't need another condition on differentials but it has to do something with the domain but in order to answer this question we need to integrate out the solution this is uh this is so what is stated above last but didn't write out in the 2nd line we also want to find a potential by integration and this is why we need a line integral to integrate out of ICT fuse and here's a question it is sufficient yes thanks show why did erectus I should thank you OK so it's necessary civil question is is it sufficient if the air if ever vector field with rotation 0 always which is itself a joint uh can we integrated out in order to get a potential in general because the answer will be no and that may it but for good domains it will be years and we need to discuss all that and it will take uh a few sessions to do that OK so uh let me since we want to integrate out all of ICT to few X 2 potential I need to um discuss line integrals and to this in order to discuss line integrals let me start with a curve so this is all the reminder um to acres of and because of is a map from an interval curve C is and is a map from a compact interval I hate to say some uh uh subset of Rn new Mac or curve and uh this is a map and I should like sorry there were is a map see in the where uh um I is a context interval and that would be my standing assumption for the entire section so an instance led out his company to it which I need in order to uh for the intervals to come out finally and omega R. N. can and segment is the OK when I mean as you saw from example we need a close curse as bad uh um play a particular role cell that's why they're knowledge and see if it is closed and if 1 endpoint equals the initial Poland any equals C of these and also then we say c is a loop in that case or seeing is that loop or Ramesh last who came to and never definition and then and continuous curves are not good enough for us in order to integrate ever in the line integral as you know the ad the attention vector comes up so um on the other hand and we can do with a little less than differential and that is piecewise differential that was that also and came up at the end of analysis to solve seat in a continuous curve for whom accompanied interval true um while many at this the piece wise we is piecewise differential below uh SHT advisor disfluency about and and sometimes I write his knowledge and when I'm lazy I want to ride a PC piecewise differential from to all major as annotation while in which case while if the curve has subsections where is a C 1 curve when I want to call it piece wise as C 1 you so if a is there exists a partition the partition of I um crowded denoted a equals say 18 or want less than t 1 less and less and G. N. equals the saturated see restricted to 1 of these intervals uh is a it is um differential all such see element of C 1 for each interval say the chain to the team J that's 1 there are only the football uh um what is the lowest index is
at the at the at points which are contained in 2 such intervals like here here I have to search tangent vectors at L all interior uh petition points OK and everything like integration is is nice in this class and I can simply sum up uh intergrowths of for these sub intervals and another notation is of the trace traces of the is what we actually see here trace of seed is C of section the set C of i and write it out OK that is a subset of omega the what OK any whenever I don't comment on it my extending assumption will be that a curve C is piecewise differential you some general standing assumption the assumption is that C is growth curves curves of piecewise differential piece wise differential you were it into it and and and at the the you what their experiences that they have no some of you are I great OK so I need some the um comments um OK so um the 1 common is that what's good about C 1 piecewise C 1 the about piecewise C 1 well the good thing is if you have if you have 2 curves where the end point of the 1st 1 is the initial point of the 2nd 1 man uh is an obvious way to cont catenate these 2 curves but defensibility will in general be lost at the point where you concatenate however in the piecewise C 1 category this is no problem so we can actually concatenated codes can calling continent in denying uh OR-join curves see such coders and so that at some point I will need the notation which um for which I used to sum all cases little little um text in real time to write this out but let me do this notation is so it's not really definition it's this notation yeah let C. on from say a i b i 2 are in what will make this is inconsistent better should be omega hadn't look let's write out the tool kernels so i equals 1 to and then the I define L 2 curves with end point of the 1st 4 but to right it's with the uh see 1 of the 1 equals C tool of a 2 endpoint of C 1 is initial point of uh C 2 and then among man right uh I use some notation and C 1 + C 2 is a curve and now I need to concatenate also the domains of definition or do something about them so it is this way is to define it on the sum of that's the 1st interval and now I and the length of the segment the the 2 major there t goes to 0 t goes to 0 Ivars C 1 of T if I'm less than B 1 less or equal than B 1 of the Clinton a 1 b 1 and now for to set it equal to be tool be tool I call it C 2 uh then I need to edge to edge to my interval of C 2 the appropriate amount namely uh this here so that will be to be in the the 1 that uh so that's the initial point and the end point is what's written out here being 1 + P 2 minus P the a 2 and if you check and that it then you figure that had the 1 where you go from many of which is the 1st point of the concatenated curve which goes on the same curve at P 1 this is the tool uh uh no it's I should should receive a D 1 class B 2 minus a 2 year if the plant is in here then I see this is exactly this is exactly the 1st point that uh should be the 1st point of C 2 I don't see but I really don't like these things OK so I think this formula is correct and it's obvious what it does OK and also um OK so let um before a comment on this let me also say that for any c for any NEC even continuous uh I let um maybe to Rn omega I let's and I called the inverse curves the pendulous curve simply the curve matter run backwards and so that is C minor perhaps of go and see my right it was an minors from AB who a major is given by running backwards namely minus of t equals see also the miners 2 minus a OK so if you plot in see is a minus a 0 this will be a b and if you plot and B. This is a B minus B so this goes away and uh it starts today so the suspect running backwards from glucose OK and um so this much about rotation invariant obviously clearly clearly um the concatenation of C 1 curve of piecewise C 1 curve stays piece wise and C 1 so so CI I in P C 1 and then see 1 class who also
in P C 1 2 C by taking the petition appropriate petition OK this so and also for K and this is 1 thing and also if c is in piecewise C 1 then C is the sum of finitely many different mix of different uh of finitely many as C 1 curves c j equals C 1 right since if if c is piecewise C 1 I have this um partition of the defining interval from a to B and then I call this C 1 V C 2 and so forth this also offers OK so let me know and let me know come to think of integral so let me say that I will always denote uh viscous standard scalar product In this notation Standard scalar product on other end so 2 vectors x and y have a scalar product sum of x i y times y long and then as you know I can ride out the coerced into her how to finish oop uh um laws that in B a uh continuous rectitude new tools Our and uh vector the the so be a vector field dust mathematically not mean any more than a is n-dimensional subset of Rn and uh if I have seen a state C 1 curves and from I could do you OK If reduce domain and then you could have and then some of them can because of the of the line integral then in the the integral x the S as last term over C is given by the integral from a wealth of ideas the I wish he won a to be here from a to B to you so uh the and ability to integrate from a to B the scalar product of x along because of and start with the tangent vector c prime of T and integrate of DT and uh high-quality the line integral in this course line integral so there's lots of other names for this guy tend to avoid I realize it's better to avoid the name path integral since of physics uh it also has a specific meaning and sense of part finer profitable so let me rather line integral uh the kindergarten what also lenient into life in German how can I is the line integral also x overseas OK and it's uh um running out of space is that in case as the piecewise C 1 curve so I have to take the sum the the and yeah yeah in the the the the OK let's try this out of this In cared that OK if the In the piecewise cases were then arrived C equals while I don't want you to some sigh perhaps here we have seek their cj is in C 1 and then I said and this for a while and I went and said and set the integral of xt s over is piecewise C 1 curve is the sum of the intervals of from i a i j equals 1 to K x of the shape of the uh CGA prime the dt and on my indices correct so I have and easier this correct OK yeah video that's the obvious definition so uh it doesn't matter if uh as long as I have also finite sum here as long as my petition is finally many singular points that the tangent vector uh is does not exist so that it is ambiguous and then I can write of the sum and there's no problem the OK and for sure I mean what this does let me also remind you what this does for a given vector feud x is uh it integrates out the component of this vector fields in the direction in the direction of the of the tangent right so here I have almost the full component of eggs and here for instance it may be it may be a better predictor love then I don't have any contribution so what I do here with my line is real is I integrate the contribution of X in the direction of the prime question if an intro psych missing yes what do I do anything else sorry the idea that was why it became some noisy here OK next time you you do this more critically it's to intervene OK the OK so we integrate out the component of X in the direction of the prime right that's the intuition here and of course this is a this is very common in physics for work integral and so forth maybe uh note a consequence if I have seen equals C 1 plus the tool and I have the definition of my piecewise uh piecewise uh and differential uh like this then I can say that the integral of editors solve an integral of C over XT as is equal to C 1 6 years plus C to fix this yeah this is Of course this annotation here is hidden physics notation with a standard scalar product here you I I guess you realize what we do mn and the mathematics as well as physics we want to be a downward compatible annotation and so this is what how
this comes about OK the so the standard example would be so a standard examples this will work interval in physics the on work integral W equal to minus interval of force this talk will over the past the and if you walk in a hilly area you know this corresponds exactly to the height difference here so this can integrate out so if in a uh and in the good field and this physicist you know what a good field is it's called a conservative field for good fields uh this will be interoperable and you get for instance and if this is in it is they what correspond to a mass and then this would be high OK so that's also an intuition for mathematicians and let me also give you some trivial examples of an example for instance going back to my vector field uh OK it's gone out to my rotation vector field xt equal x off IX Y equals minus Y X you if I integrate so remember that's this field it looks like that if care to sufficient and if I integrate say from on 1 hand from 1 to minus 1 here uh it's obvious so integral takes the over C where's the of t equals say 100 of this it's it's uh warned of minus T 0 for t and similar to mind members starts at 1 and ends up with minus 1 so my say this a C 1 here cell cycle like you OK and then the what should I get well hopefully hopefully I should get 0 1 seconds c prime c prime time points horizontal to the left and x points perpendicularly up or down so I guess uh so what I should get is um the well uh harder to this so I need to plug in this part right so I get it 0 and I don't care what they did here times uh times the derivatives which is minus 1 uh 0 and so indeed this is 0 and on the other hand is I a integrate over and sorry if I integrate over a semicircle say like see tool where true see true of t equals cosine t sign t say and T. ranges from 0 to time the what man I should get something since now a vector field is actually tangential and actually coincides with this unit vector C to so let's compute this so here I need x of x of a of cosine T sign t which is minus cosine t minus sign to a plus sign the cosine so minus don't do this correctly it's minus uh minus y so minus sign t cosine t OK so this is X this is uh this is x and CEO of T C 2 of T and as the prime of t is the derivative is minus sign t cosine T also computationally we see that these are the same unit vectors so um no annotation is for this should be the integral xt s oversee tool is interval from 0 to time and here I should like here to tools cake that can know everything is standard in any case this is the scalar product is 1 so this gives me pi meaning that the path integrals for to pass between the same endpoints will in general depend on vectors he no surprise the yeah so past integral the line integral sorry line integral how all depends on an Thomas and not an end points alone when yeah which is in very problem we're dealing dealing with it it if I integrate APP vector field there man uh then uh of I get different values depending on past this is no surprise since I told you but a necessary condition uh the necessary condition uh we have what is the um symmetry of the Jacobian of X and as I told you before this is a this is a vector field on which it has a non-symmetric Jacobian this is the rotation field if you like so here this is sort of uh out of our scope anyway this is not the kind of vector field we really want to consider OK so practice is a good moment to stop uh and come to to should to break instead of these things so the I apologize for this which was incorrect and hopefully no is correct namely a namely here every argument should be t plus a tool minus B 1 which is when I implied in be 1 I should be and a tool and in fact I am B. 1 minus P 1 and if the plaque in this whole thing I should be dead be tool which helps it's easier for you to see them for me this Council's uh this states and this Council's years now it's correct yeah OK sorry the actual foreigners really not important this but these should be correct nevertheless that OK so what should we um which should be said about the line integral while the main property it has a floor depends on the path chosen between 2 end points it does not depend on the parametrization of the past it's parameterization independent in cell line interval integral part of it is parameterized parametrization nation the independent was a little care at least indeed 10 then so in the following sense the the proposition to it um so let so I say from alpha beta the to make the S P E A C 1 addition while is in a way so that's just fancy notation for a unified being differential phi inverse being differential and uh uh what fighter differential will fight in because is exists and phi inverses
differential and in particular um in particular such a map has either a positive derivatives throughout or a negative derivatives throughout it and so then I stayed there the line integral and then what she but let's let's keep it then and the line integral over although the air re parameterized curves of the vector field X is the same as the line into the original line into no and 2nd yeah which I didn't mind out x is in C 0 vector field this should go some there uh and so can and segments um while there and this is this is not complete if I actually it only holds if 5 provided 5 prime is positive so by British should line if 5 prime is positive then so in yeah or you right provided 5 prime is positive and if I prime is negative then I get a sign change of interval interval and display perhaps to write it out is to say that the interest curve has a line integral which differs by sign from the original line into the like so the the OK and let me the truth for this that just by coagulation so source of all amino to that and but as I told you that influence the wonderful morphism 5 prime it always has 1 sign so this is what I did this is less well um so this means really uh it's a no tentative I've its 5 prime positive or negative throughout OK so let's let's just um deal with both cases and write on X xt S over of the re parameterized curves and again this is the calculation I did last term but I guess it's good to see that so what is this well see after is defined on L so they turn and I have to integrate x the overall at 4 . si after sigh and perhaps I write it this way and avoid the also t and I write C after 5 prime and this is sort of t of t heightened wanted to put it in and now what is a what does the chain will say well this is the prime after 5 times the 5 prime so this is equal to no change here but now I write OTA-C triumph after sigh times times 5 prime and since the scalar product so it's 1st this here D but since the scalar product is linear I can actually move this right this is at each point is 5 prime of t is a scalar so it goes out of the uh scalar product is like OK and now I'm in good shape for substitution reduced uh is all this off to the side and here are the uh you have the 5 orders so this is the same as uh writing out and perhaps make a case distinction ouseful 5 prime larger than 0 we do need most days should be OK interval also OK now I have sigh of L so sigh of beta In case 5 prime is positive then this is a monotone increasing function sole file for us as a 5 is B and K and I do what I do is substitution yeah this was channeled through a strong capital ones general and now I do substitution so now I simply have X after C thanks after c n c prime itself and uh whatever he tall audience or something and so this is exactly so no I integrate from a this is a from a to B the curve C so this is space this is exactly the interval of makes the US policy and and in case 5 prime is negative for all t of the only difference is that the and in this case phi is monotone decreasing so file the lower end point is the upper end point here and vise versa so I get to same thing if have alpha 5 beta but now this is and this is a and this is b yeah its 5 pages a 5 offers B in the orientation reversing case and what is here stays that's just but like this same thing and so on if I want to write it out in standard form from a to B minutes gets me a minus sign over the x dx OK so the particular case is actually that I reverse orientation uh and go to from C to C minus with uh the parameterization wrote on the board uh the 1st the in the 1st hour and then um then 5 prime end and then you get directly that this is minus uh than than the left-hand side here I should go here or here see after 5 where phi just reverses the interval that can be seen my nose in this lower case and so I get this statement here so this is the entire proof so what's what's the what's the whole substance of approved it is that the path integral is defined in a way that the chain rule that gives me an extra factor which can go out Red and that makes a path dependent so if you like the entire reason for defining uh for defining via a line integral like this is that you need to this derivative here in order to be part of to be parametrization independent if you go with double speed uh for the same curve you don't want to double because interval but once the same coefficient of this of this makes good for all came cell now I have a I could continue right away but let me go on a little digression so there's no serious theorem today I apologize but he is a digression on 1 forms and actually um this is not absolutely necessary for and a photo complex analysis so that's why a quality that aggression here a moment necessary for complex analysis how does I do it since I believe it's good for you and it's both for you good for you in both cases if you're physicists and if few mathematicians all so it just gives you some extra inside so when I discussed the gradient last term um it's to what we the 1 any
mountains of hits home care so this is from the course notes last term and this is not what is a big enough issue should I and make collapse it next their set of slides and the larger the no protest so must be very good OK and so what they did do when they're what do we do when we discussed the gradient well the but the point I wanted to um pass over to you is that the gradient is um comes uh is to be it comes from an iso morphism of a vector space and its dual vector space right to so uh uh let me remind that we recall what the dual vector space is the shift vector species space V which in fact could also be
infinitely them into have infinite dimensions then the dual vector space is the space of
linear functionals side of vector space to our linear OK I but now this this is an easy way to get from a vector
space to the space of linear functionals which is to say uh to look at the scalar product and with a fixed vector so foot each lecture you fix a vector v you fix a direction and you always take the scalar product with this fixed vector the that gives you a linear functional obviously it's linear and so this is a good way to go from a vector space to the dual space and fact looking at coordinate representation it's not hard to show just look at the the based run for bases is not hard to show that you can define a vector for any given the um dual vector for any linear functional just do it on basis vectors to define the ice component like in the ice spaces like to define the i i component of the and then you get that for each but this section isomorphous and it's not only that for each vector you get a their functional but also for each functional unit of vector is matter it's a linear isomorphous vector spaces know when to vector spaces are isomorphic you have 2 choices either you say these are the same or use a huge keep on telling its they are different so I take the 2nd point of view is a different vector space that are isomorphic but it's a different vector space now remember the example for an example that the differential dfs at a given point x is actually a functional in case if a scalar-valued right and if you apply this as morphism and then you get a vector of the pH of the differential and that's the gradient and my whole point was that you can see em what kind of victory you are you have by looking it's a notation as vectors are column vectors linear functionals of row so the D S of scalar-valued function is represented by the role namely the Jacobian has 1 row the the gradient however is a column so you take your 1 the role of the of the Jacobian and and rotate its to become the gradient this only depends on the scalar product now and for and for the line integral uh we have woops for line
integrals this is exactly as it goods to
correct setting I would say the the that so let me talk about visas 1 forms and some of the earliest slide it on the board ends the instances of the boy self you are man is the space I D a little sloppy here is the space of vectors uh an hour I write them out as columns and then I have power start this which is the space of linear functionals of so the space of linear functions over space of rows and row vectors if you could Lanitis coordinate take at n this can also be written as the maps which associated uh the from the vector space and to this scalar product the dotted with uh sorry it's not not to their this is wrong notation I 1 I want to um he should be the dot eventually for this cell
the space of linear functionals that v is indirect yet way to parameterize all these linear functionals is take its expected take the scalar product with its expected OK and an example here is the the the differential at a given point for a scalar-valued OK the so this is a short summary of what from the slide and now what we need which is something which you can actually see also but we also have a point x here in question so the f is a linear functional for each x now I want to take along the x and that's uh what makes a 1 form so loops Berlekamp seem on the end but that need write in between here so so this is sort of the point and fury and now uh its I have point dependent independent their vector spaces each vector spaces him and then and for instance I would call about not I would not uh no I would not take vectors but vector fields Net vector fields which are maps say x from you to our and so I have it he for each point in you I have a vector space and R. N. and it depends if I'm given the vector field it depends uh this vector depends continuously on the parameter of point x and you similarly cell I have a cold vector fields collective fields which I will denote by omega and may go to our and stuff and this is what I want to decide now here defined as follows world all in the summer one-forms loops er so the definition and the to nation in yeah a once-for-all all and you can also call it a co vector field coal vector field or sometimes it's called obsession um on this set the you in Rn on your and as a matter of in semantic all major from you tool are in style that gives me for each X that gives me a linear functional like the jacobian gives you for each x D F a D is a that years at X is a linear functional groups home care have sources P is mapped tool and so I carry the point dependency and by an index that's the usual entation and here this slot is for the vector which uh since it's a linear functional I can plug in a vector it uh into only get paid omega p Lu and a all OK and now this 1 we crime and I want to use a continuous for candidate where omega this is called tinyos a and is in a simple way to define this is to say that uh foot for each C 0 vector field negative Fiat eggs which I can block on you that when you I can plug it in here and I get a real valued function and this real valued function should be continuous speech the function this function p maps tool omega P of X of p is continuous the and 2 standard example the standard example would be um the differential here is test is from you 2 are then D. F it is a wonderful it's the co vector field it's a linear functional each um at each point but also the main OK so why is this interesting um tell me let me say 1 would before also trips his girls near her this isomorphism which I denoted as a with this by Ihler product
can be stated here as well right so let me just say a never example is if I have a vector field is the from Europe to our the is a vector field vector fields then then the scalar product G. uh 0 error the S P with uh something he finds the 1 format but to use area 1 yeah so same same I can plug in India vector field and that means if I plug in if I take sang the 2nd vector field then it would give me a real numbers so at each point p this gives me a linear function of OK so the same and in fact this is also there is an isomorphism in and Frank tool is then I from office in appropriately defined I haven't said I haven't told you what an iso morphism of spectacle diffuse really is but uh so all so let me no bimodal careful here OK so now I they become to the question why is this interesting well the various places the mathematics they need this but um but she also need this in physics uh and uh this is 1 of reasons why do this year it shall have you have you attended quantum theory quantum mechanics yet no OK then this is something for the future which I will tell you here right now we have that so in in quantum mechanics quantum the the mechanics OK the scope of I covered is is not sufficient actually you need to deal with infinite dimensional um vector spaces here cell can be a vector space vector spaces infinitely dimensional the OK don't worry about that uh um and it's not a problem here and for what I am telling you but it is a strange way to transit uh between the dual space and the vector space this laughing at Her who did this and I'm not sure have the following way of uh and so of dealing with in fact I should say here there are so-called Hilbert spaces so they are but usually they're very good so they uh those the in the vector space of the star is always represented represented in this form here as a vector field by v and V using this isomorphism yeah using the isomorphism None of the scalar product at don't no star and so the OK but nevertheless physicists distinguish vectors and call and have this notation that they call should they do it like this and OK the that if you look at this map here the vector W is associate to what they write the scalar uh product in this is strange form with uh with a bomb the times W this is just the scalar product no skip forward and then physicists call this a bra Victoria and this they can't victory and it's unclear what the difference is in fact what the difference is and so let me tell you this uh what I hope I can do something good for you but you will see yet the Braddock Tracy stands for a dual vector and the dual weight vector is represented by the scalar product yeah so you can represent as the dual vector by a vector of mainly by writing if as I did yeah take a fixed vector here and take a scalar with a scalar product with it however In fact really it is a you rector here so here this is just to get over the distinction off dual space and vector space which I consider important but physicists usually uh slide on and of rock M. this here stands for the dual vector about the dual operator is represented and by a scalar vector with fixed vector them non so here here everyday use vector dual vector is represented die vector yeah and this is the dual vector and this is why there is still a physicist still make a distinction and still save as 1 of the left and 1 of right here this is a scalar product you might think it's symmetric but in this form it's not symmetric and this comes from the fact that it is in fact when when you will see this in physics and quantum mechanics and please remember that 1 of the sides uh denotes a dual vector and the other 1 is the true vector this is the true vector and this is the d stands for dual vector OK so this actually comes out in and in physics OK but no it's not necessary to make the distinction professors don't make the yeah you can get along without but there's something you miss and the something can be that the scalar product and is there while there is no natural scalar product for instance yeah it the so that is the interesting question in physics then if you come across this uh where does this scalar product come from so let me go on trips I erase this here and and toughest list them it this can on all the proof cannot go on so like it's so have the OK so this is about them the physics which you haven't studied up to now but in a minute I will also tell you something about classical high school physics um the so would point why do I why do i point is out now i . results since curve line integrals naturally can be defined for 1 forms use so for 1 forms there is a natural line integral along some and for what this is is not that much to do it just take a couple of see 1 node piecewise C 1 why not well I I don't care right now so a b to you and take um see curve and take go let major but from the usual the other 2 are in stock to the dual space be the one-form the the 1 false will carry what can I do that and then our d Find the Indigo oversee of all major I don't know sometimes was something it is simply the integral from a to B of all major also uh at 4 point z of t also c prime of t the who OK so yeah this is very natural thing to do am this is all made of this takes vectors at each point and the vector i want to plug in a c prime why do I want to plug in the prime well by Vista very calculation here let me just do the essence of this calculation what is the essence it's that channel and substitution correspond nicely for of this interval to come out of the same it's so then it the integral of omega is also is also parameterization invariant yeah and 100 CVS well just let's look at omega of C after firing of CDI after 5 try and the and note that this is what causes same thing by the chain rule the same thing as before all of this is the prime after 5 times 5 prime chain move now only is a linear function of just as the scalar product was linear function and remember very plan put this out the so this the since all omega is linear at each point I can write this out in CD after finding c prime of defined times 5 prime so no it's multiplication in our near there there this takes values as a linear functional I can make multiplied with a linear was a real number at each point OK and now just write integral signs in front of and and you substitution and figure that or make out over overseas after sorry is the same as only go off to see just by substitution this interval here it is well if you're right of integral in front of here by substitution it's the same as the interval although only got a efficacy no this may seem I mean if I have 1 version of it while I was a scalar product why do I need another 1 well and there is also some significance to physics what these 2 things here I don't need a scalar product him for mode no scalar product involved product involved so as the mathematicians I like I like to a uh it to figure out the least amount of structure I need for defining something yeah when designing the path integral uh with the scalar product I need I need a strange structure namely the scalar product yeah I don't need and I get out I get along without it In a physics it's not always clear what a given scalar product is right it's not natural I mean all world is not given for natural scalar product I would say I what what is the scalar product here no no idea here but what is the time derivative of the what's attend vector of a curve to curve that's a natural object the so my claim is and whenever whenever can there is a line integral in physics In physics all in fact in fact and the the quantity to be integrated to be integrated is a wonderful is anyone from when k so what's the standard uh what's the standard line integral in physics well it's work interval mobile force here so here for instance is a bloody arrears gravitation acting on that yeah so you feel here's the force vector pointing down well that's Bush's right there is no force vector In fact operators is is you can test you can test here pass into a line integrals yet what you can do is you can test in DX and DY d z whatever it's are you can test what the gain in work is yet so if I do this begin work good so if I do this OK it's neutral here so actually what I do is when I have for us and is I evaluate such an interval here near in the direction of various the primes the the natural choice for the prime is take a bases where you take the XTY disease or something yeah so In fact you can load you you believe forces awake vector but in fact physical vector yeah example Example floors what is a yeah and all in all want you know is only no line integrals you know the gain in work line into yeah if you want to make the distinction or not is another issue yeah perhaps to get along with the leaving it's 4 it's a vector of an article vector that's fine yeah but but in fact in but there are situations where you may have to make the distinction and at the latest this comes to you when you uh look at tensor fields in physics and then there's different ways of uh of recorded transformations mn you need to make the distinction is why is it important mathematics um what in mathematics all I want to say is whatever follows in this section on line integrals is more appropriately be used as stated and full when the writers out that's just between here the so this is the last statement I want to make perhaps I can so that in here yeah mathematics now yeah the entire remainder the remainder all of this section the section works for 1 forms in place of vector fields a place where ah sorry of vector fields the yet in effect it works better I would say and if you want to convince yourself that this is all then inference and look at the book by critics barrier which is not an advanced text analysis to he does that in this setting I'm a sort of afraid to uh that uh you will get lost if I do all these complications so I will stick to the standard path integrals with and the scalar products but uh I hope I got my message through OK thank you for listening to you in a week from now