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On the Visualization of Differential Forms Dan Piping December 1998 Some people are a little nervous of differential forms. They're used to vectors having 3 components but how do you interpret an

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In this lecture we talk about regular differential forms so the space of rational differential forms on the curve C this is generated as Omega subscript C, so this is generated by symbols of the form D F where F lies in the function field of the curve, so this is subject to standard relations which we have studied in differentiation D of F plus G is TF plus DG then you have product rule d FG SMITE GT f and the differential of a constant is zero so this differential form this is written in this form Omega is summation F IDG where F I and Hi they lie in the function field of the curve so let us give some examples to make this more clear so on a 2 you have bases DX and Dy so notice that this a 2 is generated by KM comma Y, so differential form is like this Phi 3 × D of X Y wouldIeyasuu use the product rule here, and then you get 3x squared DX 3 XY DX so notice that this and this both of them lie in K × Y so on a and the basis is DX 1 2 DX sensor again corresponding to a, and we have the falling ring K x1 x2 all the way to X n so on n the basis is DX 1 to DX n so your differential forms looks like this so basically what means the functions lie in ring key n, so this is what is K of an n, so this was K of a 2 so again this is nothing but direct sum like this so this is nothing but f 1 DX 1 plus f 2 DX 2 all the way to f n DX n, so this is an n so what is omega X u, so this is set up regular differential forms on open set of the variety X so say you have a fine variety V which is generated by polynomials g1 to GT, so this is the ideal corresponding to this via ITV so the for the differential form you have this KV DX 1 all the way to K V DX n and in the denominator you take the differentials, so that just means you have a function f 1 here which F 1 lies in coordinate ring KV f 2 DX 2 f 2 lies in coordinate ring KB, and then you have F and D X and F and lies in the coordinate ring KV so notice that if F is 0 on V so that is say F basically is taken care off in OF e F is 0 and V that it means it has been taken care of here in OF e because he when you construct the coordinate ring this F would be automatically 0 than OF is also 0 and DN has taken care of here so f is 0 here and OF would be 0 in what the part you modular out so your differential form, so this looks precisely what we have seen in algebraic geometry you know in algebraic geometry we had this thing we had the space say if it was a 2 you had the space k XY and then you modular out the corresponding idea and that gives you K of V and that is precisely what is coming here and then whatever here you have you take a differential to get this Omega V so this will be more clear in an example so say you have me minus X square and a 2 so your KT of V we already know if it is an 2 this will be K X comma Y and then you modular out our ideal which is why minus X square and the differential would be KV d OK v Dy and you take its differential which is Dy minus 2x DX which comes right here so again this KV...

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