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  1. P

    A Connection between Reciprocal Space and Cotangent Space

    Thanks. Dont worry about the "maths speak". The rigour of your arguments reminds me to try to be more careful in how I phrase my arguments. Yes I enjoyed the theory of complex numbers when I was at University so I understand what you say.
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    A Connection between Reciprocal Space and Cotangent Space

    Try this link; http://bilimneguzellan.net/fuyye-serisi/
  3. P

    A Connection between Reciprocal Space and Cotangent Space

    Or maybe its because the exponential may be written as a sum of sines and cosines
  4. P

    A Connection between Reciprocal Space and Cotangent Space

    Is this anything to do with the Argand diagram representation of a complex number ?
  5. P

    A Connection between Reciprocal Space and Cotangent Space

    Please explain the "is looked at a map to the circle " ?
  6. P

    A Connection between Reciprocal Space and Cotangent Space

    OK tried reading it again and its WAY over my head !
  7. P

    A Connection between Reciprocal Space and Cotangent Space

    Thanks. I am aware of this article and have read it but as you correctly say it is a bit abstract for me to understand. Perhaps I should tr to understand as you say it discusses general duality and fourier transforms. Perhaps if I understood it I may have a clue !
  8. P

    A Connection between Reciprocal Space and Cotangent Space

    Yes I am familiar with the concepts of the Fourier Transform used here. Perhaps I am mistaken and there is no connection andI should treat the Real and Reciprocal spaces as seperate from the tangent and cotangent spaces.
  9. P

    A Connection between Reciprocal Space and Cotangent Space

    Talking about "connections" - I understand a connection in the Framework of General Relativity where the Christoffel Symbols are the "connection" relating a vector defined at one point in a smooth manifold to its image defined at a second point in the manifold. Hope I got that right at least. So...
  10. P

    A Connection between Reciprocal Space and Cotangent Space

    OK ! I understand that COMPLETELY ! Also, as you call yourself a "physics novice" I would call myself a "mathematics novice". I did a 4-year physics degree with 3 years applied mathematics and 2 years mathematics. However that was some 30 years ago. Today I dabble in physics for the fun of it...
  11. P

    A Connection between Reciprocal Space and Cotangent Space

    I would say that my intention is to work with a smooth manifold on which an inner product is defined. Also a metric but I guess that to define an inner product we must have a metric defined (or am I missing it again?).
  12. P

    A Connection between Reciprocal Space and Cotangent Space

    Sorry - that`s the "amateur physicist" in me trying to talk to the mathematician in you. I have a degree in Physics (1985) but dont work in an academic environment so my language could be very rusty. my language is certainly not as rigorous and precise as yours. I am actually a radiation...
  13. P

    A Connection between Reciprocal Space and Cotangent Space

    OK I will find a definition and send it your way. Thank you.
  14. P

    A Connection between Reciprocal Space and Cotangent Space

    Thank you Mathwonk for your explanation. I should have mentioned that an inner product was defined. I will need some time to study your response. It clarifies many things for me. I guess when you say ""maps in" , or vectors, are dual, or reciprocal, to "maps out" " you are speaking in the...
  15. P

    A Connection between Reciprocal Space and Cotangent Space

    In trying to answer my own question, and bearing in mind that I am no expert, We recognise that for each vector (in the tangent space) there is an associated 1-Form (in the cotangent space). One possible 1-Form is the Gradient function (I hope). Now Gradient means "rate of change over a unit...
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    A Connection between Reciprocal Space and Cotangent Space

    Good day Mathwonk. Thank you for your comment. You are correct that the term "Reciprocal Space " is not a common term in differential geometry. I am playing with ideas and concepts here. When you do this you may often try to match two things which seem very different. If we accept that the...
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    A Connection between Reciprocal Space and Cotangent Space

    [Moderator's note: Link to reference protected by copyright removed.]
  18. P

    A Connection between 1-Forms and Fourier Transform

    Thank you for this. This is clear now.
  19. P

    A Connection between 1-Forms and Fourier Transform

    OK. I may be answering my own question. We have vectors and we have 1-forms. They are in no way Fourier Transforms of one another. However the Fourier Transform, together with the Inverse Fourier Transform defines a relation between them...
  20. P

    A Connection between 1-Forms and Fourier Transform

    Hi All. I hope this question makes sense. In the case of Fourier Transforms one has the complex exponentials exp(2..π i. ξ.x) In 3-D, if we single out where the complex exponentials are equal to 1 (zero phase), which is when ξ.x is an integer, a given ( ξ1,ξ2,ξ3).defines a family ξ.x= integer...
  21. P

    I "Adding" a Vector Space and its Dual

    Fresh_42. I think you understand this whole thing way better than I do and probably my questions are not making sense to you. The problem is to convince me of my ignorance. You have changed my perspective on this and I need to think deeply on my new perspective. Its late here so I am retiring...
  22. P

    I "Adding" a Vector Space and its Dual

    OK perhaps, to explain, If instead of involving V with V*. If we had two vector spaces V and W being subspaces of R4 we might expect V to have dim 2 and perhaps W to have dim 2. Then, if I am correct, the Direct Sum V and W would give a vector space of dim 4. So if we could have defined a metric...
  23. P

    I "Adding" a Vector Space and its Dual

    OK then if V has 3 dim and so does V* does it make sense to define a metric tensor for the Direct Sum Space and what would be its Dimension ?
  24. P

    I "Adding" a Vector Space and its Dual

    OK Fresh_42, you have REALLY got me thinking now about my original question which may well have been misguided by my ignorance. You said earlier; Yes, but then you have V⊗V∗≅L(V,V)≅M(n,R)V⊗V∗≅L(V,V)≅M(n,R) and dimension n2n2. Yes this has dimension nxn rather than 2n. I need to rethink my...
  25. P

    I "Adding" a Vector Space and its Dual

    So can I form the following then, in any way, V = (v1,v2,v3) and V* = (v1*,v2*,v3*) and the "Direct Sum" or whatever Sum = (v1,v2,v3,v1*,v2*,v3*)
  26. P

    I "Adding" a Vector Space and its Dual

    OK thats correct as you say
  27. P

    I "Adding" a Vector Space and its Dual

    Maybe I dont need to form any kind of Sum. Maybe I just need to recognise that I have a Tensor V(n,n) consisting of the n contravariant components together with the n covectors (or 1-Forms). The these n 1-Forms Map the n contravariant elements to the set of Real numbers ?
  28. P

    I "Adding" a Vector Space and its Dual

    So the 1-Form 0 maps the 0 vector to 0 ?
  29. P

    I "Adding" a Vector Space and its Dual

    I guess I am asking if we can do the following; V = (v1,v2,v3) and V* = (v1*,v2*,v3*) and the Direct Sum = (v1,v2,v3,v1*,v2*,v3*) Thanks.
  30. P

    I "Adding" a Vector Space and its Dual

    Thank you very much for your reply. The second option (factoring out) seems a bit complicated to me. The first option, I think, at least as you have written it, seems to be the answer. I say this because you say the Direct Sum will produce a Space of Dim 2n from 2 spaces each of dim n. I...
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