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[Mathematica Help] An integration involving many vector variables

  1. Jan 7, 2010 #1
    I have an expression which has six terms. I am posting one of the terms, in its basic form:

    [tex]
    f1= \int \frac{d\vec{k}}{( \xi [\vec{k}] + i d - \xi [\vec{k}-\vec{b}] ) ( \xi [\vec{k}] + i c + i d - \xi [\vec{k}-\vec{a} - \vec{b}]) }
    [/tex]

    Then there are f2, f3,f4..f6. They are all complex conjugates.

    The six terms together constitute the expression [tex] F(\vec{a},\vec{b},c,d,\vec{k})[/tex]. I am to integrate and hence evaluate and visualize the function [tex] F(\vec{a},\vec{b},c,d,\vec{k})[/tex] in Mathematica. I was able to do the one-dimensional case correctly, where all the variables can be treated as scalars. However I am having trouble doing the 2-D case.

    In fact, I am a little unsure about how the math works when vectors are involved, and also how to make mathematica evaluate this integration for me. Here is what I tried.

    I declared
    k={kx,ky} ... a={ax,ay} ... etc

    The function [tex] F(\vec{a},\vec{b},c,d,\vec{k})[/tex] now becomes a list as:

    [tex] F(\vec{a},\vec{b},c,d,\vec{k}) = \{ F(ax,bx,c,d,kx) , F(ay,by,c,d,ky) \} [/tex]


    Now if I use: Integrate[F, k]
    I should expect an output of the form {Expr1,Expr2}
    However, I get an error... "Integrate::ilim: Invalid integration variable or limit(s) in {kx,ky}."

    Can somebody explain what I am doing wrong here?

    Even more fundamentally, I somehow doubt if this approach to carrying out the integration is correct. Can someone hint how such integrations involving vector variables and complexes (iotas) are solved?
     
  2. jcsd
  3. Jan 9, 2010 #2

    Dale

    Staff: Mentor

    If I define:
    f[t_] := {t, Sqrt[t + 1], -Exp[t]}
    and then compute
    Integrate[f[t], t]
    I get the expected result
    {t^2/2, 2/3 (1 + t)^(3/2), -E^t}

    So Mathematica can evaluate the integral of vector valued functions. Are you perhaps trying to integrate over a surface or a field where the function is not just vector valued but also takes vector arguments?
     
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