[Mathematica Help] An integration involving many vector variables

[elduderino]

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

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

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

The six terms together constitute the expression $$F(\vec{a},\vec{b},c,d,\vec{k})$$. I am to integrate and hence evaluate and visualize the function $$F(\vec{a},\vec{b},c,d,\vec{k})$$ 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 $$F(\vec{a},\vec{b},c,d,\vec{k})$$ now becomes a list as:

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


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?

 

[Dale]

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?

 

[oswaler]

I would suggest approaching this problem by breaking it down into smaller, more manageable parts. First, it may be helpful to review the mathematical principles behind vector integration and complex conjugates. This will give you a better understanding of the problem and how to approach it.

Next, I would suggest consulting Mathematica's documentation or seeking help from a mathematician or experienced Mathematica user. They may be able to provide specific tips or functions to use for integrating expressions involving vector variables.

Another approach could be to simplify the expression by hand before attempting to integrate it in Mathematica. This may involve using properties of vector operations or simplifying the complex conjugates.

Overall, it is important to have a solid understanding of the problem and the tools available in Mathematica before attempting to integrate such complex expressions. It may also be helpful to break down the problem into smaller steps and test each step to ensure it is working correctly before moving on to the next step.