# [Mathematica Help] An integration involving many vector variables

1. Jan 7, 2010

### 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?

2. Jan 9, 2010

### 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?