We begin by choosing any vector function $\textbf{a}:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$ and then choose an antisymmetric function $\textbf{b}:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$ (antisymmetric meaning that $\textbf{b}(\textbf{x},\textbf{y})=-\textbf{b}(\textbf{y},\textbf{x})$).
I need to evaluate the following integral
$\int_{\mathbb{R}^n}(\textbf{a}(\textbf{x},\textbf{y})+\textbf{a}(\textbf{y},\textbf{x}))\cdot\textbf{b}\mathop{d\textbf{y}}$
This is the general problem, and I am just trying to figure out how to do some examples with this framework. I have picked a few examples, but they all seem to go to infinite or 0, but I should be getting out a function of $\textbf{x}$.
Here is an example that I (made up) tried: $\textbf{a}:\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^4$ and $\textbf{b}:\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^4$ as well. Take $\textbf{a}=\big<{\textbf{x},3\textbf{y}}\big>$ and $\textbf{b}=\big<{\textbf{x}-\textbf{y},\textbf{y}-\textbf{x}}\big>$. Plugging this into the integral we wish to evaluate and then doing the dot product we get
$2\int_{\mathbb{R}^2}\textbf{y}\cdot\textbf{y}-\textbf{x}\cdot\textbf{x}\mathop{d\textbf{y}}$
Does this not just diverge? Even if it didn't I wouldn't be sure of how to evaluate it. Do you just split it up into its components and take the integral $dy_1dy_2$? Apparently in the end I should get a function mapping $\mathbb{R}^2 \to \mathbb{R}$, but I don't see how that is going to happen.