Yet another cross-product integral

[Johan_S]

TL;DR Summary

I am trying to figure out how to do a more complex cross-product integral and get stuck, and since my books are 1000 km away I turn to here

I am trying to integrate a cross product and I wonder if the following is true. It does not feel like it is true but it would be very nice if it was since otherwise I have a problem with the signs...

This is my first time posting here, so I just pasted in the LaTeX code and hope that it is parsed...

$\int\overline{r} \times \frac{d\overline{p}}{dt} \; d\overline{\phi} = - \int\overline{r} \times \frac{d\overline{\phi}}{dt} \; d\overline{p}$

 

[anuttarasammyak]

What kind of vector $\bar{\phi}$ is ?

I observe two vectors in your integral
$$\int \mathbf{A} d\mathbf{B}$$.
Do you mean inner product
$$\int \mathbf{A} \cdot d\mathbf{B}$$ ?