# Simplifying a Vector Integral

While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ and $x_i , x_l$ are vectors. The first term in the simplification comes from the fact that $x_i$ is not equal to $x_l$, and we are differentiating $P_N$ wrt $x_i$ (over which we do not integrate) ; so the differentiation can be taken out of the integral. However the domain of integration has boundaries $|x_i-x_l|= \sigma$ which depend upon $x_i$, so there is some second term. This is the image:

My question is: where does the second term come from? This book is mathematically very involved (for e.g. it uses at some places, volume and surface area of $n>3$ dimensional spheres) and its seems that I need to have some background/prerequisite for reading this book. I'll be very grateful if someone can suggest a maths book as prerequisite based on what I told you and which will help me to understand the integration.

• It's vector integration by parts. Jul 6, 2014 at 6:23
• How? the second term is a summation.. Jul 6, 2014 at 6:24