# Question about integral used in Lattice Boltzmann text

1. Jul 9, 2010

### Monty Hall

I would have asked in math, but I was hoping the context of lattice boltzmann may make my question clearer. Given f is the number density of particles, v velocity, and u equilibrium velocity.

In a book(http://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf equation 3.14), he derives mass conservation/continuity equation in terms of distribution function f:

NOTE: He's using Einstein index notation.

\begin{align} \partial_{t}\int f d v + \partial_\alpha\int f v_\alpha d v + F \int \partial_v f d v & = \frac{1}{\tau}\int (f^0 - f)d v \\ \partial_t n + \partial_\alpha(n u_\alpha) & = 0 \end{align}

where I'm using equilbrium distribution f0 in place of f:
$$f^0(v)=\frac{n}{(2\pi\theta)^{3/2}}\exp{-\frac{(v-u)^2}{2\theta}}$$
I'm reading along and get the first 2 terms of the lhs and rhs to be n, n u_alpha, and 0 respectively, but the third term on the lhs looks like it evaluates to zero. My question is, it looks like he's integrating a derivative of f wrt to a vector. I'm assuming this is a directional derivative and looked it up in wikipedia. However, when I throw it into mathematica (using the equilibrium distrubtion f0) and triple integrate, I don't get zero. Anybody shed some light on how he gets this? (If you can show me the integral in mathematica - so I can see that it works - that would be a bonus)

Last edited: Jul 9, 2010