next up previous contents
Next: Curvilinear Coordinates Up: Math Previous: Special Symbols   Contents


Leibnitz's Rule for Differentiating Integrals

This is Leibnitz's rule for differentiation of integrals. I wasn't sure where else to stick it.


\begin{displaymath}
\frac{d}{d\alpha}\int_{\phi_1(\alpha)}^{\phi_2(\alpha)} F(x...
...{d\phi_2}{d\alpha}
- F(\phi_1,\alpha)\frac{d\phi_1}{d\alpha}
\end{displaymath} (2.6)

We generalize this to 3 dimensions with $alpha$ being the time,

\begin{displaymath}
\frac{d}{dt}\int_{V(t)} F(\mathbf{x},t) dV
= \int_{V(t)}\...
...tial t} dV
+ \int_{A(t)} F \mathbf{u_A} \cdot  d\mathbf{A}
\end{displaymath} (2.7)


where $V(t)$ is the volume with surface area $A(t)$ and $\mathbf{u_A}$ is the velocity of the boundary. This is particularly useful for conservation laws.



Ben Breech 2003-01-14