# Closed-form Jacobian of SE(3) element with 6-degrees of freedom

Let $$A$$ and $$B$$ be two rigid transformations in 3D space that transform things from global to local coordinates. Let their relative transformation be expressed by $$W=AB^{-1}$$. $$W$$ can also be expressed by an element of SE(3): $$\omega=\ln(W)$$ where $$\ln()$$ is defined to be the inverse of the exponential map.

I'd like to know the closed form solutions for computing the 6x6 Jacobians

$$\dfrac{\partial\omega}{\partial\alpha}$$ and $$\dfrac{\partial\omega}{\partial\beta}$$

where $$\alpha$$ and $$\beta$$ are the six degrees of freedom of $$A$$ and $$B$$ respectively. In plain English: How does $$\omega$$ change when I wiggle $$A$$ and $$B$$? It might be that there's a very simple solution to this and I'm just not seeing it.