Question about the Jacobian of the exponential map

[xianshunw]

Thanks for your excellent work. But I have some doubts about the shape of the jaobian of the exponential map.

Let's take the $SE3$ for example. Its exponential function map $\mathbb{R}^{6}$ to $\mathbb{R}^{4\times4}$. Thus I think the jacobian matrix of this fun should be in $\mathbb{R}^{16\times6}$ or $\mathbb{R}^{6\times16}$. But you state that:

The Jacobian of the exponential map $\mathbf{J}_{l}=\frac{\partial }{\partial \mathbf{x}}Exp(\mathbf{x})$ is referred to as the left-Jacobian

And the left Jacobian of $SE3$ should be a $6\times6$ matrix. Where does the shape difference comes from. Is there any misunderstood about your statements?