Please consider figure 1 
which displays 3 spherical caps slightly overlapping on the unit sphere $S2$ with a spherical triangle intersection area hightlighted in green. Let $\vec{U} = [u_x,u_y,u_z]$; $\vec{V}=[v_x,v_y,v_z]$ and $\vec{W}=[w_x,w_y,w_z]$ be the 3 unit vectors giving the positions of the spherical caps, which have a height $h$ and a spherical angular radius of $r_a$ where $h$ is inferred to be the distance from the origin $O$ to the center of the circle creating the cap. (Note: $\arccos(h) = r_a$)
We have three angles to consider, $\angle\alpha$ is the angle between $\vec{U},\vec{V}$; $\angle\beta$ between $\vec{U},\vec{W}$ and $\angle\gamma$ between $\vec{V},\vec{W}$, as shown in Fig 1.
Can we find a formula giving the solid angle of the highlighted green area in terms of $h$, $\vec{U}$, $\vec{V}$, $\vec{W}$, $\alpha$, $\beta$ and $\gamma$ and/or their dot products and square root radicals involving those? I am trying to find a formula which can be differentiated.
For the area of a digon, created by 2 overlapping spherical caps, I found a nice differentiable formula:
$ area(\vec{A},\vec{B},h) = 2\pi - 4\pi h - 2\: \text{acos} \left(\frac{\vec{A} \bullet \vec{B} - h^2}{1-h^2}\right) + 4 h\: \text{acos} \left(\frac{h \vec{A} \bullet \vec{B} - h}{\sqrt{1-h^2} \sqrt{1 - (\vec{A} \bullet \vec{B})^2 }}\right)$ where $\vec{A}\bullet\vec{B}$ denotes the usual dot product.
I am aware of a reference 3 spherical caps overlap which derives this solid angle on pages 2046-2048, but unfortunately even the simple two caps overlap formuli on pg 2046 seems to be in error.
It should be possible to find this formula, using the 4 terms $h$, $U$, $V$, and $W$ alone since the angles $\alpha$, $\beta$ and $\gamma$ can be found from the dot products of the vectors. It is necessary to find a diffentiable formula, since this is needed to create a Jacobian later on for optimization.
