I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors aren't true elements of $R^3$ but elements of a Lie group, say SO(3) or SU(2), I wonder how to formulate Stokes' theorem (or Maxwell's equations). The derivative of a Lie group would be an element of the Lie algebra, thus one would put two incompatible quantities into one equation.
Let me add one more explanation why there should be a solution to the problem: For small values, elements of SO(3) (threedimensional rotations) correspond to "vectors": express the rotation axis as direction of the vector and the twisting angle as its length (the Euler axis representation). But for finite rotations, this does not work. Mind also that finite rotations do not commute, while vector addition does.
To summarize, I'd like generalize (or better formulate) Stokes' theorem and Maxwell's equations for elements of the Lie-group SO(3) (or its double cover SU(2)).