Darboux derivative

The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes' theorem.
Formal definition.
Let formula_1 be a Lie group, and let formula_2 be its Lie algebra. The Maurer-Cartan form, formula_3, is the smooth formula_2-valued formula_5-form on formula_1 (cf. Lie algebra valued form) defined by
for all formula_8 and formula_9. Here formula_10 denotes left multiplication by the element formula_8 and formula_12 is its derivative at formula_13.
Let formula_14 be a smooth function between a smooth manifold formula_15 and formula_1. Then the Darboux derivative of formula_17 is the smooth formula_2-valued formula_5-form 
the pullback of formula_3 by formula_17. The map formula_17 is called an integral or primitive of formula_24.
More natural?
The reason that one might call the Darboux derivative a more natural generalization of the derivative of single-variable calculus is this. In single-variable calculus, the derivative formula_25 of a function formula_26 assigns to each point in the domain a single number. According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a linear map from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates two pieces of data: the image of the domain point "and" the linear map. In single-variable calculus, we drop some information. We retain only the linear map, in the form of a scalar multiplying agent (i.e. a number).
One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of formula_27 under addition. The tangent bundle of any Lie group can be trivialized via left (or right) multiplication. This means that every tangent space in formula_27 may be identified with the tangent space at the identity, formula_29, which is the Lie algebra of formula_27. In this case, left and right multiplication are simply translation. By post-composing the manifold-type derivative with the tangent space trivialization, for each point in the domain we obtain a linear map from the tangent space at the domain point to the Lie algebra of formula_27. In symbols, for each formula_32 we look at the map
Since the tangent spaces involved are one-dimensional, this linear map is just multiplication by some scalar. (This scalar can change depending on what basis we use for the vector spaces, but the canonical unit vector field formula_34 on formula_27 gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by formula_36.
Uniqueness of primitives.
If the manifold formula_15 is connected, and formula_38 are both primitives of formula_24, i.e. formula_40, then there exists some constant formula_41 such that 
This constant formula_44 is of course the analogue of the constant that appears when taking an indefinite integral.
The fundamental theorem of calculus.
The structural equation for the Maurer-Cartan form is:
This means that for all vector fields formula_46 and formula_47 on formula_1 and all formula_49, we have
For any Lie algebra-valued formula_5-form on any smooth manifold, all the terms in this equation make sense, so for any such form we can ask whether or not it satisfies this structural equation.
The usual fundamental theorem of calculus for single-variable calculus has the following local generalization.
If a formula_2-valued formula_5-form formula_54 on formula_15 satisfies the structural equation, then every point formula_56 has an open neighborhood formula_57 and a smooth map formula_58 such that
i.e. formula_54 has a primitive defined in a neighborhood of every point of formula_15.
For a global generalization of the fundamental theorem, one needs to study certain monodromy questions in formula_15 and formula_1.