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| The '''Darboux derivative''' of a map between a [[manifold]] and a [[Lie group]] is a variant of the standard derivative. In a certain sense, 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]].
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| ==Formal definition==
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| Let <math>G</math> be a [[Lie group]], and let <math>\mathfrak{g}</math> be its [[Lie algebra]]. The [[Maurer-Cartan form]], <math>\omega_G</math>, is the smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form on <math>G</math> (cf. [[Lie algebra valued form]]) defined by
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| :<math>\omega_G(X_g) = (T_g L_g)^{-1} X_g </math>
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| for all <math>g \in G</math> and <math>X_g \in T_g G</math>. Here <math>L_g</math> denotes left multiplication by the element <math>g \in G</math> and <math>T_g L_g</math> is its derivative at <math>g</math>.
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| Let <math>f:M \to G</math> be a [[smooth function]] between a [[smooth manifold]] <math>M</math> and <math>G</math>. Then the '''Darboux derivative''' of <math>f</math> is the smooth <math>\mathfrak{g}</math>-valued <math>1</math>-form
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| :<math>\omega_f := f^* \omega_G,</math>
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| the [[pullback (differential geometry)|pullback]] of <math>\omega_G</math> by <math>f</math>. The map <math>f</math> is called an '''integral''' or '''primitive''' of <math>\omega_f</math>.
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| ==More natural?==
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| 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]] <math>f'</math> of a function <math>f: \mathbb{R} \to \mathbb{R}</math> 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).
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| 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 <math>\mathbb{R}</math> under addition. The [[tangent bundle]] of any [[Lie group]] can be trivialized via left (or right) multiplication. This means that every tangent space in <math>\mathbb{R}</math> may be identified with the tangent space at the identity, <math>0</math>, which is the [[Lie algebra]] of <math>\mathbb{R}</math>. 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 <math>\mathbb{R}</math>. In symbols, for each <math>x \in \mathbb{R}</math> we look at the map
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| :<math>v \in T_x \mathbb{R} \mapsto (T_{f(x)} L_{f(x)})^{-1} \circ (T_x f) v \in T_0 \mathbb{R}.</math>
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| 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 units|canonical unit]] vector field <math>\frac{\partial}{\partial t}</math> on <math>\mathbb{R}</math> gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by <math>f'(x)</math>.
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| ==Uniqueness of primitives==
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| If the manifold <math>M</math> is connected, and <math>f,g: M \to G</math> are both primitives of <math>\omega_f</math>, i.e. <math>\omega_f = \omega_g</math>, then there exists some constant <math>C \in G</math> such that
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| :<math>f(x) = C \cdot g(x)</math> for all <math>x \in M</math>. | |
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| This constant <math>C</math> is of course the analogue of the constant that appears when taking an [[indefinite integral]].
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| ==The fundamental theorem of calculus==
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| Recall the '''structural equation''' for the [[Maurer-Cartan form]]:
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| :<math>d \omega + \frac{1}{2} [\omega, \omega] = 0.</math>
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| This means that for all vector fields <math>X</math> and <math>Y</math> on <math>G</math> and all <math>x \in G</math>, we have
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| :<math>(d \omega)_x (X_x, Y_x) + [\omega_x(X_x), \omega_x(Y_x)] = 0.</math>
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| For any Lie algebra-valued <math>1</math>-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.
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| The usual [[fundamental theorem of calculus]] for single-variable calculus has the following local generalization.
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| If a <math>\mathfrak{g}</math>-valued <math>1</math>-form <math>\omega</math> on <math>M</math> satisfies the structural equation, then every point <math>p \in M</math> has an open neighborhood <math>U</math> and a smooth map <math>f: U \to G</math> such that
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| :<math>\omega_f = \omega|_U,</math>
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| i.e. <math>\omega</math> has a primitive defined in a neighborhood of every point of <math>M</math>.
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| For a global generalization of the fundamental theorem, one needs to study certain [[monodromy]] questions in <math>M</math> and <math>G</math>.
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| ==References==
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| * {{cite book|author=R. W. Sharpe|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program|year=1996|publisher=Springer-Verlag, Berlin|isbn=0-387-94732-9}}
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| * {{cite book|author=[[Shlomo Sternberg]]|title=Lectures in differential geometry|year=1964|publisher=Prentice-Hall|id=LCCN 64-7993|chapter=Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.}}
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| [[Category:Differential calculus]]
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| [[Category:Lie groups]]
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