Suppose we have an n-dimensional [[manifold]], M and a target [[manifold]] T. Let <math>\mathcal{C}</math> be the [[configuration space]] of [[smooth function]]s from M to T.
Before we go on, let's give some examples:
* In [[classical mechanics]], M is the one-dimensional [[manifold]] <math>\mathbb{R}</math>, representing time and the target space is the [[tangent bundle]] of [[space]] of generalized positions.
* In [[Field_theory_(physics)|Field Theory]], M is the [[spacetime]] [[manifold]] and the target space is the set of values the fields can take at any given point. For example, if there are m [[real number|real]]-valued [[scalar]] fields, φ<sub>1</sub>,...,φ<sub>m</sub>, then the target [[manifold]] is <math>\mathbb{R}^m</math>. If the field is a [[real]] vector field, then the target [[manifold]] is [[isomorphic]] to <math>\mathbb{R}^n</math>. There's actually a much more elegant way using [[tangent bundle]]s over M, but for the purposes of this proof, we'd just stick to this version.
