Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 175is dual to the space of vector–fields X k (M).In particular, the coordinate 1−forms dx 1 , ..., dx n are locally definedat any point m ∈ M by the property that for any vector–field X =(X 1 , ..., X n) ∈ X k (M),dx i (X) = X i .The dx i ’s form a basis for the 1−forms at any point m ∈ M, with localcoordinates ( x 1 , ..., x n) , so any 1−form α may be expressed in the formα = f i (m) dx i .If a vector–field X on M has the form X(m) = ( X 1 (m), ..., X n (m) ) ,then at any point m ∈ M,α m (X) = f i (m) X i (m),where f ∈ C k (M, R).Suppose we have a 1D closed curve γ = γ(t) inside a smooth manifoldM. Using a simplified ‘physical’ notation, a 1–form α(x) defined at a pointx ∈ M, given byα(x) = α i (x) dx i , (3.37)can be unambiguously integrated over a curve γ ∈ M, as follows. Parameterizeγ by a parameter t, so that its coordinates are given by x i (t). Attime t, the velocity ẋ = ẋ(t) is a tangent vector to M at x(t). One caninsert this tangent vector into the linear map α(x) to get a real number.By definition, inserting the vector ẋ(t) into the linear map dx i gives thecomponent ẋ i = ẋ i (t). Doing this for every t, we can then integrate over t,∫ (αi(x(t))ẋ i) dt. (3.38)Note that this expression is independent of the parametrization in termsof t. Moreover, from the way that tangent vectors transform, one candeduce how the linear maps dx i should transform, and from this how thecoefficients α i (x) should transform. Doing this, one sees that the aboveexpression is also invariant under changes of coordinates on M. Therefore,a 1–form can be unambiguously integrated over a curve in M. We writesuch an integral as∫∫α i (x) dx i , or, even shorter, as α.γγ