Higher dimensions Clause Samples

Higher dimensions. We have seen that we can integrate one-forms on 1-dimensional manifolds. We need higher-rank forms that we can integrate on higher-dimensional manifolds in a coordinate-independent man- ner. Consider the integral of a real-valued function, f : Rn → R, over a region U in Rn. Under a coordinate transformation g : Rn → Rn, we have2 ∫ ∫ f = U g−1(U) (f ◦ g) det (Dg) . (5.6) A rank n form field takes n vector field arguments and produces a real-valued manifold function: ω (v, w,... , u) (m). By analogy with the 1-dimensional case, higher-rank forms are linear in each argument. Higher-rank forms must also be antisymmetric under interchange of any two arguments in order to make a coordinate- free definition of integration analogous to equation (5.3). Consider an integral in the coordinate system χ: ∫ χ(U) ω (X0, X1,.. .) ◦ χ−1. (5.7) Under coordinate transformations g = χ ◦ χj−1, the integral be- comes ∫ χt(U) ω (X0, X1,.. .) ◦ χj−1 det (Dg) . (5.8) Using the change-of-basis formula, equation (3.19): X(f) = Xj(f)(D(χj ◦ χ−1)) ◦ χ = Xj(f) .D .g−1ΣΣ ◦ χ. (5.9)