Exterior Derivative Clause Samples

Exterior Derivative. The intention of introducing the exterior derivative is to capture all of the classical theorems of “vector analysis” into one unified ▇▇▇▇▇▇’▇ Theorem, which asserts that the integral of a form on the boundary of a manifold is the integral of the exterior derivative of the form on the interior of the manifold:4 ∫ ∫ dω. (5.22) As we have seen in equation (3.34), the differential of a function on a manifold is a one-form field. If a function on a manifold is considered to be a form field of rank zero,5 then the differential operator increases the rank of the form by one. We can generalize this to k-form fields with the exterior derivative operation. Consider a one-form ω. We define6 dω(v1, v2) = v1(ω(v2)) − v2(ω(v1)) − ω([v1, v2]). (5.23) More generally, the exterior derivative of a k-form field is a k + 1- form field, given by:7 dω(v0,... , vk) = (5.24) Σk Σk . i ((−1) vi(ω(v0,... , vi−1, vi+1,... , vk))+ (−1)i+j ω([vi, vj], v0,... , vi−1, vi+1,... , vj−1, vj+1,... , vk))} . j=i+1 This formula is coordinate-system independent. This is the way we compute the exterior derivative in our software. 4This is a generalization of the Fundamental Theorem of Calculus. 5A manifold function f induces a form field ˆf of rank 0 as follows: ˆf()(m) = f(m). 6The definition is chosen to make ▇▇▇▇▇▇’▇ Theorem pretty. 7See ▇▇▇▇▇▇, Differential Geometry, Volume 1, p.289.