Differential forms

Skew-symmetric covariant tensor fields of order r, with an exterior derivative d that takes an r-form to an (r+1)-form and needs neither a metric nor Euclidean parallelism to be defined. The corpus holds one account, chapters 10 and 13 of bowen-wang-vectors-tensors-vol-2, built on the exterior algebra of that book's Volume 1 (not held).

The operator

Section 51 defines d on forms in a coordinate chart and says it generalizes the classical curl. Because d is independent of the metric it also applies to forms tangent to a hypersurface bowen-wang-vectors-tensors-vol-2. Two properties carry the theory: every exact form (A = dB) is closed (dA = 0), and the converse holds locally.

Closed versus exact

The Poincaré lemma (Thm 52.2): on a star-shaped domain a closed r-form is exact. A ball is star-shaped; the open region between two concentric spheres is not, and the book uses that gap to say what is needed (every closed hypersurface of the relevant dimension must retract). The equivalence for 1-forms needs only a simply connected domain. This is the same mathematics as the curl-free-means-gradient fact in vector-calculus, with the dimension and the domain condition stated.

Frobenius, in two forms

A smooth D-dimensional distribution (a field of D-dimensional subspaces) has integral hypersurfaces if and only if it is closed under the Lie bracket (§50, Thm 50.1). The dual statement in forms (§52) says the same through d. A one-dimensional distribution always has integral curves; higher dimensions need the condition bowen-wang-vectors-tensors-vol-2.

Three-dimensional vector analysis, redone

§53 and 54 return to R³, where the exterior product becomes the cross product, and rebuild lamellar (v = grad f, curl v = 0) and related field classes from the Poincaré lemma bowen-wang-vectors-tensors-vol-2.

Integration and Stokes

§70 integrates N-forms, using the fact that the surface-area density transforms exactly like the strict components of such a form, so the integral does not depend on the chart. §71's generalized Stokes theorem relates the integral of dZ over a domain to the integral of Z over its boundary, and the book reads off the classical Stokes and divergence theorems as formulas (71.28) and (71.30) bowen-wang-vectors-tensors-vol-2. vector-calculus ends where this begins: Corral states those theorems in R³ and stops.

tensor-analysis · euclidean-manifold · vector-calculus · topology · mathematics