Differential Forms

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Differential forms are a coordinate independent approach to multivariable calculus. The modern theory of differential forms is due to Cartan, and has many applications in geometry, topology and physics, for example differential forms can be used to represent classical electrodynamics. Formally, a differential k form is a smooth section of the k’th exterior power of the cotangent bundle. Being a coordinate independent theory it is generally covariant by definition, and general relativity in physics is based on general covariance in the Riemannian manifold. Exotic non-orientable manifolds such as the Mobius strip, Klein bottle and chiral manifolds, exist in pure mathematics, but it is doubtful whether these have any meaning in physics. Cartan geometry in the Riemannian manifold is very well known, and is correct in that manifold. I give a couple of self checking proofs below. Any criticism of a self checking proof can be discarded as unscientific, and people who make such criticisms can be discarded by the scientific readership of ECE.

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