Gradient 1-form

Geometric interpretation of a 1-form α as a stack of planes of constant value, each corresponding to those vectors that α maps to a given scalar

A pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

The cross product of two vectors is not a (true) vector, but rather a pseudovector. In reality the cross product is only defined in three dimensions. In all other dimension one must use the wedge product which results in a bivector.

In three dimensions, the dual of a bivector is a vector therefore the bivector (2-blade) is a pseudovector. In four dimensions, however, the dual of a trivector is a vector therefore trivectors are pseudovectors.

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