Locally Orthonormal FrameΒΆ

At each point we will also store a matrix such that that will map to a locally orthonormal basis. The hodge star operator will be implemented in that basis as

Coordinate bases for vectors and forms

Non-coordinate orthonormal bases for vectors and forms

Transformations to orthornormal basis

where and are transposed inverses of each other

The above property is required because the pairing of one forms with vector fields remains invariant

Relationship to the metric

Transforming one-forms

Transforming two-forms