Geometric Calculus - Relation To Differential Forms

Relation To Differential Forms

In a local coordinate system (x1, ..., xn), the coordinate differentials dx1, ..., dxn form a basic set of one-forms within the coordinate chart. Given a multi-index i₁, ..., i_k with 1 ≤ i_p ≤ n for 1 ≤ p ≤ k, we can define a k-form

We can alternatively introduce a k-grade multivector A as

and a measure

$\begin{align}\mathrm{d}^kX &= \left(\mathrm{d}x^{i_1} e_{i_1}\right) \wedge \left(\mathrm{d}x^{i_2}e_{i_2}\right) \wedge\cdots\wedge \left(\mathrm{d}x^{i_k}e_{i_k}\right) \\ &= \left( e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_k} \right) \mathrm{d}x^{i_1} \mathrm{d}x^{i_2} \cdots \mathrm{d}x^{i_k}\end{align}.$

Apart from a subtle difference in meaning for the exterior product with respect to differential forms versus the exterior product with respect to vectors, we see the correspondences of the differential form

its derivative

and its Hodge dual

embed the theory of differential forms within geometric calculus.

Read more about this topic: Geometric Calculus

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