Garrity's Formula: Ordering Covariant Alternating Tensors

[Rasalhague]

I have a question about Garrity's formula at the top of p. 125, here, for a function from the set of 2-form fields to the set of tangent vector fields, together with the formula on p. 123 for the exterior derivative of a 1-form field and Theorem 6.3.1 on p. 125 (Garrity: All the Mathematics you Missed).

These relations seem to depend on a convention for ordering the vectors of a standard basis for /\²(R³). Is that right? Is there a convention for ordering the vectors of a standard basis for /\^k(Rⁿ). If so, what is the convention?

 

[Rasalhague]

To elaborate: where the manifold is R³, Garrity gives the following rule for transforming a 2-form field into a tangent vector field:

$$T_2(\alpha_i \, e^i)=\alpha_3 \, \partial_1 - \alpha_2 \, \partial_2 + \alpha_1 \, \partial_3 = (\alpha_3,-\alpha_2,\alpha_1),$$

where eⁱ is a constant basis field for /\²(R³), consisting of wedge products of all permutations of pairs of coordinate basis 1-form fields with indices in ascending order: (1,2), (1,3), (2,3). He gives the following rule for transforming a 1-form field into a tangent vector field:

$$T_1(\omega) = (\omega_1,\omega_2,\omega_3),$$

and the following rule relating the exterior derivative of a differentiable 1-form field to the curl of its corresponding tangent vector field:

$$T_2(\mathrm{d}\omega) = \text{curl}(T_1(\omega)).$$


Now, consider a differentiable 1-form field

$$\omega = \omega_i \, \mathrm{d}x^i.$$

By the coordinate-basis definition of the exterior derivative,

$$\mathrm{d} \omega = \mathrm{d}\omega_i \wedge \mathrm{d}x^i$$

$$=\mathrm{d} \omega = \left ( \partial_k \omega_i \, \mathrm{d}x^k \right ) \wedge \mathrm{d}x^i$$

$$=\left ( \partial_1 \omega_2 - \partial_2 \omega_1 \right ) \mathrm{d}x^1 \wedge \mathrm{d}x^2 + \left ( \partial_1 \omega_3 - \partial_3 \omega_1 \right ) \mathrm{d}x^1 \wedge \mathrm{d}x^3 + \left ( \partial_2 \omega_3 - \partial_3 \omega_2 \right ) \mathrm{d}x^2 \wedge \mathrm{d}x^3.$$


The foregoing implies Garrity is using the order

$$(e^1, e^2, e^3) = \left ( \mathrm{d}x^1 \wedge \mathrm{d}x^2 \right, \mathrm{d}x^1 \wedge \mathrm{d}x^3, \mathrm{d}x^2 \wedge \mathrm{d}x^3 ).$$

I notice that this differs in order from the dual basis to the coordinate basis, when /\²(R³) is "considered as" the dual space to /\¹(R³), i.e.

$$( \mathrm{d}x^2 \wedge \mathrm{d}x^3, \mathrm{d}x^3 \wedge \mathrm{d}x^1, \mathrm{d}x^1 \wedge \mathrm{d}x^2).$$

The Garrity basis, in this case, is ordered "alphabetically". Is that convention followed for /\^k(Rⁿ) generally? Are there standard names for each of these bases: the "Hodge dual basis" and the "alphabetic basis"?

EDIT: Oh, I see he calls the (Hodge) dual basis to {dxⁱ} the "natural basis". I don't think he gives a particular name to his alphabetic basis. Perhaps he considers the latter too natural to need a name : )