A general 2-form - Colley Chapter 8, Section 8.1

[Math Amateur]

I am reading Susan Colley's book: Vector Calculus ... and am currently focussed on Section 8.1: An Introduction to Differential Forms ... ...

Colley, on page 491 in Example 3 gives a formula for a general 2-form as follows:

I am trying to understand what Colley describes as 'the somewhat curious ordering of the terms' ... ... with the terms $dy \ \wedge \ dz , dz \ \wedge \ dx$ and $dx \ \wedge \ dy$ occurring in that order ... ... ?... Now when Colley comes to defining a general differential k-form, not two pages further on from Example 3, we find (page 493):

So ... from the general k-form above, for a general 2-form we have$\omega = \sum_{ 1 \le i_1 \lt i_2 \lt 2 } F_{ i_1 i_2 } dx_{i_1} \ \wedge \ dx_{i_2}$$\omega = F_{12} \ dx_1 \ \wedge \ dx_2 \ + \ F_{13} \ dx_1 \ \wedge \ dx_3 \ + \ F_{23} \ dx_2 \ \wedge \ dx_3$or if we write $x_1$ as $x$, $x_2$ as $y$, and $x_3$ as $z$ then we have ... ...

$\omega = F_{12} \ dx \ \wedge \ dy \ + \ F_{13} \ dx \ \wedge \ dz \ + \ F_{23} \ dy \ \wedge \ dz$How do we match this general form with that stated two pages earlier in Example 3 ...Hope someone can help ...

Peter

 

[Samy_A]

Hmm, I think I share your confusion.

Example 3: $F_1(x,y,z)dy\wedge dz+F_2(x,y,z)dz\wedge dx+F_3(x,y,z)dx\wedge dy$
should have been written as $F_3(x,y,z)dx\wedge dy-F_2(x,y,z)dx\wedge dz+F_1(x,y,z)dy\wedge dz$
to match the standard form with strictly increasing indexes.

EDIT: maybe later in the text they will write 2-forms as $F_{12}dx \wedge dy+ F_{23}dy \wedge dz +F_{31}dz\wedge dx$, and that's what they are hinting at.

 

[lavinia]

Two forms can be added. Addition is commutative so the order of addition does not change the form.

One can multiply forms by arbitrary functions and it doesn't matter what you call them. Indexing is just notation. You can index any way that you want.

In the coordinate system, $(x,y,z)$ on the domain $U$ dx^dy ,dx^dz and ,dy^dz form a basis for all 2 forms. The general 2 form is a linear combination of them with the coefficients being arbitrary functions. Note that at each point of $U$ one just has a linear combination of basis vectors of a vector space.

So one can write the general 2 form in the first notation as

$F_{1}(x,y,z)$dy$∧$dz + $F_{2}(x,y,z)$dz$∧$dx + $F_{3}(x,y,z)$dx$∧$dy or as

$F_{12}$dx$∧$dy + $F_{13}$dx$∧$dz + $F_{23}$dy$∧$ dz in the second notation.

Both are the same provided that the corresponding $F$'s are equal.

Note also the if one reverses the order of wedging that the form gets multiplied by -1. So dx$∧$dy = -dy$∧$dx
This relation shows that there are only three basis vectors, not 6.