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I've been browsing through MTW recently and I found something that puzzles me:

They claim that if you have two form, call it [itex]\mathbf{T}[/itex], it's value, say [itex]\mathbf{T}(\mathbf{u} , \mathbf{v} ) [/itex] can be represented geometrically as follows: take two vectors [itex]\mathbf{u}[/itex] and [itex]\mathbf{v}[/itex]; the surface containing those two is [itex]\mathbf{u} \bigwedge \mathbf{v}[/itex] (I don't get this, why isn't it just the vector product [itex] \mathbf{u} \times \mathbf{v}[/itex]?) and the value of the two form is just the number of tubes the "egg-crate" structure cuts through this parallelogram. I don't get this.

They also state that the a basis two-form, say [itex]\mathbf{d}x \bigwedge \mathbf{d}y[/itex] can be represented by just crossing the surfaces of each basis one-form. This is also confusing.

They claim that if you have two form, call it [itex]\mathbf{T}[/itex], it's value, say [itex]\mathbf{T}(\mathbf{u} , \mathbf{v} ) [/itex] can be represented geometrically as follows: take two vectors [itex]\mathbf{u}[/itex] and [itex]\mathbf{v}[/itex]; the surface containing those two is [itex]\mathbf{u} \bigwedge \mathbf{v}[/itex] (I don't get this, why isn't it just the vector product [itex] \mathbf{u} \times \mathbf{v}[/itex]?) and the value of the two form is just the number of tubes the "egg-crate" structure cuts through this parallelogram. I don't get this.

They also state that the a basis two-form, say [itex]\mathbf{d}x \bigwedge \mathbf{d}y[/itex] can be represented by just crossing the surfaces of each basis one-form. This is also confusing.

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