I was under the impression that all 2-forms are the wedge $(\wedge)$ of two 1-forms. Is it possible to have a 2-form that you can't write as $A\wedge B$ with $A,B$ 1-forms?
Can you find a 2-form not written as the wedge of two 1-forms?
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exterior-algebra
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0The question in your title is the opposite of the question in the body of your thread. Mariano is responding to the latter. – 2010-10-06
1 Answers
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Yes, it is possible. (And you should find an example yourself: I will not deprive you of the joy of finding it :) )
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1Would dA^dB+dC^dD (in R^4) be an example? It's a two form but it can't be written as A^B – 2010-10-06
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4@JimJones: your example works, but you should not be satisfied until you have a proof of this. Just because you try to write it as the wedge of two one-forms and fail is of course not enough... – 2010-10-06
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1@Mariano, awesome answer – 2010-10-06
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0I still just don't understand – 2010-10-06
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0@JimJones. I assume that $A,B,C,D$ in your first formula are functions... (Maybe I could add to Mariano's answer: simply look at the definition of 2-forms and remember something about linear independence.) – 2010-10-06
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0A,B,C,D are suppose to be 1-forms... is that wrong? – 2010-10-06
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0@JimJones: you don't have to understand: you have to compute. Find the general form of a $2$-form which is a wedge product of two $1$-forms, and next show that not all $2$-forms are of that form. – 2010-10-06
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0The definition I was given in class is "a k-form is a function taking p to a k-linear alternating map on TR^n. These look like linear combinations of dx^1^dx^2^..." I was under the impression that the definition of a 2-form is the wedge of two one forms... that's why I'm lost – 2010-10-06
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0If you think a 2-form is *that* then what you need is to read up a bit on what forms are :) – 2010-10-06
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0@JimJones. If $A,B,C,D$ are 1-forms, then $dA, dB, dC, dD$ are 2-forms. Then $dA\wedge dB$ and $dC\wedge dD$ are 4-forms. – 2010-10-06
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0@JimJones. As for your "look like linear combinations": that's ok. And you are making linear combinations of *which* kind of vectors? Can you find a linear combination of $dx\wedge dy, dy\wedge dz, dz \wedge dx$ equal to zero -different from the trivial one? – 2010-10-06