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I am looking for a word or phrase that is similar in meaning to the "support of a function", but in the context of differential forms.

The "support of a function" is the subset of the domain of a function where the function is non-zero.

In my case, consider the bilinear functional $(dx \wedge dy)$ acting on vector pairs in $xyz$-space.

I would like to say something like, "The support of $(dx \wedge dy)$ is the $xy$ plane.", because the result of the calculation is zero unless both vectors have a no-zero projection on the $xy$-plane.

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The notion of support of a differential form $\omega$ already exists. Unfortunately it means the closure of the set of points on which $\omega$ is non-zero. (Recall, a differential form is a smooth assignment of alternating multilinear forms to each point of a manifold...) – Zhen Lin Jan 16 at 9:59
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The more fundamental space associated with $dx\wedge dy$ is actually the $z$-axis, which is the set of vectors $v$ such that $dx\wedge dy(v,\cdot)=0$. This is sometimes called the kernel, or characteristic space, or annihilator of the differential form; notice that it's defined without reference to the inner product. If you're willing to rely on the inner product of $\mathbb R^3$, you can single out the $xy$-plane as the space orthogonal to the kernel. But a different inner product would result in a different such $2$-dimensional space, while the kernel wouldn't change. – Jack Lee Jan 17 at 5:15

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