# Tag Info

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Yes, it is possible. For example, Serge Lang exhibits the basics of a theory of differential forms on Banach manifolds in Chapter V of his Differential and Riemannian Manifolds. (Non-)Separability is not an issue. According to Lang, a $p$-form on a Banach space $E$ is simply a continuous alternating $p$-linear map on $E$. This yields a notion of $p$-forms ...

2

Yes. Since the product of a form and a smooth function is again a form, extend the form to a small tubular neighborhood of your submanifold and multiply by a bump function. Georges Elencwajg has suggested I explain further. To extend a $p$-form $\omega\in\Omega^p(M)$ to a $p$-form on $\mathbb{R}^n$, we need to first define a $p$-form on ...

1

It seems to me that patches or coordinate charts require some choice of basis. However, the real thing which defines a hyperplane is that it has one less dimension than the ambient vector space. However, manifold dimension is not quite restrictive enough as there are plenty of curved spaces of codimension $1$. We should also insist that it is possible to ...

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You need a $1$-form $\gamma$ so that $d\alpha$, $d\beta$, and $d\gamma$ are all in the ideal generated by $\alpha$, $\beta$, and $\gamma$. So the major HINT is: Compute $d\alpha$ and $d\beta$. Can you see an obvious $\gamma$ so that $d\alpha,d\beta\in (\gamma)$?

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Doesn't the "region of integration" of a Riemann integral have to be connected? Depends on definitions, as always. If one develops Riemann integration exclusively for $\int_a^b f(x)\,dx$, then yes. On the other hand, it takes one line to say that if a set $E$ is the union of finitely many disjoint closed intervals $[a_i,b_i]$, then we define  ...

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In your example, let's use $x$ and $y$ as coordinates for $M$. The outward vector on the boundary should be tangent to M. A suitable choice is $-\partial_x$, considered as a coordinate basis vector on $M$. The minus sign is there because $M$ is on the postive $x$ side, so the outward vector should point to negative $x$ values. The induced orientation on the ...

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