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A differential $2$-form can be understood in terms of integral operators, specifically what happens when you integrate them over (directed) 2-surfaces. Stoke's theorem, the higher-dimensional analog of the fundamental theorem of calculus, tells us that $$\int_S d\omega = \int_{\partial S} \omega$$ where $\partial S$ is the boundary of $S$. (if $f$ is a ...

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To develop some intuition, you might find it helpful to go back to the physics that inspired the definitions of grad, div and curl: electromagnetism. There are nice physical reasons why we would expect $d^2$ to be zero on both $0$-forms/scalar fields, $\nabla \times \nabla \phi = \vec{0}$, and $1$-forms/vector fields, $\nabla\cdot\nabla\vec{A} = 0$. ...

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Henri Cartan, one of the greatest 20th century mathematicians wrote: ...utinam intelligere possim ratinationes pulcherrimae quae e propositione concisa DE QUADRATO NIHILO EXAEQUARI fluunt In the unlikely event that you don't speak Latin, o amice, this means: ...if I could only understand the beautiful consequences following from the concise proposition ...

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The general statement has to do with $n$ forms on a compact $n$-dimensional manifold $M$ without boundary. It is a general fact that for such manifolds the $n$th de Rham cohomology group is $H^n_{dR}(M) = \mathbb R$. Now if $\alpha$ is an $n$-form that represents a non-zero class in $H^n_{dR}(M)$ (i.e. $\alpha$ is not exact), then every other $n$th degree ...

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On 1-dimensional manifolds, this is just the holonomy (for circles) or parallel transport (for intervals). One integrates the connection with the path-ordered exponential. It seems that there is a Stokes' theorem for higher gauge theory: There is a notion of 2-holonomy of surfaces for 2-connections, see for example An Invitation to Higher Gauge Theory or ...

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