in 2d, "curl"(grad(f)) = 0 "curl" is the operation that green's tehorem talks about.(adsbygoogle = window.adsbygoogle || []).push({});

In 3D, curl(grad(f)) = 0 and div(curl(F)) = 0.

We may consider vector calculus in 4 spatial dimensions, for vector fields F:R^4 -> R^4. what is "curl" like in 4D, since curl is actually only difined in 3D. I think there would be no curl in 4D because there's no cross product in 4D. instead there would be 2 operators related by Stokes's theorem for general manifolds.

my conjecture is that in 4 dimensions, some other operator D2 exists such that div(D2(F)) = 0. Also there exists another operator D1 such that D2(D1(f)) = 0 and D1(gradient(f))=0. Div would still be from an operator from a vector field to a scalar field.

the divergence theorem would be relating 4-volumes to 3-volumes(boundary of 4-volume)

Do they exist? what is the nature of such an operator (in cartesian) and how can i visualize it?

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# Vector calculus in higher dimensions

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