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What is the least restrictive set of conditions needed to utilize the formula ##\int\limits_{\Omega}\mathrm{d}\alpha=\int\limits_{\partial\Omega} \alpha##?
Could you please expand on this? Thinking of forms as distributions feels foreign, and I don't see where that line of thought would go.Ben Niehoff said:You can even relax smoothness of the manifold (and the forms!) if you are careful about using Dirac delta functions. Generally, since a form is something you integrate, they should be thought of as distributions.
Stokes' Theorem is a mathematical theorem that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface.
The general conditions for applying Stokes' Theorem are that the vector field must be continuously differentiable and the surface must be a closed, smooth, and oriented surface.
Stokes' Theorem is a generalization of the Fundamental Theorem of Calculus, which only applies to line integrals in a one-dimensional space. Stokes' Theorem extends this concept to higher dimensions by incorporating surface integrals and closed surfaces.
Stokes' Theorem has applications in physics, engineering, and fluid dynamics, where it is used to calculate flux through a closed surface and study the behavior of vector fields. It is also used in electromagnetism to relate electric and magnetic fields.
One limitation of Stokes' Theorem is that it can only be applied to smooth and closed surfaces. Additionally, the vector field and surface must meet certain criteria for the theorem to be valid, which may not always be the case in real-world scenarios.