I have a profound understanding of the classical Stokes theorem (aka Curl theorem). However, I am a little confused, how the theorem is written on Wikipedia $$ \iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d}^2 \mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{\Gamma}. $$ Could someone explain the deeper meaning of the square on the left side?
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That $d^{\color{red}{2}}\Sigma$ only indicates that it is a surface integral which should be clear from the context. Write $dS$ if you like that better. – Kurt G. Jun 24 '23 at 19:38
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Seems a bit strange. Math should not depend on context. – ConvexHull Jun 24 '23 at 19:52
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Moreover, $\Sigma$ is already defined as surface, see the link. – ConvexHull Jun 24 '23 at 19:53
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Notation isn't math. It helps to express mathematifal statements in more or less convenient ways that depend on what people are used to. There is never a notation that explains itself. Either the author does that or it is clear from context. – Kurt G. Jun 24 '23 at 20:30
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Wise words. However, it is neither described on the page nor it is clear from context. As I already said, in the picture $\Sigma$ is defined as surface. For me, it is simply wrong or missleading. – ConvexHull Jun 24 '23 at 20:42
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Then you may want to discuss that with those who wrote that Wikipedia article. – Kurt G. Jun 24 '23 at 20:48
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Anyway, thank you for your reply. – ConvexHull Jun 24 '23 at 20:52
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Personally I prefer this notation because I want to see the vector normal to the surface. – Kurt G. Jun 24 '23 at 20:54
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D'accord. Looks more reasonable. – ConvexHull Jun 24 '23 at 20:58
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But then came Elie Cartan and found something even better. :) – Kurt G. Jun 24 '23 at 21:04
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I like Elie, especially for $v \times w = \left(\star(v^\flat \wedge w^\flat )\right)^\sharp$. – ConvexHull Jun 25 '23 at 08:27
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He was an absolute genius, and the gem Hopeful Whitepiller found in that link is marvellous. I cannot stop laughing about that Absinthe. – Kurt G. Jun 25 '23 at 10:06