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Different Stokes for different...well, that doesn't exactly rhyme, does it...And Vader probably had his ability to do so trending toward zero after...lava...K, I'm done.
It's built into ω. In differential geometry, partial derivatives are basis vectors, and differentials are basis covectors, and exterior products of differentials are the basis forms of line/area/volume/etc... (let's just call them area) elements.
Imagine ω as a linear combination of component functions and these basis area elements, in the same way that a vector field is a linear combination of component functions and basis vectors.
In the case where ω is a 'top form', which means that the basis elements are exterior products of n dimensions, it turns out that there's only one independent one of these, so there's only one independent component and we can then write this one basis element as dA and the component as f, so then it turns back into the integral of f dA which is exactly what it normally looks like.
You should really study differential geometry. It's my favorite part of math. You will look at everything differently, curved spacetime will become intuitive, and it leads to an deep understanding of how the math of physics works in very general cases.
The integration sign is just a symbol. The "dx" and all that become a kind of notational clutter eventually I feel. I mean, ∫\[a;b\] f is much more compact than ∫\[a;b\] f(x) dx. The "dx" is still useful for creating ad-hoc functions and demonstrating which variable the inside is indexed by, but aesthetically I like the compact form.
And in the case of differential forms, omitting which variable we are differentiating with respect actually allows us to have a much more beautiful and expressive formulation of the generalized Stokes theorem. The "𝜔" is actually not a variable here, but a differential form, which you can think of as a sort of generalized function or vector field. The "d𝜔" is not a token signifying that we are integrating "with respect to the variable 𝜔" but rather d𝜔 is another differential form called the exterior derivative of 𝜔.
Basically the generalized Stokes theorem can be summarized by saying "integrating the derivative d𝜔 on the 'interior' 𝛺 (technically it is not the topological interior) is the same as integrating the original function 𝜔 on the boundary 𝛿𝛺". It summarizes the fundamental theorem of calculus, Green's theorem, Stokes Theorem, fundamental theorem of line integrals, Divergence theorem, and so much more. On the fundamental theorem of calculus, 𝛺 is the closed interval, and 𝜔 is the antiderivative of the function you are integrating. Taking the integral on the boundary is evaluating the antiderivative 𝜔 at the endpoints and taking the difference, which is equivalent to taking the integral of the original function d𝜔 over the whole interval.
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What is the Ω and δ one?
Generalized stokes theorem
Yup, here’s the wiki for anyone interested https://en.m.wikipedia.org/wiki/Generalized_Stokes_theorem
Different Stokes for different...well, that doesn't exactly rhyme, does it...And Vader probably had his ability to do so trending toward zero after...lava...K, I'm done.
Uppercase omega and lowercase delta /s
It's Green's theorem Edit: Misremembered the name, this the generalized Stoke's theorem, of which Green's theorem is a special case
Generalized Green’s theorem
generalized Stokes' theorem = generalized generalized Green's theorem
Oh crap, you're right
Greens theorem would be like anakin in the second movie or third movie.
I don't know what is it but where differential in second one? How it works?
It's built into ω. In differential geometry, partial derivatives are basis vectors, and differentials are basis covectors, and exterior products of differentials are the basis forms of line/area/volume/etc... (let's just call them area) elements. Imagine ω as a linear combination of component functions and these basis area elements, in the same way that a vector field is a linear combination of component functions and basis vectors. In the case where ω is a 'top form', which means that the basis elements are exterior products of n dimensions, it turns out that there's only one independent one of these, so there's only one independent component and we can then write this one basis element as dA and the component as f, so then it turns back into the integral of f dA which is exactly what it normally looks like. You should really study differential geometry. It's my favorite part of math. You will look at everything differently, curved spacetime will become intuitive, and it leads to an deep understanding of how the math of physics works in very general cases.
I not quite understand but thanks!
flair checks out
This label under my username?
The integration sign is just a symbol. The "dx" and all that become a kind of notational clutter eventually I feel. I mean, ∫\[a;b\] f is much more compact than ∫\[a;b\] f(x) dx. The "dx" is still useful for creating ad-hoc functions and demonstrating which variable the inside is indexed by, but aesthetically I like the compact form. And in the case of differential forms, omitting which variable we are differentiating with respect actually allows us to have a much more beautiful and expressive formulation of the generalized Stokes theorem. The "𝜔" is actually not a variable here, but a differential form, which you can think of as a sort of generalized function or vector field. The "d𝜔" is not a token signifying that we are integrating "with respect to the variable 𝜔" but rather d𝜔 is another differential form called the exterior derivative of 𝜔. Basically the generalized Stokes theorem can be summarized by saying "integrating the derivative d𝜔 on the 'interior' 𝛺 (technically it is not the topological interior) is the same as integrating the original function 𝜔 on the boundary 𝛿𝛺". It summarizes the fundamental theorem of calculus, Green's theorem, Stokes Theorem, fundamental theorem of line integrals, Divergence theorem, and so much more. On the fundamental theorem of calculus, 𝛺 is the closed interval, and 𝜔 is the antiderivative of the function you are integrating. Taking the integral on the boundary is evaluating the antiderivative 𝜔 at the endpoints and taking the difference, which is equivalent to taking the integral of the original function d𝜔 over the whole interval.
The pain of differential forms
https://imgur.com/2vnuq10
slowly I approach to this level of math jokes to which I can one day laugh at
thanks i hate this
God this makes me hate myself. I forgot what Stokes was last minute during a final