How to calculate the integral in the question

[final bronze]

γ is not given. Also what kinda integral is the integral mentioned in the question. All I know is to check if the integral in question is equal to 0 or not in order to check if dn = w.

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[velvet geyser]

The integral of a total differential over a closed curve, assuming the function is continuous inside the region bounded by the curve, is 0.

The physical analogy would be potential or potential energy.

[final bronze]

It doesn’t seem to be conservative

Can you expand on how it is 0

[velvet geyser]

Ah, yeah, that's true, it isn't a total differential.

As for how to check it - you can find the rotor of this field. If it's 0 (vector), then it's conservative and the differential is total.

Ahh, wait.

Sorry, I didn't pay close enough attention. This is a second differential.

Then I don't quite understand how we're integrating this over a closed curve. It is a surface integral, not a line integral.

[final bronze]

There is no parametrization given

[tardy fog]

Given a 2-form of your manifold $\eta$ and $X:R\to M$ is a 2-segment (rectangular patch of the manifold), then $\int_X\eta=\int_R X^\ast(\eta)$

[chrome yew]

Omegabet_

[tardy fog]

you're just integrating the push forward/pull back (whatever upper astericks is again lol) of the form over the euclidean region

[velvet geyser]

Hm... Yeah, seems like I haven't learned this, sorry.

[final bronze]

I’m confused

[tardy fog]

I assume you're taking differential geometry?

[final bronze]

Nope

I’m taking multivariable calculus

[tardy fog]

Then you're using atypical notation

But anyway, for a 1-form $\varphi$ and a curve $\alpha$, $\int_\alpha\varphi:=\int_a^b\varphi(\alpha'(x))\dd{x}$

[chrome yew]

Omegabet_

[tardy fog]

the analogous for 2-forms and regions holds too

[final bronze]

Seems the curve in question is not given

[tardy fog]

$\iint_R\omega:=\iint_R\omega(X_u,X_v)\dd{u}\dd{v}$, where $X$ parameterizes the region

[chrome yew]

Omegabet_

[tardy fog]

it's a loop

[final bronze]

Oh

Uh what does that look like

[tardy fog]

... a loop

draw something in the plane, and end where you started

thats a loop

[final bronze]

Would x^2+y^2=1 work

[tardy fog]

the graph of that is indeed a loop yes

but gamma is any loop

[final bronze]

Uh how do I show for any loop

[tardy fog]

wdym

[final bronze]

Is it enough to show it is not 0 for one loop

[tardy fog]

if you think the result isnt 0, sure

[final bronze]

If the result is 0 how to show it is 0 for all

[tardy fog]

but it's two questions:

1. Check if $\omega$ is exact.

2. If $\omega$ were exact, how would we use that to compute integrals of it?

[chrome yew]

Omegabet_

[tardy fog]

but yeah, the notation beind used is shit lol

[final bronze]

Is there another way to check n st dn = w exists

[tardy fog]

by definition of the differential

[final bronze]

Isn’t the way to check is to compute the integral and check if it’s 0

[tardy fog]

Thats one way yes

integrate omega over some region and hope it isnt 0

that requires finding such a region though

alternatively you know $\dd^2{\varphi}=0$, so exact implies closed

[chrome yew]

Omegabet_

[tardy fog]

hence not closed means not exact

so if $\dd{\omega}\neq 0$, $\omega$ isnt exact

[chrome yew]

Omegabet_

[tardy fog]

You can also invoke Poincare's Lemma if you've covered it (forms defined on R^n are exact iff closed)

[final bronze]

Oh ok

Thanks

What about part 2

[tardy fog]

apply Stoke's

exact forms integrate to 0

[final bronze]

Oh ok

+close