# Second Moment of Inertia using Green's Theorem

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Green's Theorem: double intergral ( curl vector(F) dA ) = closed integral ( vector(F) dot vector(ds) ) What I need is double integral ( x^2 + y^2 dA ), right? Knowing that vector(F) = <F1, F2>.. and applying green's theorem to this problem we get: dF2/dy - dF1/x = x^2 + y^2. I just chose F1 = 0. So F2 = x^2*y + y^3/3. My areas are polygonal, so I am just going to parameterize the boundaries into segments and from there it should go alright. So my question is: Is my work correct?

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You're not going to get too far using two dimensional vectors with the curl operator. Unless, of course, you want to get an answer that has only a Z component.

Or were you just being brief?

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yeah brief.. it's actually curl(F) dot <0,0,1>

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OK. And just to double check, we're talking about plane deformation and not rotation?

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What rotation/deformation.. If u're talking about curl operator.. Well it's a differential operator. curl( F1, F2 ) = dF2/dx - dF1/dy

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Ah, no. Your topic is about the second moment of inertia (aka second moment of area) but the math looks like it's for the moment of inertia. In either case, can I assume the axis of rotation is k?

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Ah alright, got u.. It's for rotation.. and axis of rotation is indeed k.

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Alright. Last question: Are the polygons solid?

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Yes..
But in later stages in code, there's gonna be some finite element stuff, so let's build on the simple stuff first, and then I'll see how to deal with the changing stuff.

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