So I was applying some mathematical techniques I learned to physics, and one thing that captured my interest, is the power or luminosity flux of a star. So modeling the situation, taking the scalar field of Luminosity($L$), and the vector surface(stellar surface), $S$, I began by setting up a Surface (Flux, in this case) integral of
$$ \iint_{S}L\,dS $$
Where,
$$ S=\langle r\sin\theta \cos\phi,r\sin\theta \sin\phi,r\cos\theta \rangle \;,\; \theta \, \epsilon \, [0,\pi] \wedge \phi \, \epsilon \, [0,2\pi] $$ $$ L = 4\pi r^{2}\sigma T^{4} $$
$r$ in this model is the invariant stellar radius. Following logic, since $L$ has no explicit dependency on $ \theta $ or $\phi$, I rearranged as:
$$ 4\pi r^{2} \sigma T^{4} \iint_{S} dS $$
Since the integral of dS over S is merely surface area, and our surface is a sphere of radius $ r $, this expression reduces to:
$$ 16\pi^{2} r^{4} \sigma T^{4} $$
The dimensional analysis for this breaks down to $ W \cdot m^{2} $. Isn't flux supposed to be quantity per area?