A spherical hot-air balloon with radius $R$ lies on the $(x_1,x_2)$-Plane.
The balloon has an opening at the bottom right where the radius is $\frac{R}{4}$. The center of the opening is at the origin.
The velocity of the hot air leaving the balloon is given by $\vec{v} = \vec{\nabla} \times \vec{w}$, where $\vec{w} = (-x_2, x_1,0) ms^{-1}$
My question is, what does it mean exactly that the velocity is defined by the curl of a vector field?
How do I find rate of the volume of gas lost?
Stokes' Theorem should somehow help, but I fail to identify the elements to plug into the theorem.
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It's a physics problem, but since my doubts are purely of mathematical nature, I think it's appropriate to have it here.
Hint: the volumetric flow rate is given by integrating velocity over the area so it’s given by $\iint_R v \cdot n \, dA = \iint_R (\nabla \times w) \cdot n \, dA$. By Stokes theorem this is $\oint_{\partial R} w \cdot T \, ds$, where $\partial R$ is just the circle with radius $\frac{R}{4}$ centred at the origin. This makes the assumption that the opening lies in the $xy$-plane but without further information, I think that’s the best assumption we can make.