Flux across a hemisphere, with and without the Divergence Theorem, Multivariable Calculus
We compute the vector surface integral for F = (y,x,z) across the upper hemisphere of radius 3 in two different ways. First, we compute the flux directly using the vector surface integral process: we parametrize the surface, use a cross product to compute the orthogonal vector field generated by the parametrization, set up and evaluate the double integral for the flux, etc.
Then, we rework the problem using the Divergence Theorem, which is a simpler computation, which requires enclosing the region fully. We add a "lid" to the hemisphere to form a closed solid region. By computing the divergence of the vector field and integrating it over the solid region, we ultimately find the volume of the enclosed region.
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