= Solution
At every regular point of the <double circular cone>, its normal is perpendicular to $\mathbf F$ by part (a). Therefore the integrand of the <flux integral> vanishes pointwise:
$$
\boxed{\int_S\mathbf F\cdot d\mathbf S=0.}
$$
The vertex, if included, is a single point of zero surface area and does not alter the <surface integral>. The answer is independent of both the subset and its orientation. Tangency, rather than a cancellation between different parts of the surface, explains the zero flux.
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