Let \(\mathbf{F}=\left\langle y,-x, x^{2}+y^{2}ightangle\), and let \(\mathcal{S}\) be the portion of the paraboloid \(z=x^{2}+y^{2}\) where \(x^{2}+y^{2} \leq 3\).
(a) Show that if \(\mathcal{S}\) is parametrized in polar variables \(x=r \cos \theta, y=r \sin \theta\), then \(\mathbf{F} \cdot \mathbf{N}=r^{3}\).
(b) Show that \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}=\int_{0}^{2 \pi} \int_{0}^{3} r^{3} d r d \theta\) and evaluate.