Calculate the flow rate of a fluid with velocity field \(\mathbf{v}=\left\langle x, y, x^{2} yightangle\) (in meters per second) through the portion of the ellipse \(\left(\frac{x}{2}ight)^{2}+\left(\frac{y}{3}ight)^{2}=1\) in the \(x y\)-plane, where \(x, y \geq 0\), oriented with the normal in the positive \(z\)-direction.