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Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$ with upward orientation. The vector field is $\mathbf{f}(x,y,z)=\langle xy, 4x^2, yz \rangle$.

Using the surface integral to evaluate flux,

$$\iint_{\Sigma} \mathbf{f} \cdot d\sigma= \iint_R f(x(u,v)),y(u,v),z(u,v)) \left|\left| \frac{\partial r}{\partial u} \times \frac{\partial r}{\partial u} \right|\right| du dv$$

What would $u$ and $v$ be? I'm not sure how to parametrize this.

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$\Sigma$ is the graph of the function $z = xe^y$. That is $(x,y,z) = (x,y, \varphi(x,y)) = G(x,y)$. So \begin{equation} \int_{\Sigma}\mathbf{f}\cdot d\sigma = \int_{x}\int_{y}f(G(x,y))\cdot \left(\frac{\partial G}{\partial x}\times \frac{\partial G}{\partial y}\right)dx dy \end{equation} What should $\varphi$ be? Once you have determined that and the bounds of integration for $x$ and $y$, the computation of the integral follows as usual.

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