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Consider the following parameterization $φ$ of the parametric surface $S = φ(R)$ with$$ R = [0,2π] × [0,1] ⊂ \mathbb{R}^2, $$\begin{align*} φ:&& R &\longrightarrow \mathbb{R}^3\\ && (u,v) &\longmapsto (x(u,v),y(u,v),z(u,v)) = (v \cos u, v \sin u, v^2), \end{align*} and the vector field\begin{align*} \boldsymbol{F}: && \mathbb{R}^3 &\longrightarrow \mathbb{R}^3\\ && (x,y,z) &\longmapsto (x,y,x+y+z). \end{align*}

  1. Find a unit normal $\boldsymbol{n}$ on $S$ from the given parameterization.
  2. With the orientation given by $\boldsymbol{n}$, compute both sides of Stoke's theorem for the vector field $\boldsymbol{F}$. (We have to remember here that the orientation of $∂S$ is given by the condition that $\boldsymbol{n} × \boldsymbol{T}$ points towards the surface, where $\boldsymbol{T}$ is the unit tanget vector of $∂S$ defining its orientation.
  3. Let $W = \{ (x,y,z) \mid x^2 + y^2 \leqslant z,\ z ∈ [0,1]\}$ be the solid enclosed by $S$ and the disk $D = \{ (x,y,1) ∈ \mathbb{R}^3 \mid x^2 + y^2 \leqslant 1 \}$. For the vector field $\boldsymbol{F}$ and the solid $W$ verify the divergence theorem.
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    $\begingroup$ Don’t apologize; learn. Go check a few other posts. You can click edit and see the code used there to produce the desired symbol(s). Otherwise, you can look up LaTeX. Not all commands are the same here (thisbis MathJax, but the most relevant commands do apply). $\endgroup$ – Clayton Jun 5 '18 at 5:30
  • $\begingroup$ Here’s the MathJax tutorial and here’s how to ask a good question. Please continue to contribute to our wonderful site! $\endgroup$ – gen-z ready to perish Jun 5 '18 at 5:55
  • $\begingroup$ Please show your try at the question so that the community can help you :) $\endgroup$ – Saad Jun 5 '18 at 6:45
  • $\begingroup$ I have no idea how to input my attempt $\endgroup$ – Z. Aslam Jun 5 '18 at 7:39

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