Problem Zero: Read the section and make your own summary of the material.

Given is the text and step 2 contains the summary.

$$\begin{gathered} 1.\mathit{S}\mathit{u}\mathit{r}\mathit{f}\mathit{a}\mathit{c}\mathit{e}\mathbf{}\mathit{A}\mathit{r}\mathit{e}\mathit{a}\mathbf{:}ThesurfaceareaofasmoothsurfaceSis \\ {\int }_{S}1dS. \\ \left(a\right)IfSisgivenbyz=f(x,y)for(x,y)\in D\subseteq {\mathrm{ℝ}}^2,then \\ dS=\sqrt{{\left(\frac{\partial z}{\partial x}\right)}^2+{\left(\frac{\partial z}{\partial y}\right)}^2+1}dA. \\ \left(b\right)IfSisparametrizedby\mathit{r}(u,v)for(u,v)\in D,then \\ dS=\left|\left|{\mathit{r}}_u\mathbf{\times }{\mathit{r}}_v\right|\right|dA=\left|\left|{\mathit{r}}_u\mathbf{\times }{\mathit{r}}_v\right|\right|dudv. \end{gathered}$$

$$\begin{gathered} 2.\mathit{S}\mathit{u}\mathit{r}\mathit{f}\mathit{a}\mathit{c}\mathit{e}\mathbf{}\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{r}\mathit{a}\mathit{l}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{a}\mathbf{}\mathit{M}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{v}\mathit{a}\mathit{r}\mathit{i}\mathit{a}\mathit{t}\mathit{e}\mathbf{}\mathit{F}\mathit{u}\mathit{n}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{:} \\ Theintegraloff(x,y,z)overasmoothsurfaceSis \\ {\int }_{S}f(x,y,z)dS=\int {\int }_{D}f\left(x\right(u,v),y(u,v),z(u,v\left)\right)\left|\left|{\mathit{r}}_u\mathbf{\times }{\mathit{r}}_v\right|\right|dA \\ =\int {\int }_{D}f\left(x\right(u,v),y(u,v),z(u,v\left)\right)\left|\left|{\mathit{r}}_u\mathbf{\times }{\mathit{r}}_v\right|\right|dudv, \\ whereDisthedomainofthesmoothparametrizationofSby\mathit{r}(u,v). \end{gathered}$$

$$\begin{gathered} 3.\mathit{F}\mathit{l}\mathit{u}\mathit{x}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{a}\mathbf{}\mathit{V}\mathit{e}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\mathbf{}\mathit{F}\mathit{i}\mathit{e}\mathit{l}\mathit{d}\mathbf{}\mathit{A}\mathit{c}\mathit{r}\mathit{o}\mathit{s}\mathit{s}\mathbf{}\mathit{a}\mathbf{}\mathit{S}\mathit{u}\mathit{r}\mathit{f}\mathit{a}\mathit{c}\mathit{e}\mathbf{:} \\ If\mathit{F}(x,y,z)isacontinuousvectorfielddefinedonanorientedsurfaceS \\ givenby\mathit{r}(u,v)for(u,v)\in Dwithunitnormalvector\mathit{n},thenthe\mathbf{}\mathit{f}\mathit{l}\mathit{u}\mathit{x}of \\ \mathit{F}(x,y,z)throughSis \\ {\int }_S\mathit{F}(x,y,z).\mathit{n}\mathbf{}dS. \\ \left(a\right)IfSisthegraphofz=z(x,y),then \\ {\int }_S\mathit{F}(x,y,z).\mathit{n}\mathbf{}dS=\int {\int }_D\left(\mathit{F}(x,y,z).\mathit{n}\right)\left|\left|z_x\times z_y\right|\right|dydx. \\ \left(b\right)IfSisparametrizedbyuandv,then \\ {\int }_S\mathit{F}(x,y,z).\mathit{n}\mathbf{}dS=\int {\int }_D\left(\mathit{F}(x,y,z).\mathit{n}\right)\left|\left|{\mathit{r}}_u\times {\mathit{r}}_v\right|\right|dudv. \end{gathered}$$