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pretty basic question but I can't seem to work it out:

Question:

Let S be the triangle with vertices $\mathbf{a}=(1,2,3),\mathbf{b}=(1,1,1),\mathbf{c}=(3,1,2)$ with unit normal chosen such that the third component of the unit normal is positive. Let

$$ \mathbf{F}=y^{2}\mathbf{i}+x\mathbf{j} $$

(i) Find a parametric representation of S

(ii) Evaluate the flux integral $\int_{S}\mathbf{F.}d\mathbf{S}$

Solution (attempt):

I have parametrized the triangle as

$$ \mathbf{S}=(1,1,1)+u(-2,-1,0)+v(2,0,1)\text{ with }0\leq u\leq1,0\leq v\leq1-u $$

I have also found the normal vector (with appropriate orientation) to be

$$ \mathbf{N}=(2,-4,2). $$

Now, I want to evaluate the iterated integral:

$$ \intop_{0}^{1}\intop_{0}^{1-u}\mathbf{F\cdot N}dvdu $$

but I am unsure how to change $\mathbf{F}$ into a function of $u$ and $v$.

Any help would be appreciated.

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Your coefficient of $u$ should be $(0,1,2)$, your normal vector is not correct either. It should be $\vec{r}_u \times \vec{r}_v$.

To write $F$ in terms of $u,v$, notice that from your triangle

$$x=1+2v,y=1+u,z=1+2u+v$$

You can plug these into $F$.

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