Mathematics: Walk me through the steps on how to calculate the volume

I m not sure what to make of the h and i in your question, so I ll just ignore them (and the 3). Looks like we haveand a surface parameterzied bywith and . Thenso that the flux is given by the integral...

Walk me through the steps on how to calculate

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gabrieljerron

19.05.2021

Mathematics

I'm not sure what to make of the "h" and "i" in your question, so I'll just ignore them (and the 3). Looks like we have

$\vec f(x,y,z)=(x+y,x-y,x^2+y^2-2z)$

and a surface parameterzied by

$\varphi(u,v)=(u+2v,u-2v,u^2+2v^2)$

with $0\le u\le1$ and $0\le v\le1$ . Then

$\varphi_u\times\varphi_v=(4u+4v,4u-4v,-4)$

so that the flux is given by the integral

$\displaystyle\iint_S\vec f\cdot\mathrm d\vec S=\int_0^1\int_0^1(2u,4v,4v^2)\cdot(4u+4v,4u-4v,-4)\,\mathrm du\,\mathrm dv$

$=\displaystyle8\int_0^1\int_0^1(u^2+3uv-4v^2)\,\mathrm du\,\mathrm dv=\boxed{-2}$

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Walk me through the steps on how to calculate the volume

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