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mathmari

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I want to calculate $\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )$ using the fomula of Stokes, when $\sigma$ is the curve that is defined by the relations $x^2+y^2=1$ and $x+y+z=1$.

Is the curve not closed? Because we have an integral of the form $\int_{\sigma}$ and not of the form $\oint_{\sigma}$ ? (Wondering)

We have that $$\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )=\int_{\sigma}\left (-y^3, x^3, -z^3\right )\cdot \left (dx, dy, dz\right )=\int_{\sigma}f\cdot d\sigma$$ with $f(x,y,z)=\left (-y^3, x^3, -z^3\right )$, right?

From the formula of Stokes we have that $$\int_{\sigma} f\cdot d\sigma=\iint_{\Sigma}\left (\nabla \times f\right )\cdot N\ dA$$ Which is the surface $\Sigma$ ? Do we take the intersection of the given relations $x^2+y^2=1$ and $x+y+z=1$ ? (Wondering)