Transform the surface integral $\int_S \text{rot}\vec{F}\cdot \vec{dS}$ in a line integral using the Stokes Theorem and then calculate the line integral for: $\vec{F}(x,y,z)=(y,z,x)$, where $S$ is the part of the paraboloid $z=1-x^2-y^2$ with $z\ge 0$ and the normal vector has the non-negative $z$ component. It $\vec{F}(x,y,z)$ is conservative.


$\text{rot}\vec{F}= (-1,-1,-1)$.

And how should I understand the expression $\vec{dS}$ and what does it mean that $\vec{F}(x,y,z)$ is conservative?


The expression $$\int_S\operatorname{rot}(\vec{F})\cdot d\vec{S}$$ simply means $$\int_S(\operatorname{rot}(\vec{F})\cdot \hat{n})\, dS$$ where $\hat{n}$ is the unit vector normal to the surface.

A conservative vector field is a vector field such that any line integral $$\int_L \vec{F}\cdot d\vec{l}$$ is identically zero whenever $L$ is a closed loop.


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