0
$\begingroup$

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.

First,

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

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

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.