Recent questions in Vector Calculus

​​Let $C$ be the positively oriented boundary of the domain bounded by the curves $y=2 x^{2}$ and $y^{2}=4 x$. Then the value of the line integral$$\oint_{C}\left(2 y^{2}...

Consider the vector field $\vec{F}=\left(2 x+y^{2}\right) \hat{\imath}+(2 x y+3 y) \hat{\jmath}$ and $\alpha_{m}=\int_{C_{m}} \vec{F} \cdot d \vec{r}$, $m=1,2$, where $C_...

The value of $m$ for which the vector field$\vec{F}(x, y)=\left(4 x^{m} y^{2}-2 x y^{m}\right) \hat{\imath}+\left(2 x^{4} y-3 x^{2} y^{2}\right) \hat{\jmath}$is a conserv...