Calculate the circulation of the vector $\vec{A} = (2 + y)\vec{i}_x$ along the path designated by the edges (length l) of a square, two of which lie on the positive axes x and y. Choose right-handed circulation around the z-axis.
What I believe is we need to find $\nabla$x $\vec A$, the curl of vector $\vec A$. I'm new to the vector field analysis.
I believe you are asked to compute the line integral: $$ \oint_C \vec{A} . d \vec{x},$$ where $C$ is the square with vertices $(0,0,0), (l,0,0), (l,l,0), (0,l,0)$.
You can calculate this explicitly, one edge at a time. For example, the edge from $(0,0,0)$ to $(l,0,0)$ can be parametrised as $$ (x,y,z) = (t,0,0), \ \ \ \ \ \ 0 \leq t \leq l.$$ So the vector field along this edge is $$ \vec{A} = 2 \vec{e}_x,$$ and the line element along this edge is $$ d \vec{x} = \vec{e}_x dt.$$ Therefore, $$ \int_{(0,0,0) \mapsto (1,0,0)} \vec{A}. d \vec{x} = \int_0^l (2 \vec{e}_x).(\vec{e}_x dt ) = \int_0^l 2 dt = 2l.$$ I'll leave you to evaluate the integral along the other three edges.
Alternatively, if you wish, you can use Stoke's theorem, which says: $$ \oint_C \vec{A}. d \vec{x} = \int_A (\nabla \times \vec{A}) . \vec{\hat n} d^2 S.$$
Here, $A$ is the square $$0 \leq x \leq l, \ \ \ \ 0 \leq y \leq l,$$ and $\vec{\hat n}$ is the unit normal to the square, $$\vec{\hat n} = (0,0,1).$$
Perhaps you could try evaluating this both ways. You should get the same answer!