Solving a complex line integral via Stokes' Theorem

Question

I haven't a clue how to set up my double integral to solve this question

The answer, according to the textbook, is:

Since $curl F = 1/(1+y^2)i + 2ze^{x^2}j + y^2k$

Ummm... What? Am I missing something? Where did that come from?!

Answer 1

Compare $$\oint z^2e^{x^2}dx+xy^2dy+\arctan y \, dz$$ with $$\oint F_x \,dx+F_y \,dy+F_z \,dz$$ to see that the vector field in this case is given by $\vec F = \left( z^2e^{x^2},xy^2,\arctan y\right)$.

Now compute the curl, e.g. via $$\mbox{curl} \vec F =\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \tfrac{\partial}{\partial x} & \tfrac{\partial}{\partial y} & \tfrac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \tfrac{\partial}{\partial x} & \tfrac{\partial}{\partial y} & \tfrac{\partial}{\partial z} \\ z^2e^{x^2} & xy^2 &\arctan y \end{vmatrix} = \cdots$$