= Solution
{c}
The middle component suggests a <scalar potential> with $-\phi_y=2y\cosh^2(xz)$, so $\phi=-y^2\cosh^2(xz)+h(x,z)$. The <chain rule> and $2\sinh t\cosh t=\sinh(2t)$ give
$$
-\phi_x=y^2z\sinh(2xz)-h_x,\qquad
-\phi_z=y^2x\sinh(2xz)-h_z.
$$
Matching the other two components requires $h_x=h_z=0$. Therefore
$$
\boxed{\phi(x,y,z)=-y^2\cosh^2(xz)+C.}
$$
Its negative <gradient> is the given <vector field> in all three components; the additive constant has no effect on that <gradient>.
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