= Solution
Assume the continuously differentiable <irrotational vector field> is defined on all of $\mathbb R^3$, as in this <vector calculus> setting. A constructive <vector-field potential> with the requested negative-gradient convention is
$$
\phi(x)=-\int_0^1 x_jE_j(tx)\,dt.
$$
Since zero <curl> means $\partial_iE_j=\partial_jE_i$, differentiation under the integral gives
$$
\begin{aligned}
\partial_i\phi
&=-\int_0^1\left[E_i(tx)+t x_j\partial_iE_j(tx)\right]dt\\
&=-\int_0^1\frac d{dt}\left[tE_i(tx)\right]dt=-E_i(x).
\end{aligned}
$$
Thus $E=-\nabla\phi$. This is the <Poincaré lemma> in its straight-line form, and works on any <star-shaped set> about the integration base point. Zero <curl> on a domain with holes would not by itself guarantee a global <vector-field potential>; the full-space hypothesis matters.
For the given field, the second component requires $\phi_y=ye^{-x^2z}$. Integration in $y$ gives $\phi=\tfrac12y^2e^{-x^2z}+C(x,z)$. Its $x$ derivative is $-xy^2ze^{-x^2z}+C_x$, and its $z$ derivative is $-\tfrac12x^2y^2e^{-x^2z}+C_z$. Comparison with the other two components forces $C_x=C_z=0$. Hence
$$
\boxed{\phi(x,y,z)=\frac12y^2e^{-x^2z}+C.}
$$
Its negative <gradient> is exactly the given field. Because this <vector-field potential> is smooth everywhere, commutation of mixed <partial derivatives> gives $\nabla\times E=-\nabla\times\nabla\phi=0$, verifying irrotationality. All such <vector-field potentials> differ by a constant on the connected domain.
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