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The displayed vector calculus identity follows by contracting the two alternating tensors in the component formula for curl. If both fields have zero divergence, only the two directional derivatives remain. It rewrites the transport terms in the adjoint linearized Navier-Stokes evolution.

For a smooth differential one-form $\alpha ={\alpha }_i(x)d{x}^i$ with $d\alpha =0$, take a manifold chart identifying a neighborhood with a ball centered at zero, and set $\Phi (x)={\int }_0^1{x}^i{\alpha }_i(sx)ds$. Closedness means ${\partial }_j{\alpha }_i={\partial }_i{\alpha }_j$. ConsequentlyThus $d\Phi =\alpha$ in any dimension. This is the one-form case of the Poincare lemma, and applies to spacetime as well as three-dimensional vector calculus. A global potential need not exist on a domain with nonzero periods.

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 isSince zero curl means ${\partial }_i{E}_j={\partial }_j{E}_i$, differentiation under the integral givesThus $E=-\nabla \varphi$. This is the Poincare 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 ${\varphi }_y=y{e}^{-x^{2}z}$. Integration in $y$ gives $\varphi =\frac{1}{2}{y}^2{e}^{-x^{2}z}+C(x,z)$. Its $x$ derivative is $-x{y}^{2}z{e}^{-x^{2}z}+C_x$, and its $z$ derivative is $-\frac{1}{2}{x}^2{y}^2{e}^{-x^{2}z}+C_z$. Comparison with the other two components forces $C_x=C_z=0$. HenceIts negative gradient is exactly the given field. Because this vector-field potential is smooth everywhere, commutation of mixed partial derivatives gives $\nabla \times E=-\nabla \times \nabla \varphi =0$, verifying irrotationality. All such vector-field potentials differ by a constant on the connected domain.