= Solution
On the simply connected domain $\mathbb R^3$, a continuously differentiable <differential one-form> is <exact differential>[exact] if its mixed partial derivatives agree, equivalently if its associated vector field has zero <curl>. Here one can verify exactness directly by observing that
$$
F(x,y,z)=(x^2+z^2)e^{(x+y)z}
$$
satisfies
$$
\frac{\partial F}{\partial x}=u,
\qquad
\frac{\partial F}{\partial y}=v,
\qquad
\frac{\partial F}{\partial z}=w.
$$
Thus $u\,dx+v\,dy+w\,dz=dF$. By the <fundamental theorem for line integrals>, its integral is <path independent> and equals
$$
F(1,0,1)-F(-1,0,1)=2e-2e^{-1}.
$$
Consequently, for every path with the stated endpoints,
$$
\boxed{\int_{(-1,0,1)}^{(1,0,1)}(u\,dx+v\,dy+w\,dz)=4\sinh1}.
$$
Back to article page
