Compute $∫_C −e^y\sin(x) dx + e^y\cos(x) dy + dz$, where $C$ is an arbitrary smooth path from $(0, 0, 1)$ to $(π, π, 0)$. Make sure to check you satisfy the hypotheses of any theorems you use.
I went about this through parameterization, however, I got an integral that was rather complicated which made me think there may be a better way to go about this.
The function $f(x,y,z) = e^y\cos x+z$ is a potential for the field $(-e^y\sin x,e^y \cos y, 1)$, so by the fundamental theorem of calculus we have $$\int_C -e^y\sin x\,{\rm d}x + e^y\cos x\,{\rm d}y + {\rm d}z = f(\pi,\pi,0)-f(0,0,1).$$