I have a fundamental question regarding path independence that should help me to better connect the dots from multivariable calculus to vector calculus.
Given some arbitrary curve $C$ and function $f(x,y,z)=(g(x,y,z),0,0)$ where $g: \mathbb{R}^3\to\mathbb{R}$. If I want to evaluate the line Integral $\mathop{\int}\limits_{C} f \,d(x,y,z)$, could I directly conclude that the integral has to be path dependent. From a purely geometric interpretation, I would assume it is impossible to have path independence. Further, I believe we could not find $\varphi$ such that $f=grad$ $\varphi$ simply because $f$ has only one non zero component.
Please correct me if I am wrong. I would appreciate any help in order to get a better grasp on how to geometrically interpret this situation.
The condition of grad(phi) comes from exact differential,that is if f=Mdx+Ndy and if del(M)/del(y)=del(N)/del(x),only then f can be written as a gradient fuction,so check that first.