I recently started learning multivariable calculus and I came across some questions regarding conservative vector fields and it's a bit confusing.
If I took the derivative of my potential function (as in $\nabla f(x,y,z)$), it will give me a vector field. But if I integrated my vector field, would it give me the potential function as well, or would it give me a completely different function? If I encounter a problem where I have to find the potential function of a conservative vector field, should I approach the problem as an initial value condition problem except with multiple independent variables?
First of all, I can't imagine the problem graphically. What does it mean for a vector field to be conservative? I don't get why my professor said it has to be a closed path in order to find the potential. What's the meaning of this symbol $\oint$? How does this relate to other real-world applications? I don't really understand much of this at all.
Note: I don't have a good background in physics, so it doesn't really help to say "know your physics" as others have said to me. And I just started learning line integrals, so I would really love some insight that allows to me view the problem more easily.