Does fundamental theorem of calculus apply to closed curves? If yes why the closed line integral of a function is not zero?

Question

Why closed curve integral $$\oint A.dl$$ doesn't give us zero in many physics related examples?

Closed line/loop/curve/contour have same starting and end points and according to fundamental theorem of calculus we have $$\int_{a}^{b}g'=g(b)-g(a)$$

Doesn't the above relation tell us that integral of a function along closed line must be zero? What I'm missing?

My secondary question is how does this relate to analytic function or contour integral?

Answer 1

In the first integral, $\oint A \cdot \mathrm d l$, you are integrating over a vector field. In the second integral, $\int_a^b g' \mathrm d x$, you are integrating over a scalar field. The fundamental theorem of calculus only applies to scalar fields. However, there is a special subset of vector fields called "conservative vector fields" which sort of obey the fundamental theorem of calculus. You can look up more about them here: https://en.wikipedia.org/wiki/Conservative_force