Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces

[Falgun]

I have seen conservative vector fields being defined as satisfying either of the two following conditions:

1. The line integral of the vector field around a closed loop is zero.
2. The line integral of the vector field along a path is the function of the endpoints of the curve.

It is apparent to me how 2 implies 1 but what I cant understand is how 1 implies 2?

More specifically why is ∮F⃗ .d⃗ r =𝑓(𝑏)−𝑓(𝑎) for some scalar function f?

Why not something like f(a⃗ .b⃗ ) instead?

Additionally when we calculate a line integral we do it assuming a Euclidean metric. How would the line integral be modified while working with a different metric say the Minkowski metric?

 

[renormalize]

1. The line integral of the vector field around a closed loop is zero.
2. The line integral of the vector field along a path is the function of the endpoints of the curve.

It is apparent to me how 2 implies 1 but what I cant understand is how 1 implies 2?

Apply Stokes theorem to the zero closed-loop integral and conclude that the curl of the vector-field must vanish, i.e., the vector must be the gradient of a scalar field, the integral of which depends only on the endpoints.

 

[Falgun]

Apply Stokes theorem to the zero closed-loop integral and conclude that the curl of the vector-field must vanish, i.e., the vector must be the gradient of a scalar field, the integral of which depends only on the endpoints.

How do we show that a vector field whose curl is zero is necessarily a gradient field?

 

[renormalize]

How do we show that a vector field whose curl is zero is necessarily a gradient field?

See, e.g., this discussion: https://people.math.harvard.edu/~knill/teaching/summer2004/handouts/conservative.pdf