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A line integral measures how a vector field contributes along a curve from one point to another.

The Fundamental Theorem for Line Integrals applies when the vector field comes from a potential function.

In other words, the vector field F is the gradient of a potential function g of two or three variables.

The theorem states that, for such a field, the line integral along a smooth curve depends only on the values of the potential function at the endpoints.

So, if two different smooth curves, C_1 and C_2, connect the same two points, the line integral is the same on both curves.

As a result, the line integral becomes path-independent, simplifying complex calculations.

In physics, this theorem applies to conservative force fields such as gravity. When an object moves through such a field, the work done by gravity equals the line integral of the gravitational force along its path. Because gravitational force equals the negative gradient of a potential energy function, the work depends only on the initial and final positions.

For example, whether a ball falls straight down or follows a curved path, the work done is the same.