Fundamental Theorem for Line Integrals

For a conservative field, a line integral collapses to a two-number subtraction: $\int_C \nabla f\cdot d\mathbf{r} = f(B)-f(A)$ , depending only on endpoints.

The Fundamental Theorem for Line Integrals needs $\nabla f$ continuous along $C$ , but nothing about $C$ itself beyond piecewise smoothness — it can loop, self-intersect, or double back, and the conclusion $\int_C \nabla f\cdot d\mathbf{r} = f(B)-f(A)$ is unaffected as long as the actual start and end points are $A$ and $B$ . In particular, if $C$ is closed ( $A=B$ ), the right-hand side is $f(A)-f(A)=0$ automatically, no matter how convoluted the loop is.

Examples

For $\mathbf{F}=(2x,2y)=\nabla f$ with $f(x,y)=x^2+y^2$ , and any curve $C$ from $(0,0)$ to $(1,1)$ : $\int_C \mathbf{F}\cdot d\mathbf{r} = f(1,1)-f(0,0) = 2$ , regardless of the curve's shape.