Fundamental Theorem for Line Integrals

The central idea

If $f$ has continuous gradient on a region containing a piecewise $C^1$ curve $C$ from $A$ to $B$, then $\int_C\nabla f\cdot d\mathbf r=f(B)-f(A)$. Thus gradient fields are path independent. Conversely, path independence on a connected open region yields a potential under standard hypotheses; zero curl is sufficient on simply connected domains.

Definitions, hypotheses, and notation

Potential functions are unique up to a constant on a connected domain. A component-by-component construction must retain an unknown function of the variables held constant during integration. Dropping that function can produce a candidate whose other partial derivatives fail.

Topology matters for the curl test: $\langle-y/(x^2+y^2),x/(x^2+y^2)\rangle$ has zero scalar curl away from the origin but nonzero circulation around loops enclosing the hole. The fundamental theorem is safe once an actual global potential has been exhibited on the relevant region.

Conceptual meaning

Along a parametrized path, the chain rule gives $d[f(\mathbf r(t))]/dt=\nabla f\cdot\mathbf r'$. Integrating collapses all intermediate changes to endpoint difference. Closed-loop work is therefore zero for a globally conservative field.

A dependable method and decision rule

1. Identify or construct a potential $f$ with $\nabla f=\mathbf F$.
2. Check domain conditions and that the curve stays within the potential's region.
3. Verify every component of the gradient.
4. Evaluate $f$ at terminal point minus initial point.
5. Use path independence only after conservativeness is justified.

Fully worked example

Common mistakes and why they fail