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

Graphical or geometric meaning

A potential is a height landscape, and the field is its steepest-ascent gradient. Work along any route records total height change; detours contribute rises and falls that cancel.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate the potential and recover the full field.
• Compute a simple alternate-path integral.
• Verify reversing endpoints negates the work.

Verify potential and domain before using endpoints

If $\mathbf F=\nabla\phi$ on a domain containing the path, then $\int_C\mathbf F\cdot d\mathbf r=\phi(B)-\phi(A)$. Find a potential by integrating one component and matching the others, keeping “constants” that may depend on remaining variables. Equality of cross-partials is a useful local test under continuity assumptions, but on a domain with holes it may not guarantee a global potential. Endpoint subtraction applies only after conservativeness on the relevant domain is established. Reversing the path swaps endpoints and changes the sign. For a closed path in a conservative region, the integral is zero; a nonzero closed-loop integral therefore disproves the existence of a single-valued potential on that region.

Practice

1. Find a potential for $\langle y,x\rangle$.
2. Find its work from $(0,0)$ to $(2,3)$.
3. What is work around a closed loop in a conservative field?

Connections and next steps

• Line Integrals
• Gradient
• Green's Theorem