A counterexample to path independence

The video exposes a hidden topological prerequisite in the theorem of path independence for line integrals. It starts with the standard condition ($\partial Q/\partial x = \partial P/\partial y$) and tests it against a rotational vector field undefined at the origin. Although the partial derivatives match perfectly, calculating integrals along two different semi-circular paths yields contradictory results ($\pi$ vs $-\pi$), and the closed loop integral is non-zero ($2\pi$). The video concludes by correcting the theorem statement: path independence holds only in simply connected domains. A visual comparison between a solid disk (simply connected) and an annulus (multiply connected) illustrates why the presence of a singularity ('hole') invalidates the naive application of the theorem.

Before you watch

• Multivariable Calculus
• Partial Derivatives
• Line Integrals of Vector Fields
• Topology Basics (Connectedness)