A counterexample to path independence

To calculate the line integral of $\vec{F} = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$ along a semicircle of radius $R$ centered at the origin, parameterize the path using polar coordinates. For the upper semicircle from $(-R, 0)$ to $(R, 0)$, use $x = R\cos t, y = R\sin t$ with $t$ going from $\pi$ to $0$.

Conditions: The path is a semicircle centered at the origin.; The radius $R$ is constant.; The field is the standard vortex field.

The corrected theorem states that for a vector field $\vec{F} = (P, Q)$ with continuous first partial derivatives, the line integral is path-independent if and only if the domain $D$ is simply connected AND $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$ holds throughout $D$. Simple connectivity ensures that every closed curve can be continuously shrunk to a point, eliminating topological obstructions like holes where circulation could persist.

Conditions: The domain $D$ is open and connected.; The functions $P$ and $Q$ have continuous first partial derivatives on $D$.; $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$ for all points in $D$.; $D$ is simply connected.

The value of the closed loop integral for the vortex field $\vec{F} = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$ around any simple closed curve enclosing the origin once in the counterclockwise direction is $2\pi$. This result is independent of the shape or size of the loop, depending only on the winding number around the singularity at the origin.

Conditions: The curve is simple, closed, and oriented counterclockwise.; The curve encloses the origin exactly once.; The field is the standard 2D vortex field.

The equality of cross-partial derivatives is a necessary condition for path independence, but it is not sufficient when the domain is not simply connected. For the vortex field $\vec{F} = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$, the partial derivatives match perfectly everywhere except at the origin.

Conditions: The vector field is defined on a domain excluding the origin.; The domain is multiply connected (has a hole).; The cross-partial derivatives are equal wherever defined.

The naive application assumes that $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$ is sufficient for path independence. For the vortex field, this condition holds everywhere in the domain.

Conditions: The vector field satisfies $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$.; The domain is not simply connected.; Two distinct paths connect the same endpoints.

A vector field is conservative if and only if its line integral over every closed loop is zero. While the vortex field $\vec{F}$ has zero curl ($\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0$) everywhere it is defined, the domain excludes the origin.

Conditions: The field is defined on $\mathbb{R}^2 \setminus \{(0,0)\}$.; The curl is zero everywhere in the domain.; There exists a closed loop in the domain with non-zero circulation.

A simply connected domain is a connected region where every closed curve can be continuously contracted to a single point without leaving the region. Intuitively, it has no 'holes'.

Conditions: The domain is open and connected.; Comparison involves the ability to contract closed curves.

Green's Theorem relates a line integral around a simple closed curve $C$ to a double integral over the plane region $D$ bounded by $C$. The theorem requires that the vector field be defined and have continuous partial derivatives on an open region containing $D$.

Conditions: The curve $C$ is simple, closed, and positively oriented.; The region $D$ is bounded by $C$.; The vector field must be $C^1$ on an open set containing $D$.; $D$ must not contain singularities of the field.