Curl of a 2D Vector Field

Curl in 2D

For a 2D vector field $\mathbf{F}(x,y) = (F_x(x,y),\, F_y(x,y))$, the curl (scalar curl, or $z$-component of the 3D curl) measures rotation:

$\text{curl}\,\mathbf{F} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}$

Interpretation

• $\text{curl} > 0$: counterclockwise rotation (like a whirlpool)
• $\text{curl} < 0$: clockwise rotation
• $\text{curl} = 0$: irrotational — no net rotation (potential fields, conservative fields)

Examples

$\mathbf{F} = (-y, x)$ (perfect rotation field):

• $\partial F_y/\partial x = 1$, $\partial F_x/\partial y = -1$
• $\text{curl} = 1 - (-1) = 2$ (constant, everywhere)

$\mathbf{F} = (x, y)$ (radial outward field):

• $\partial F_y/\partial x = 0$, $\partial F_x/\partial y = 0$
• $\text{curl} = 0$ (irrotational — no rotation, just expansion)

$\mathbf{F} = (y^2, x^2)$ at $(1, 1)$:

• $\partial F_y/\partial x = 2x = 2$, $\partial F_x/\partial y = 2y = 2$
• $\text{curl} = 2 - 2 = 0$

Stokes' Theorem (2D)

The curl is related to circulation via Green's theorem:

$\oint_{\partial R} \mathbf{F} \cdot d\mathbf{r} = \iint_R \text{curl}\,\mathbf{F} \, dA$

Numerical Implementation

Use central differences to approximate the partial derivatives:

double curl_2d(double (*Fx)(double, double), double (*Fy)(double, double),
               double x, double y, double h) {
    double dFy_dx = (Fy(x+h,y) - Fy(x-h,y)) / (2.0*h);
    double dFx_dy = (Fx(x,y+h) - Fx(x,y-h)) / (2.0*h);
    return dFy_dx - dFx_dy;
}

Your Task

Implement double curl_2d(Fx, Fy, x, y, h) using central differences.