Singularity with index 2

For the first figure v(x,y) = (y,x), and (0,0) has an index of –1; note how the vector field goes anticlockwise once as we move clockwise around the circle. For the second field v(x,y) = (x2−y2,2xy), then (0,0) has index 2. This definition then generalizes to tangent vector fields on surfaces (see Poincaré-Hopf Theorem) and to higher dimensions (see degree (of a map)).