Question

I want to numerically calculate the topological charge using a 2D angle field defined on a rectangular grid. At each vertex of the grid, a value of angle is defined. Now, I want to calculate the topological charge by traversing a closed path over the grid. Such charges are quantized (0,2π,−2π,⋯⋯). A method to calculate the charge is using the line integral over the closed path using the gradient of angle. I don't understand, how to evaluate such numerical integral using a gradient of angle. For example, I want to calculate the topological charge enclosed by each smallest square by traversing the closed square path. The key is to take theprincipal valueafter you computed the differences for the gradient. Then you add up these principal values for the contour integration.

Answer 1

To calculate the topological charge using a 2D angle field defined on a rectangular grid, you can use the line integral over a closed path using the gradient of the angle.

Step-by-step explanation:

To calculate the topological charge using a 2D angle field defined on a rectangular grid, you can use the line integral over a closed path using the gradient of the angle.

The topological charge is obtained by summing the principal values of the differences in the gradient along the closed path.

For example, to calculate the charge enclosed by each smallest square by traversing the closed square path, you would calculate the differences in the gradient at each vertex and sum the principal values of these differences.

The topological charge is obtained by summing the principal values of the differences in the gradient along the closed path.