Theory of vector field rotation

The point (x0, y0) is called a nonsingular point of the field F if there exists a vicinity of the point (jc0, y0) in which the field is defined and continuous and F(x0, y0) = 0. Otherwise (x0, yQ) is called a singular point of the field F. 

Let be given in a plane a certain oriented line L not containing singular points of the field F. The rotation of the field F along the line L, Ol F is defined as an increment of the angle, divided by 2n and computed counterclockwise, which forms the vector F(x,y) with a specified direction when the point (x, y) crosses the line L according to its orientation. 

If the line L is smooth in segments and the functions P, Q are continuously differentiable in a certain vicinity of the line, then the following equation is valid  

If the line L is closed, then the vector field rotation has the following properties (1) Ol F is an integer (2) Ot F does not change when the line L or the field F are continuously deformed in such a way that the singular points of the field F during the deformation miss the line L (3) Ol F = 0 if in the region contained by the line L there are no singular points of the field F. 

If we compute 0L F for a given point (x0, y0) in such a way that L is a closed line and in the region contained by L there are no singular points of the field F except, possibly, for the point (x0, y0), then such a quantity is called the index of the point (x0, y0) and denoted as Ind F (x0, y0). 

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