To the calculation of the mapping degree of finite dimensional vector field

In this paper we propose and justify a new method calculation of the mapping degree of $n$-dimensional vector field on the unit sphere of the space $\mathrm{R}^n$, $n\geq 2$. The essence of the proposed method is that the calculation of the mapping degree of vector field is reduced to the calculation of the mapping degree of its tangent component on the components of the set, where the vector field has an obtuse angle with the unit vector field. In the special case, for the gradient of a smooth positively homogeneous function, we derive a formula for calculation of the mapping degree through the Eulerian characteristic of the set of points where the function is negative.