We give a direct geometric proof of Hopf's theorem on the sum of indices at the zeros of a vector field in a manifold M, or rather of that part of the theorem that says that the sum is the same for ...

Analytical methods in differential systems and vector fields combine classical techniques with modern computational approaches to elucidate the intricate dynamics of nonlinear systems. Researchers use ...

In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of ...