Vector Fields and Classical Theorems of Topology by Daniel

The reason for this exercise is to argue that the following equation, which we call the “Law of Vector Fields,” is fated to play a central role in mathematics. These theorems are all easy consequences of the law of vector fields. The proofs are so mechanical that one could say that the Law of Vector Fields is a generalization of each of them. The Law of Vector Fields is the following: Let M be a compact smooth manifold and let V be a vector field on M so that V (m) 6= ~0 for all m on the boundary ∂M of M . Then ∂M contains an open set ∂−M which consists of all m ∈ ∂M so that V (m) points inside. We define a vector field, denoted ∂ − V on ∂−M , so that for every m ∈ ∂−M we have ∂−V (m) = Projection of V (m) tangent to ∂−M . Under these conditions we have