For a Riemannian metric, critical points are where these vanish when mapped from a smooth function.
According to one theorem, the Radon-Hurwitz numbers govern how many of these that are non-zero can be mapped to an n-sphere. For a Riemannian metric, critical points are where these vanish when mapped from a smooth function. For a symplectic manifold, one can define the Hamiltonian type of these constructs from an energy function. It is impossible to define a continuous and non-zero one of these constructs tangent to an even dimensional (*) n-sphere according to the hairy ball theorem. Ones with vanishing curl are said to be irrotational and conservative; such ones can be derived by taking the gradient of a scalar potential function. For 10 points, name these topological constructs that assign a length and direction over all points of a subset of Euclidean space.
According to one theorem, the Radon-Hurwitz numbers govern how many of these that are non-zero can be mapped to an n-sphere.
vector fields [prompt "fields."]