C1 metric on a smooth mapping space ($d_1(f,g)={\sup }_x|f(x)-g(x)|+{\sup }_x\Vert D{f}_x-D{g}_x\Vert$)

For smooth maps from a compact manifold with a fixed Riemannian metric to Euclidean space, the displayed metric measures uniform convergence of maps and their first derivatives. For a manifold embedded in Euclidean space use its induced metric. Different fixed metrics give equivalent topologies on a compact domain.