# Definition Consider a [[Coordinate Transformations on Manifolds|coordinate transformation]] on a [[Manifold]], $ y^i = y^i(x^j) $ If the partial derivatives of order $k$ or less of all the $\{y^i\}$ with respect to all the $\{x^i\}$ exist and are [[Continuous Map|continuous]], then the [[Coordinate Chart|charts]] $(U, f)$ and $(V, g)$ are said to be *$C^k$-related*. If it is possible to construct an [[Atlas]] in such a way that every point of $M$ where every chart is $C^k$-related to every other one it overlaps with, then the [[Manifold]] $M$ is said to be a $C^k$-[[Manifold]]. > A *differentiable [[Manifold]]* is a $C^1$-manifold In most physics applications, we will assume a $C^\infty$-[[Manifold]], but sometimes it will be useful to assume a $C^\omega$-[[Manifold]]. Another definition is that a map from a manifold $M$ to a manifold $N$ is $C^\infty$ if the coordinates of a point in $N$ are infinitely differentiable functions of the coordinates of the inverse image of the point in $M$.