a subset is a smooth -dimensional manifold if locally it is the graph of a mapping expressing variables as functions of the other variables.
"locally" means that every point has a neighborhood such that (the part of in ) is the graph of a mapping expressing of the coordinates of each point in in terms of the other .
this definition says that if a function is , then its graph is a smooth -dimensional manifold in .
a manifold embedded in , denoted , is sometimes called a submanifold of . strictly speaking, it should not be referred to simply as a "manifold", which can mean an abstract manifold, not embedded in any space.