smooth manifold
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 is a smooth -dimensional manifold if locally it is the graph of a mapping expressing variables as functions of the other variables.
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.
[cite:@calc_hubbard_2015] 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.