r space E.  We then say that X is an &quot;E-manifold&quot; or that
| X is &quot;modeled&quot; on E.
|
| If E = R^n for some fixed n, then we say that the manifold is &quot;n-dimensional&quot;.
| In this case, a chart
|
| q : U -&gt; R^n
|
| is given by n coordinate functions q_1, ..., q_n.  If 'P' denotes a point of U,
| these functions are often written
|
| x_1(P), ..., x_n(P),
|
| or simply x_1, ..., x_n.  They are called &quot;local coordinates&quot; on the manifold.
|
| If the integer p (which may also