--- a/docs/src/manifolds_charts_atlases.md	Tue Feb 11 09:03:04 2025 +0100
+++ b/docs/src/manifolds_charts_atlases.md	Tue Apr 29 09:03:05 2025 +0200
@@ -1,18 +1,25 @@
 # Manifolds, Charts, and Atlases
 
 To construct grids on more complicated geometries we use manifolds described
-by one or more charts. The charts describe a mapping from some parameter space
-to the geometry that we are interested in. If there are more than one chart
-for a given geometry this collection of charts and their connection is
-described by and atlas.
+by one or more charts. The charts describe a mapping from a logical parameter
+space to the geometry that we are interested in. If there are more than one
+chart for a given geometry this collection of charts and how they are
+connected is described by an atlas.
+
+We consider a mapping from the logical coordidinates ``\xi \in \Xi`` to the
+physical coordinates ``x \in \Omega``. A `Chart` describes the mapping by a
+`ParameterSpace` respresenting ``\Xi`` and some mapping object that takes
+arguments ``\xi \in \Xi`` and returns coordinates ``x\in\Omega``. The mapping
+object can either be a function or some other callable object.
 
 For the construction of differential and difference operators on a manifold
-with a chart the library needs to know the Jacobian of the mapping as a
-function of coordinates in the logical parameter space. Internally,
-Diffinitive.jl uses a local Jacobian function, `Grids.jacobian(f, ξ)`. For
-geometry objects provided by the library this function should have fast and
-efficient implementations. If you are creating your own mapping functions you
-can implement `Grids.jacobian` for your function or type, for example
+with a chart the library needs to know the Jacobian,
+``\frac{\partial x}{\partial \xi}``, of the mapping as a function of
+coordinates in the logical parameter space. Internally, Diffinitive.jl uses a
+local Jacobian function, `Grids.jacobian(f, ξ)`. For geometry objects provided
+by the library this function should have fast and efficient implementations.
+If you are creating your own mapping functions you must implement
+`Grids.jacobian` for your function or type, for example
 
 ```julia
 f(x) = 2x
@@ -32,5 +39,3 @@
 using ForwardDiff
 Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x)
 ```
-
-<!-- What more needs to be said here? --/>