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+# 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 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,
+``\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
+Grids.jacobian(::typeof(f), x) = fill(2, length(x))
+```
+
+```julia
+struct F end
+(::F)(x) = 2x
+Grids.jacobian(::F, x) = fill(2,length(x))
+```
+
+You can also provide a fallback function using one of the many automatic
+differentiation packages, for example
+
+```julia
+using ForwardDiff
+Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x)
+```