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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 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.
+
+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
+
+```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)
+```
+
+<!-- What more needs to be said here? --/>