# 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.

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)
```