Mercurial > repos > public > sbplib_julia
comparison docs/src/manifolds_charts_atlases.md @ 1922:84e5ab97902d feature/grids/manifolds
Add sketch of a docs page about manifolds, charts, and atalses
| author | Jonatan Werpers <jonatan@werpers.com> |
|---|---|
| date | Thu, 06 Feb 2025 08:59:47 +0100 |
| parents | |
| children | 3694f11075c5 |
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| 1921:7fb713570965 | 1922:84e5ab97902d |
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| 1 # Manifolds, Charts, and Atlases | |
| 2 | |
| 3 To construct grids on more complicated geometries we use manifolds described | |
| 4 by one or more charts. The charts describe a mapping from some parameter space | |
| 5 to the geometry that we are interested in. If there are more than one chart | |
| 6 for a given geometry this collection of charts and their connection is | |
| 7 described by and atlas. | |
| 8 | |
| 9 For the construction of differential and difference operators on a manifold | |
| 10 with a chart the library needs to know the Jacobian of the mapping as a | |
| 11 function of coordinates in the logical parameter space. Internally, | |
| 12 Diffinitive.jl uses a local Jacobian function, `Grids.jacobian(f, ΞΎ)`. For | |
| 13 geometry objects provided by the library this function should have fast and | |
| 14 efficient implementations. If you are creating your own mapping functions you | |
| 15 can implement `Grids.jacobian` for your function or type, for example | |
| 16 | |
| 17 ```julia | |
| 18 f(x) = 2x | |
| 19 Grids.jacobian(::typeof(f), x) = fill(2, length(x)) | |
| 20 ``` | |
| 21 | |
| 22 ```julia | |
| 23 struct F end | |
| 24 (::F)(x) = 2x | |
| 25 Grids.jacobian(::F, x) = fill(2,length(x)) | |
| 26 ``` | |
| 27 | |
| 28 You can also provide a fallback function using one of the many automatic | |
| 29 differentiation packages, for example | |
| 30 | |
| 31 ```julia | |
| 32 using ForwardDiff | |
| 33 Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x) | |
| 34 ``` | |
| 35 | |
| 36 <!-- What more needs to be said here? --/> |