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annotate docs/src/manifolds_charts_atlases.md @ 1988:c4dc1bed0a81 feature/grids/manifolds

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Minor docs change
author Jonatan Werpers <jonatan@werpers.com>
date Tue, 15 Apr 2025 08:35:51 +0200
parents 3694f11075c5
children d89f7a1a6f37
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1922
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1 # Manifolds, Charts, and Atlases
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2
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3 To construct grids on more complicated geometries we use manifolds described
1972
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4 by one or more charts. The charts describe a mapping from a logical parameter space
1922
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5 to the geometry that we are interested in. If there are more than one chart
1988
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6 for a given geometry this collection of charts and how they are connected is
1972
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7 described by an atlas.
1922
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8
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9 For the construction of differential and difference operators on a manifold
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10 with a chart the library needs to know the Jacobian of the mapping as a
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11 function of coordinates in the logical parameter space. Internally,
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12 Diffinitive.jl uses a local Jacobian function, `Grids.jacobian(f, ΞΎ)`. For
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13 geometry objects provided by the library this function should have fast and
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14 efficient implementations. If you are creating your own mapping functions you
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15 can implement `Grids.jacobian` for your function or type, for example
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16
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17 ```julia
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18 f(x) = 2x
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19 Grids.jacobian(::typeof(f), x) = fill(2, length(x))
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20 ```
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22 ```julia
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23 struct F end
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24 (::F)(x) = 2x
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25 Grids.jacobian(::F, x) = fill(2,length(x))
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26 ```
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27
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28 You can also provide a fallback function using one of the many automatic
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29 differentiation packages, for example
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30
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31 ```julia
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32 using ForwardDiff
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33 Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x)
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34 ```