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annotate docs/src/manifolds_charts_atlases.md @ 1996:d89f7a1a6f37 feature/grids/manifolds

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Add some notation to the docs page
author Jonatan Werpers <jonatan@werpers.com>
date Wed, 23 Apr 2025 09:02:20 +0200
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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
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4 by one or more charts. The charts describe a mapping from a logical parameter
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5 space to the geometry that we are interested in. If there are more than one
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6 chart for a given geometry this collection of charts and how they are
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7 connected is described by an atlas.
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8
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9 We consider a mapping from the logical coordidinates ``\xi \in \Xi`` to the
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10 physical coordinates ``x \in \Omega``. A `Chart` describes the mapping by a
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11 `ParameterSpace` respresenting ``\Xi`` and some mapping object that takes
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12 arguments ``\xi \in \Xi`` and returns coordinates ``x\in\Omega``. The mapping
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13 object can either be a function or some other callable object.
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14
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15 For the construction of differential and difference operators on a manifold
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16 with a chart the library needs to know the Jacobian,
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17 ``\frac{\partial x}{\partial \xi}``, of the mapping as a function of
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18 coordinates in the logical parameter space. Internally, Diffinitive.jl uses a
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19 local Jacobian function, `Grids.jacobian(f, ΞΎ)`. For geometry objects provided
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20 by the library this function should have fast and efficient implementations.
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21 If you are creating your own mapping functions you must implement
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22 `Grids.jacobian` for your function or type, for example
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24 ```julia
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25 f(x) = 2x
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26 Grids.jacobian(::typeof(f), x) = fill(2, length(x))
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27 ```
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29 ```julia
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30 struct F end
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31 (::F)(x) = 2x
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32 Grids.jacobian(::F, x) = fill(2,length(x))
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33 ```
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35 You can also provide a fallback function using one of the many automatic
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36 differentiation packages, for example
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38 ```julia
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39 using ForwardDiff
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40 Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x)
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41 ```