Mercurial > repos > public > sbplib_julia

comparison docs/src/manifolds_charts_atlases.md @ 2057:8a2a0d678d6f feature/lazy_tensors/pretty_printing

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Merge default
author Jonatan Werpers <jonatan@werpers.com>
date Tue, 10 Feb 2026 22:41:19 +0100
parents d89f7a1a6f37
children
comparison
equal deleted inserted replaced
1110:c0bff9f6e0fb 2057:8a2a0d678d6f
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 a logical parameter
5 space to the geometry that we are interested in. If there are more than one
6 chart for a given geometry this collection of charts and how they are
7 connected is described by an atlas.
8
9 We consider a mapping from the logical coordidinates ``\xi \in \Xi`` to the
10 physical coordinates ``x \in \Omega``. A `Chart` describes the mapping by a
11 `ParameterSpace` respresenting ``\Xi`` and some mapping object that takes
12 arguments ``\xi \in \Xi`` and returns coordinates ``x\in\Omega``. The mapping
13 object can either be a function or some other callable object.
14
15 For the construction of differential and difference operators on a manifold
16 with a chart the library needs to know the Jacobian,
17 ``\frac{\partial x}{\partial \xi}``, of the mapping as a function of
18 coordinates in the logical parameter space. Internally, Diffinitive.jl uses a
19 local Jacobian function, `Grids.jacobian(f, ΞΎ)`. For geometry objects provided
20 by the library this function should have fast and efficient implementations.
21 If you are creating your own mapping functions you must implement
22 `Grids.jacobian` for your function or type, for example
23
24 ```julia
25 f(x) = 2x
26 Grids.jacobian(::typeof(f), x) = fill(2, length(x))
27 ```
28
29 ```julia
30 struct F end
31 (::F)(x) = 2x
32 Grids.jacobian(::F, x) = fill(2,length(x))
33 ```
34
35 You can also provide a fallback function using one of the many automatic
36 differentiation packages, for example
37
38 ```julia
39 using ForwardDiff
40 Grids.jacobian(f,x) = ForwardDiff.jacobian(f,x)
41 ```