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Project.toml

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name = "DerivableFunctions"
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uuid = "b86e32d1-1d4d-4472-88d2-1980e9d19c92"
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authors = ["Rafael Arutjunjan"]
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version = "0.1.2"
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version = "0.1.3"
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[deps]
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DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
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DataFrames = "1"
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FiniteDifferences = "0.11, 0.12"
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ForwardDiff = "0.10"
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ModelingToolkit = "5, 6, 7"
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ModelingToolkit = "5 - 8"
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ReverseDiff = "1.8, 1.9"
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Symbolics = "2, 3, 4"
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Symbolics = "2 - 4"
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Zygote = "0.6"
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julia = "1"
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docs/src/Operators.md

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Clearly, this simplified implementation features some redundant evaluations of the inverse metric and could be made more efficient.
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Nevertheless, it nicely illustrates how succinctly complex real-world examples can be formulated.
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Given the metric tensor ``g(\\theta, \\phi) = \\mathrm{diag}(1, \\mathrm{sin}(\\theta))`` induced by the canonical embedding of ``S^2`` into ``\\mathbb{R}^3`` with spherical coordinates, it can be shown that the Ricci scalar assumes a constant value of ``R=2`` everywhere on ``S^2``.
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Given the metric tensor induced by the canonical embedding of ``S^2`` into ``\\mathbb{R}^3`` with spherical coordinates, it can be shown that the Ricci scalar assumes a constant value of ``R=2`` everywhere on ``S^2``.
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```julia
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S2metric((θ,ϕ)) = [1.0 0; 0 sin(θ)^2]
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2 RicciScalar(S2metric, rand(2); ADmode=Val(:ForwardDiff)) RicciScalar(S2metric, rand(2); ADmode=Val(:ReverseDiff))
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```
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(In this particular instance, due to a term ``\\mathrm{cos}(\\theta) \\, \\mathrm{sin}(\\theta) / (\\mathrm{sin}(\\theta))^2`` in the `ChristoffelSymbol` where the ``\\mathrm{sin}(\\theta)`` in the numerator does not cancel with the identical term in the denominator, the symbolic computation does not recognize the fact that the final expression can be simplified to yield exactly ``R=2``.)
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(In this particular instance, due to a term in the `ChristoffelSymbol` where the `sin` in the numerator does not cancel with the identical term in the denominator, the symbolic computation does not recognize the fact that the final expression can be simplified to yield exactly ``R=2``.)
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```julia
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using Symbolics; @variables p[1:2]
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RicciScalar(S2metric, p)

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