### include my numbering into bibrefs

parent babc08fb
 @book{IntroRiemannLee, @book{q3-IntroRiemannLee, doi = {10.1007/978-3-319-91755-9}, year = {2018}, publisher = {Springer International Publishing}, author = {John M. Lee}, title = {Introduction to Riemannian Manifolds} } @article{Ball2011, @article{q4-Ball2011, doi = {10.1007/s00205-011-0421-3}, year = {2011}, month = may, ... ... @@ -17,7 +17,7 @@ title = {Orientability and Energy Minimization in Liquid Crystal Models}, journal = {Archive for Rational Mechanics and Analysis} } @article{Nitschke2018, @article{q5-Nitschke2018, doi = {10.1098/rspa.2017.0686}, url = {https://royalsocietypublishing.org/doi/10.1098/rspa.2017.0686}, year = {2018}, ... ...
 ... ... @@ -10,7 +10,7 @@ I always struggle with the more-dimensional chainrule and with coordinates in differential geometry. Hence I tried to understand the Jacobian matrix representation of the \newTerm{differential of $F$ at $p$} as in \textcite[Tangent Vectors]{IntroRiemannLee}. as in \textcite[Tangent Vectors]{q3-IntroRiemannLee}. We have given \begin{itemize} ... ... @@ -40,7 +40,7 @@ $dF_p(v) f = v(f ∘ F)$. Hence &= v^i \tanvecat{x^i}{p} (f ∘ F) \\ &= v^i \tanvecat{x^i}{p} (f ∘ ψ^{-1} ∘ \hat F ∘ φ) && (\text{Def. } \hat F) \\ &= v^i \rest{∂_i}{φ(p)} (f ∘ ψ^{-1} ∘ \hat F) \\ & \qquad (\text{Def. } \tanvecat{x^i}{p} \text{\textcite[A.2]{IntroRiemannLee}}) \\ & \qquad (\text{Def. } \tanvecat{x^i}{p} \text{\textcite[A.2]{q3-IntroRiemannLee}}) \\ \intertext{Now we are in a purely flat setting and can apply the calculus chain rule for $f ∘ ψ^{-1} \colon ℝ^n \supseteq ψ(V) → ℝ$ and $\hat F \colon ℝ^m \supseteq φ(U) → ℝ^n$} ... ...
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