Timeline for Suggestions for good notation
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2021 at 10:31 | comment | added | Zach Teitler | How would $D_j$ eliminate any confusion? You seem to assert that $D_1 y_2$ would be somehow unambiguous. But it is just as ambiguous as $\partial y_2 / \partial y_1$. If $D_1$ means differentiation with respect to $y_1$, holding $y_2$ constant, then $D_1 y_2 = 0$. If $D_1$ means differentiation with respect to $x_1$, holding $x_2$ constant, then $D_1 y_2 = -1$. More generally, there is no symbol or notation involving only one single coordinate function --- not $D_1$, not $\frac{\partial}{\partial y_1}$, nothing --- that can possibly convey full information about a coordinate chart. | |
| Jan 28, 2018 at 15:44 | comment | added | liuyao | In PDE, one often uses $\partial_j$, and $\partial^\alpha=\partial_1^{\alpha_1}\cdots \partial_n^{\alpha_n}$ for a multi-index $\alpha$. (They reserve $D_j$ for $-i\partial_j$ which is convenient for Fourier transform.) In some cases I seem to like writing $D_\xi$ for $\xi\in V$ (without choosing a basis), and it acts on $f(x)$ where $x$ somehow is an element of $V^*$. | |
| Nov 26, 2017 at 15:41 | comment | added | Michael Bächtold | Jacobi was aware of this problem when he introduced the $\partial$ notation. Unfortunately his choice of notation for avoiding this ambiguity might have led to more confusion. | |
| Jan 24, 2013 at 21:14 | comment | added | Marcos Cossarini | @Ben, the index in the coordinate expression $\frac{\partial f}{\partial x^j}$ for the 1-form $df$ is clearly in the low position! In fact, this is the main reason that I see for having to put the indexes of the coordinates in the high position as we do, instead of doing everything in the opposite way, which would be better in some way: we could write $f=x_1^2+x_3$ instead of $f=(x^1)^2+x^3$. | |
| Jan 24, 2013 at 21:03 | comment | added | Marcos Cossarini | Regarding differential geometry: If $f:M\to\mathbb R$ is a smooth function on a manifold and $x:M\to\mathbb R^n$ is a chart, I prefer $\left(\frac{\partial f}{\partial x}\right)_j$ or $\left(\frac\partial{\partial x}\right)_j f$ (or even $\partial_j f$ if the choice of the particular chart is clear or irrelevant). Because the notation $\frac{\partial f}{\partial x^j}$ suggests that $\frac{\partial f}{\partial g}$ could be defined using only $g$, and in fact you need to know that you are restricting to the curve along which the other coordinates $x^i$ are constant. | |
| Oct 15, 2012 at 20:39 | comment | added | user21349 | Another reason to prefer $D_j$ or $\nabla_j$ over $\partial/\partial x_j$ is that when you're expressing tensors using the Einstein "index gymnastics notation," the derivative $\nabla_j$ has a lower index, as it should, whereas the same operator in Leibniz notation is $\partial/\partial x^j$, which looks like it transforms as an upper-index tensor. | |
| Oct 20, 2010 at 22:37 | history | edited | Mark | CC BY-SA 2.5 |
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| Oct 20, 2010 at 22:29 | history | answered | Mark | CC BY-SA 2.5 |