One can find many cases that mathematicians and physicists use different notations for the same concepts. Here is a few cases I find.

Inner product of vectors: Mathematicians use $(a,b)$ or $\left<a,b\right>$ for inner product of two vectors $a,b$ while physicists use $\left<a|b\right>$ for vectors $\left.|a\right>,\left.|b\right>$.

Hermitian inner product: Mathematicians adapt rule $\left<\lambda a,b\right>=\lambda\left<a,b\right>=\left<a,\bar\lambda b\right>$ while physcists adapt $\left<\lambda a|b\right>=\lambda^\dagger\left<a|b\right>=\left<a|\lambda^\dagger b\right>$

Tensor notation: Mathematicians tend to use a single letter, say $T$ to express a tensor; while physicists tend to write down every index, say $T^i_j$, for instead of one component, but the whole body.

Summation notation: I think very rigorous mathematicians will not allow $a^kb_k$ to represent $\sum_{k=1}^na_kb_k$.

So my question is, what are other cases do you know about different notations for the same concept? Also, why (what advantages) do they use respectively?


In a physics or mathematical physics context I've often seen $\int \mathrm{d}x\; f(x)$ rather than $\int f(x) \;\mathrm{d}x$, e.g. in courses I've done on waves, diffusion, and statistical mechanics. Briefly the benefits of the former are the emphasis on integration being a linear operator, and clarity for multi-dimensional integrals. The latter is more familiar and often more convenient, especially in the single-dimensional case. However the preference for one notation or the other can't be divided strictly along maths/physics lines.

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