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I reviewed this question but sometimes I'm unsure about delta versus differential notation. Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the same thing as "$\Delta s^2=-c^2\Delta t^2+a^2(t)[\Delta r^2 + S_k^2(r)\Delta \Omega^2 ]$" ? It seems like it may, but my understanding is that the former may be more appropriate for infinitesimal distance.

Qmechanic
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bblohowiak
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    I'd say that the other question/answers you reviewed is pretty comprehensive and informative. In general, the prefix $d$ usually means an (exact) infinitesimal change and the prefix $\Delta$ simply means change that is not necessarily infinitesimal. Though you are right to ask since many authors are sloppy and abuse this notation/use them interchangeably. – joseph h Apr 17 '22 at 20:12
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    Of course anyone can use any notation for anything, but for most people $\Delta s$ is a small real number and $ds$ is a cotangent vector. – WillO Apr 17 '22 at 22:06

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The first one is exact, but the second would be an approximation for a non-infinitesimal (i.e. finite) interval.

ProfRob
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