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I thought I had pretty much figured out the difference between $\equiv$ and $=$. Then I came across this while reading about partial derivatives (in Colley's Vector Calculus): $$ \frac{\partial^2f}{\partial z^2} = \frac{\partial}{\partial z} \left(\frac{\partial f}{\partial z}\right) = \frac{\partial}{\partial z} (y^2) \equiv 0 $$ when $f(x,y,z)=x^2y+y^2z$. Why do they use $\equiv$ in stead of $=$ here?

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    $\begingroup$ In this case, it means "identically $0$" (that is, the value is always $0$), rather than "find a value where it is equal to $0$". $\endgroup$ Mar 28, 2011 at 15:20
  • $\begingroup$ I think you can safely ignore it, just imagine it is = instead. The author may be using it in some weird way that normally isn't done. $\endgroup$
    – quanta
    Mar 28, 2011 at 17:23
  • $\begingroup$ Could you please provide a reference for the above excerpt. $\endgroup$ Mar 28, 2011 at 20:26
  • $\begingroup$ @Bill: I'm very sorry for the late response. I found this in Susan Colley's Vector Calculus. $\endgroup$
    – Eivind
    Apr 22, 2011 at 15:43

1 Answer 1

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$\equiv$ is often used (between functions) to mean they are identical (instead of being equal at some point). in your example this means identically $0$. if someone write something like $f(z)=g(z)$ it might be thought these functions are equal at some point $z$ instead of every point $z$, so you could write $f\equiv g$ or $f(z)\equiv g(z)$ to mean they are equal everywhere.

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  • $\begingroup$ So $\equiv$ is stronger than $=$? $\endgroup$
    – Eivind
    Mar 28, 2011 at 15:25
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    $\begingroup$ @Eivind: It's not "stronger" or "weaker", it's meant to emphasize that this is an equality of functions rather than an equality of values of the function. When one writes, for example, "$x^2+x = x^3+x^2$", it may be unclear if one is talking about the values of $x$ will make the two expressions equal, or if one is talking about an equality of functions (which holds, for example, if you consider the two polynomials as functions on the field of two elements). Use of $\equiv$ emphasizes that you are talking about equality of functions, not values. But both are statements about equality. $\endgroup$ Mar 28, 2011 at 15:34
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    $\begingroup$ @Arturo: OK, that makes it clear. Thank you. $\endgroup$
    – Eivind
    Mar 28, 2011 at 18:39
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    $\begingroup$ You could think about it like this: $f(x) \equiv g(x) \iff f(x)=g(x) \forall x$ $\endgroup$ Apr 22, 2011 at 19:08
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    $\begingroup$ I suppose my next question would be, why don't we see ≡ more often? $\endgroup$ Nov 12, 2016 at 12:48

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