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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5In this case, it means "identically $0$" (that is, the value is always $0$), rather than "find a value where it is equal to $0$". – Arturo Magidin Mar 28 '11 at 15:20
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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. – quanta Mar 28 '11 at 17:23
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Could you please provide a reference for the above excerpt. – Bill Dubuque Mar 28 '11 at 20:26
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@Bill: I'm very sorry for the late response. I found this in Susan Colley's Vector Calculus. – Eivind Apr 22 '11 at 15:43
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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.
yoyo
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12@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. – Arturo Magidin Mar 28 '11 at 15:34
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4You could think about it like this: $f(x) \equiv g(x) \iff f(x)=g(x) \forall x$ – Fernando Martin Apr 22 '11 at 19:08
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1I suppose my next question would be, why don't we see ≡ more often? – Mateen Ulhaq Nov 12 '16 at 12:48
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because with reasonable notation otherwise you dont need it and people tend to use it wrong. for functions $f,g$, $f=g$ is precisely defined as $\forall x:f(x)=g(x)$, so no need to ever write $f\equiv g$, which is how i see it abused most often. as opposed to the answer, $f(x)=g(x)$ is not an equality between functions at all, but between values, $x$ has to be bound outside of this statement. – peter Nov 22 '21 at 02:55
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one use case remains, i've never seen it in the wild and it is similar to $\leftrightarrow$ vs $\Leftrightarrow$ in logic, where the latter automatically universally quantifies all free variables, i.e. $f(x)\equiv g(x)$ then means $f=g$, but we have the advantage that we can use it for expressions which cannot unclumisly be written as functions, e.g. $2^{x+y}\equiv 2^x2^y$ instead of $((x,y)\mapsto 2^{x+y}) = ((x,y)\mapsto 2^x2^y)$. – peter Nov 22 '21 at 03:03
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see the accepted answer here: https://math.stackexchange.com/questions/29143/when-should-i-use-and-equiv?noredirect=1&lq=1 . – peter Nov 22 '21 at 03:06