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I want to check if I understand the concept and notation of partial and total derivatives correctly.

Is it true that if I have some multivariable function f(x,y,..) I can only use the partial derivate notation ∂/∂ if I know for sure, that all other variables do not depend on x? i.e we DO NOT FIX all other variables constant by just using the partial derivative notation - they should be FIXED w.r.t x already by the construction and physical meaning of the function in order for the notation to be used?

If I do not have this apriori knowelege about the function I'm required to stick with an ordinary derivative, which in the case of multivariable function becomes the total derivative /? If so happens that no other variables depend on x, then by the chain rule the total derivative / will just become this simpler ∂/∂, in other case it will stay more complex and general expression.

Is it true then, that partial derivative notation does not automatically make sense for all multivariable functions but only the ones that have at least one completely independent variable, and only w.r.t this variable?

Can we say that partial derivative is just a special case of total derivative, a notational convenience when we have this knowelege about independence of other variables?

The total derivative notation on the other hand always make sense.

simd
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    Suppose $f(x,y,z)$ is a function of $3$ variables. If you like, $\frac{\partial f}{\partial x}(x_0, y_0, z_0)$ is the same as $\frac{dg}{dx} (x_0)$ where $g(x) = f(x, y_0, z_0)$. Does that clarify things somewhat? – Charles Hudgins Mar 22 '23 at 09:06
  • @CharlesHudgins I do understand that as a mathematical definition of the famous images with the cross-section of multivariable function in one direction as an illustration of partial derivatives. But I can't make any further connection with any of my points so far. – simd Mar 22 '23 at 09:24
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    I find your terminology confusing. If you talk about a function $f \colon \mathbf{R}^n \to \mathbf{R}$, or “$f(x_1,\dots,x_n)$”, then the variables are by definition independent; you can evaluate it at any point in $\mathbf{R}^n$ that you like, and your choice of $x_1$ doesn't affect how to choose $x_2$ (for example). If there is some dependence among the variables, then you are actually talking about a different function which is the composition of $f$ with some other function(s), such as $g \colon \mathbf{R} \to \mathbf{R}$ given by $g(x)=f(x,\alpha(x),\beta(x))$. – Hans Lundmark Mar 22 '23 at 09:52
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    @HansLundmark thank you! This is exactly the source of my confusions! I completely forgot that variables are independent by definition of the function. – simd Mar 22 '23 at 11:35
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    Of course, it's easy to be confused by things like the traditional notation for total derivatives, where (continuing my example from above) people never bother with introducting the functions $g$, $\alpha$ and $\beta$, and instead just say “$y$ and $z$ depend on $x$” and write “$df/dx$” when they really mean $g'(x)$. – Hans Lundmark Mar 22 '23 at 11:46
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    @user3537411 "I completely forgot that variables are independent by definition of the function." Yes, if the function is defined on the entire $\mathbb R^n$. – ryang Mar 22 '23 at 17:29
  • @ryang, are you saying that we can't claim that variables are independent (thus can't use partial derivative) unless it was clearly stated that function is defined on entire ℝ^? (Which probably implied in most cases even if it's not stated). But if I understand correctly, it may be a very important distinction if we're working/constructing some hypothetical physical model for which it wasn't proved yet. In this case it would be better to stick with total derivative? – simd Mar 22 '23 at 21:12
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    @user3537411 I was just trying to be clear that the last part of your sentence that I quoted refers to the fact that $f$'s domain is specified as $\mathbb R^n$ rather than a proper subset of it. To answer your main question: no, the notation $\frac{∂f}{∂x}$ does not care whether $x,y,z$ depend on one another; consider the multivariable chain rule, for example. – ryang Mar 23 '23 at 06:36
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    An analogous question is being asked at Is posterior probability affected by a hidden observer?; I'm comparing a conditional probability and a partial derivative. – ryang Mar 23 '23 at 13:57
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    @ryang an interesting comparison, thank you – simd Mar 24 '23 at 00:18

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