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AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation.

From purely a physical phenomena to math mapping POV: Why does a partial derivative show up? That is, how do we justify that we need something like a partial derivative to address something that ordinary derivatives are deficient at?

Let's assume that the vibrating string was the genesis of partial differentials (AFAIK Bernoulli (one of them) had invented this concept sooner but let's assume otherwise for now). The question then is what aspect of capturing the vibrating string problem's as math equations leads one to "invent" partial derivatives?

I've been pondering over this for a while ever since I learned about the history and derivation of the 1D wave equation. I don't think I understand the "need for partial derivates" well enough to answer this question.

PS: Decided to post in the Physics.SE vs. Math.SE since I'd like to understand PDEs from the POV of physical phenomena and build intuition about their need.

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    $\begingroup$ I'm not really sure I understand what you're asking. Any time you have a function of more than one argument and want to answer the question "How does this function vary if I vary one of the arguments and keep the others constant?", the answer to that is a partial derivative (that's it's definition!). Why do you think that's somehow uniquely motivated by the wave equation, and where is the mystery in an equation that relates precisely such rates of partial variation needing that concept? $\endgroup$ – ACuriousMind Dec 13 '18 at 21:01
  • $\begingroup$ Simply put the "wave" depends on two variables, a space coordinate and time. How else can you express these dependencies without partials. $\endgroup$ – ggcg Dec 13 '18 at 23:46
  • $\begingroup$ I think I'm trying to understand the "history of partial differentiation" and I believe it originates in the world of "mathematical physics". I intend to "see the need" for something like a partial just for the sake of the understanding to stick vs. saying - here is the definition and that's what it is. No. The definition was "made up" because of a need to have something like this. I would like to recreate that need in a classroom setting for example (hypothetically speaking), when introducing PDs for the first time. $\endgroup$ – PhD Dec 14 '18 at 1:18
  • $\begingroup$ Try to define the ordinary derivative on a function of multiple variables without an appeal to partials...it's impossible. Taking differentiation into higher dimension requires us to consider components independently. $\endgroup$ – Keefer Rowan Dec 14 '18 at 1:54
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    $\begingroup$ If you are really more interested in the history of partial differentiation, this question might be a better fit for History of Science and Mathematics than our site. If you agree, simply flag your post with a custom moderator flag asking us to migrate it to HSM. $\endgroup$ – ACuriousMind Dec 14 '18 at 17:07
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I suppose deflection is a function of time and position, and hence the partials are needed to make the notation unambiguous.

A small change in displacement is defined as

$$ {\rm d}y = \frac{\partial y}{\partial x} {\rm d}x + \frac{\partial y}{\partial t} {\rm d}t $$

without the partials, the above would be confusing.

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