Timeline for Partial derivative in Newtons Second law
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2021 at 5:34 | vote | accept | I am Me | ||
| Jan 17, 2021 at 5:35 | |||||
| Jan 14, 2021 at 14:59 | comment | added | Vercassivelaunos | @MoreAnonymous: Which is why I formulated it as a rule of thumb. But even the heat equation can be understood in the form $\frac{\mathrm d u}{\mathrm dt}=\Delta u$, where $\Delta u$ is the trace of the second total derivative of $u$ as a function of space, and $\frac{\mathrm du}{\mathrm dt}$ is the total derivative of $u$ as a function of time. Of course, we have to disconnect the parameter space into a spacial and a temporal part to really understand these as total derivatives, but it's still better than only seeing the partial derivatives without any interconnection between them. | |
| Jan 14, 2021 at 14:48 | comment | added | Vercassivelaunos | The total derivative of $f$ as used in physics is the total derivative of the function $f\circ Q$ as used by mathematicians. This often coincides with what physicists call the gradient, especially if $M$ represents space. | |
| Jan 14, 2021 at 14:48 | comment | added | Vercassivelaunos | Now we want to examine some quantity $f$ which depends on multiple quantities $(Q_1,\dots,Q_n)\in\mathbb R^n$, which in turn depend on our parameters in $M$.That is, $f:\mathbb R^n\to\mathbb R$ is a function of the quantities $Q_i$, and we have a function $Q:M\to\mathbb R^n$ which describes how the quantities depend on the parameters. Then we can express $f$ in terms of the parameters mathematically via the composition $f\circ Q$. continued | |
| Jan 14, 2021 at 14:48 | comment | added | Vercassivelaunos | @nasu: Depends on the context. But essentially what mathematicians mean when they say total derivative. Essentially, we have some space $M$ of "main" quantities which consists of the parameters relevant to our problem. For the mechanics of point masses, this would usually just be time. For wave or continuum mechanics, it might be spacetime. For electrostatics, it might only be the space coordinates. Whatever fits the problem we're trying to solve. continued | |
| Jan 14, 2021 at 14:34 | comment | added | More Anonymous | @vercassivelauno there are many equations in physics which use partial derivatives for example the of heat flow. | |
| Jan 14, 2021 at 14:24 | comment | added | nasu | What do you mean by total derivative? The gradient of the function? | |
| Jan 14, 2021 at 12:45 | history | answered | Vercassivelaunos | CC BY-SA 4.0 |