35 questions linked to/from Is the world $C^\infty$?
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### Classical Mechanics: Continuous or Discrete universe? [duplicate]

The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a ...
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### How "smooth" is the evolution of the universe? [duplicate]

Mathematicians have developed different definitions for how "smooth" a function might be. A function (e.g. from or to $\mathbb{R}^n$ or $\mathbb{C}^n$) might be continuous, or once-...
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### Infinite differentiablity in Nature? [duplicate]

I'm working through understanding Clay Institute's paper addressing their Navier-Stokes millennium prize. I can't find any resources online stating if there are any infinitely differentiable examples ...
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### Why don't we prove that functions used in physics are continuous and differentiable?

I have studied physics up to 12th grade and I noticed that whenever new equations are introduced for certain entities, such as a simple harmonic wave, we never prove that it's continuous everywhere or ...
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### Why does Taylor’s series “work”?

I am an undergraduate Physics student completing my first year shortly. The following question is based on the physical systems I’ve encountered so far. (We mostly did Newtonian mechanics.) In all of ...
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### In GR, why should the spacetime manifold be differentiable?

In general relativity (GR), spacetime is viewed as a differentiable manifold of dimension $D$ with a metric of Lorentzian signature $(-,+,+,...,+)$. My question is why differentiable?
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### Why is analyticity a good mathematical assumption in general relativity?

In general relativity, real-variable analytic continuation is commonly used to understand spacetimes. For example, we use it to extend the Schwarzschild spacetime to the Kruskal spacetime, and also ...
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### Is it guaranteed that wavefunction is well behaved everywhere?

I don't really know much about Quantum mechanics, but would like to know one simple fact. The state function $\Psi(r, t)$ whose magnitude gives the probability density of the position of the particle ...
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### Use of the mathematical concept 'function' in theoretical physics

The mathematical concept of function is used in physics to represent different physical quantities. For example the air pressure variation with time and space is called an acoustic wave. We use a ...
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### Are velocity and acceleration smooth quantities?

My thinking: acceleration corresponds to a force which is instantaneous, so the acceleration of a rigid body can be rather spiky (non-smooth) velocity (angular velocity) describes the ratio of ...
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### Is it normal for physical functions to lack a 2nd derivative?

My question is about the appearance of a non-analytic function in the formula for the resistive force in air or other medium. Considering the 1-dimensional case as covered by Walter Lewin in his 8.01 ...
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### Why do we consider potential energy function $U(x)$ differentiable?

Recently when skimming through my physics-text I encountered an interesting definition of Force $$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$ We were taught that some functions are continuous but not ...
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### Can motion graphs have sharp edges?

I'm pretty convinced but I need to make sure I'm right about this. By a sharp edge, I mean a point in the graph where the curve is not differentiable. in motion graphs, (x,y,z) coordinates depend on ...
In electromagnetism, we say that any conservative electric field $\vec{E}(\vec{r})$ is associated to a scalar potential $V(\vec{r})$ such that $\vec{E}(\vec{r}) = -\nabla V(\vec{r})$. If the electric ...
"A four-dimensional differentiable (Hausdorff and paracompact) manifold $M$ will be called a space time if it possesses a pseudo-Riemannian metric of hyperbolic normal signature $(+,-,-,-)$ and a time ...