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3

Essentially, I would like to prove $$ \sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$ \rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. As mentioned in the comments, you need to introduce a measure on the LRS to get the dimensions to work out. To put it another way, your $f(k)$ on the LHS can't be the same as your ...


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A very good book along the lines you seem to want is Gallavotti's Statistical Mechanics - A Short Treatise, which can be downloaded from here. He covers many of the classical topics, with a detailed discussion of foundational issues, the role of ergodicity/mixing, etc. From a very different point of view, with a colleague, we are writing a mathematically ...


2

My personal favorite is "Mathematical Foundations Of Statistical Mechanics" by A. I. Khinchin (a mathematician) and G. Gamow. The content remains mathematically rigorous throughout, but nonetheless very readable. In chapter two, both the Liouville and Birkhoff theorems are derived, followed up by a long discussion on metric decomposability of phase space and ...


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Mathematics is a tool for physics, to model pre-existing data and predict future observations and measurements. Physics is one of the areas where mathematics is used in this fashion. If one considers measuring the fields in flat land, as the Nile valley in Egypt, as data, then the origin of western axiomatic mathematics was also from fitting observations, ...


1

To avoid to decide if his derivative $\dot u(x)$ is a covariant or a contravariant object (or perhaps to go for the contravariant one). Seriously. Of course not rigorously, nor even formally. Duality will enter scene in the XIXth-XXth centuries. We got used to integrate a density across a path, or to multiply vector and covectors from the tangent and ...


6

The places in physics where commutation of partial derivatives tends to be important are in the identities of vector calculus. The situations where these identities might seem to break down is when there is some kind of topological winding. Then the partial derivatives commute at almost all points except some small set where they are undefined but still can ...


1

Singularities in functions often lead to non commuting second derivatives. As for a Physical interpretation I think the following exercise may help: The partial derivative can be from First Principles can be written as df(x,y)/dx = (f(x+h,y)-f(x,y))/h i.e the function is incremented by h and then the derivative is found. (x,y+h). .(x+h,y+h) (x,y). ...


2

A discontinuity in the flow of water could be a wall, or a clog that water is still getting around, but not flowing directly through. In finite electric current it could be a substance with a different conductivity, notably zero or ∞. In theory, the mixed partial second derivatives would not be generally equal, just on the cusp of a boundary such as these. ...


2

If you want to be physical, you'd have to have a physical interpretation of the derivatives. If you've already taken two derivatives you can ask yourself whether it is possible to take the gradient of those second derivatives. If so, then the second derivatives commuted, if not then the second derivatives are weird (if something wasn't weird you could take ...


5

The general requirement you are looking for is that the particular function be of class $C^1$, where ...if all order $p$ partial derivatives evaluated at a point $\mathbf a$: $$\frac{\partial^p}{\partial x_1^{p1}\partial x_1^{p2}\cdots\partial x_n^{pn}}f\left(\mathbf x\right)\vert_{\mathbf x=\mathbf a}$$ exist and are continuous, where $p1,\,p2, ..., ...



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