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Yes, it looks like it's integrated (and sometimes it is even called integrating), but really it is due to the Euler theorem. Suppose $X$ is a first order homogeneous function of $n_1, n_2, \dots$ (this is equivalent to saying it is extensive) and that $\mathrm{d}X(\mathbf{n}) = \sum_i \frac{\partial X(\mathbf{n})}{\partial n_i}\textrm{d}n_i$. This is to ...

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