Where in Eulers writings can I find a proof of his homogeneous function theorem: $y$ is a homogeneous function of degree $k$ in $x_1,\ldots,x_n$ iff $ky = \sum_{i=1}^n x_i\frac{\partial y}{\partial x_i}$? None of the sources I've seen cite him.

I'm curious because in his Introduction to the analysis of the infinite he defines a homogeneous function as one "in which each term has the same degree" and goes on to discuss several examples (not just polynomials). But I don't see him writing down the "modern" definition $$f(\lambda{\bf x})=\lambda^k f({\bf x})$$ and wonder how he would prove the result without this. (I know he used function notation, but very rarely.)


1 Answer 1


Lucky you! (or me :-) ). This question was answered a while back on Math.SE

I found his original thoughts in the translated version of "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1", chapter 7. The translation is called "Foundations of Differential Calculus" and a link is found here https://link.springer.com/book/10.1007%2Fb97699 .

  • $\begingroup$ Thanks. So it starts in par. 220 and indeed he does not use $f(x)$ notation! $\endgroup$ Jan 22, 2020 at 8:56
  • $\begingroup$ Though as far as I can tell from that book he only proves the direction $\implies$. $\endgroup$ Jan 22, 2020 at 10:10

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