What is the meaning of the notation $\Phi_\mu g=\frac{\partial(f_1,...,f_{\rho},g)}{\partial(x_{\mu_1},...,x_{\mu_\rho},x_k)}$?

Question

I was reading a paper by Whitney H. titled, "Elementary Structure of Real Algebraic Variety", published in the Annal of Mathematics, Vol. 66, No. 3, November 1957.

In section 6 of the paper, he defines $\Phi_\mu g=\frac{\partial(f_1,...,f_{\rho},g)}{\partial(x_{\mu_1},...,x_{\mu_\rho},x_k)}$. Here, $f_1,\dotsc,f_\rho$ are polynomials with independent differentials. Now, I haven't come accross and am unsure of what the notation in that definition means. It would be great if somebody could help.

Answer 1

This is a community-wiki post to clear this question from the unanswered list. We note the comment of Mohan is correct:

The notation $\frac{\partial(f_1,\ldots, f_n)}{\partial(x_1,\ldots, x_n)}$ usually means, the determinant of the $n\times n$ matrix whose $ij^{\mathrm{th}}$ term is $\frac{\partial f_i}{\partial x_j}$. - Mohan

There was some mention in the comments that this may refer to the matrix, but as the text of the paper immediately before this reads "define the polynomial...", we see that this interpretation is not applicable and Mohan's comment resolves the issue.