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I don't quite understand why a standard Jacobian matrix $$JF(x) = \left[{\partial F_i\over \partial x_j}\right]_{ij}$$

is alternant matrix.

Because I think Jacobian matrix uses the same $\alpha$, or the same variate in the expression, so it is not a alternant matrix.

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Hint

Define $$f_i(x)={\partial F_i(x)\over \partial x}$$and try to represent $J\ F(x)$ as a matrix with entries being of form $f_i(x_j)$.

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