Notation for second derivative of vector argument

Question

In my work I use derivatives of a scalar function f of a (column) vector argument x. I use the notation $\frac{df}{dx}$ for the row vector first derivative, as well as, $\frac{d^2f}{dx^2}$ for the second derivative matrix. I have been strongly suggested by my advisor that this is not an appropriate notation and I should use $\frac{d^2f}{dxdx^T}$ instead. However, what should I do with third derivative then?

Answer 1

If you're willing to use it, this is solved by index notation (if not, then feel free to disregard this answer). In that case, the derivatives are simply $$ \frac{df}{dx^i}, \frac{df^2}{dx^idx^j}, \frac{df^3}{dx^idx^jdx^k}, \cdots $$ where $i,j,k$ are simply indices into the components of the $x$ vector.

In index notation, an upper index corresponds to the entries what you would normally consider a column vector, and a lower index corresponds to the entries of a row vector. A matrix has one upper and one lower index, and anything with more than one index either up or down isn't really representable in regular matrix-and-vector arithmetic (something with two upper indices, for instance, has in a sense two column dimensions and no row dimensions rather than one column dimension and one row dimension).

The upper indices in the derivative "denominator" are flipped and considered overall to be lower indices. Your teacher's desire to write $\frac{df^2}{dxdx^T}$ is, in this framework, considered a misguided attempt at consolidating two indices of the same kind into the familiar matrix, even though it's not entirely right (it can still be written down as an $n\times n$ table, but calling it a matrix isn't entirely right).