$D_j f$ to denote the partial derivative of a function between Euclidean spaces, w.r.t. the $j$'th coordinate. For some reason Jacobi's notation $\frac{\partial f}{\partial x_j}$ has become more popular. Jacobi's notation tends to cause much ambiguity and confusion, a point which is emphasized in the book "Multidimensional Real Analysis" by Duistermaat & Kolk. For instance (this example is taken from their book), let $e_1,e_2$ be the standard basis for $\mathbb{R}^2$ and define a new basis by $e'_1 = e_1 + e_2, e'_2 = e_2$. The passage from one basis to another is as follows: If $x_1 e_1 + x_2 e_2 = y_1 e'_1 + y_2 e'_2$ then $y_1 = x_1, y_2 = x_2 -x_1$. Now the meaning of $\frac{\partial y_2}{\partial y_1}$ is ambiguous: If one interprets $y_1$ and $y_2$ as independent coordinate functions, then $\frac{\partial y_2}{\partial y_1} = 0$. On the other hand, $\frac{\partial y_2}{\partial y_1} = \frac{\partial (x_2 -x_1)}{\partial x_1} = -1$, right? This was the result of much confusion for me when I was taught multivariate calculus and the notation $D_j f$ would have eliminated this confusion entirely.

$D_j f$ to denote the partial derivative of a function between Euclidean spaces, w.r.t. the $j$'th coordinate. For some reason Jacobi's notation $\frac{\partial f}{\partial x_j}$ has become more popular. Jacobi's notation tends to cause much ambiguity and confusion, a point which is emphasized in the book "Multidimensional Real Analysis" by Duistermaat & Kolk. For instance (this example is taken from their book), let $e_1,e_2$ be the standard basis for $\mathbb{R}^2$ and define a new basis by $e'_1 = e_1 + e_2, e'_2 = e_2$. The passage from one basis to another is as follows: If $x_1 e_1 + x_2 e_2 = y_1 e'_1 + y_2 e'_2$ then $y_1 = x_1, y_2 = x_2 -x_1$. Now the meaning of $\frac{\partial y_2}{\partial y_1}$ is ambiguous: If one interprets $y_1$ and $y_2$ as independent coordinate functions, then $\frac{\partial y_2}{\partial y_1} = 0$. On the other hand, $\frac{\partial y_2}{\partial y_1} = \frac{\partial (x_2 -x_1)}{\partial x_1} = -1$, right? This was the source of much confusion for me when I was taught multivariate calculus and the notation $D_j f$ would have eliminated this confusion entirely.