Remembering which way Jacobians go – Taking derivatives of vectors with respect to vectors

Jessica YungMathematics

Matrices of the derivative of vectors with respect to vectors (Jacobians) take a specific form:

\frac{\partial\underline{f}}{\partial\underline{x}}(\underline{x_e}, \underline{u_e}) = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & ... & \frac{\partial f_n}{\partial x_n} \end{bmatrix}

Here, note that

  1. each column is the partial of f with respect to one component x_j, whereas
  2. each row is the partial of f_i with respect to the x_j. That is, the rows ‘cover’ the range of f.

You can then easily remember that C: the columns are components (of the inputs), and R: the rows cover the ranges.