Exterior derivative

The operator $d$ that turns a $k$ -form into a $(k{+}1)$ -form -- grad, curl, and div are all just $d$ in disguise.

The exterior derivative $d$ raises a $k$ -form to a $(k+1)$ -form, generalizing the ordinary differential. In $\mathbb{R}^3$ , applying $d$ to a 0-form (function) gives the gradient, applying it to a 1-form gives the curl, and applying it to a 2-form gives the divergence -- one operator, three classical disguises.

Examples

$d(f) = f_x\,dx+f_y\,dy+f_z\,dz$ -- the gradient, restated as a 1-form.