What is Gradient, divergence, curl, Laplacian? — Physics

Definition 11.1 University Physics — Year 2 · Chapter 11 — Maxwell’s Equations

For a scalar field $f$ and a vector field $\vect A$ , in Cartesian coordinates with the symbolic vector $\vect\nabla = (\partial_x, \partial_y, \partial_z)$ :

\begin{align*} \operatorname{\vect{grad}}f &= \vect\nabla f = (\partial_xf,\ \partial_yf,\ \partial_zf) ,\\ \operatorname{div}\vect A &= \vect\nabla\cdot\vect A = \partial_xA_x + \partial_yA_y + \partial_zA_z ,\\ \operatorname{\vect{curl}}\vect A &= \vect\nabla\wedge\vect A = (\partial_yA_z - \partial_zA_y,\ \partial_zA_x - \partial_xA_z,\ \partial_xA_y - \partial_yA_x) ,\\ \Delta f &= \operatorname{div}\operatorname{\vect{grad}}f = \partial_x^2f + \partial_y^2f + \partial_z^2f , \end{align*}

and $\Delta\vect A$ is the Laplacian taken on each Cartesian component. Their meaning is independent of coordinates: $\operatorname{\vect{grad}}f \cdot\dd\vect l = \dd f$ is the change of $f$ along $\dd\vect l$ ; $\operatorname{div} \vect A$ is the outgoing flux of $\vect A$ through the surface of a small volume, per unit volume; $\operatorname{\vect{curl}}\vect A\cdot\vect n$ is the circulation of $\vect A$ around a small loop of normal $\vect n$ , per unit area.

The two operators of the Maxwell equations, read geometrically: the divergence measures how much a field flows out of a small volume, the curl how much it circulates around a small loop.

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