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Physics · Glossary

What is Gradient, divergence, curl, Laplacian?

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

For a scalar field ff and a vector field A→\vect A, in Cartesian coordinates with the symbolic vector ∇→=(∂x,∂y,∂z)\vect\nabla = (\partial_x, \partial_y, \partial_z):

grad→⁡f=∇→f=(∂xf, ∂yf, ∂zf),div⁡A→=∇→⋅A→=∂xAx+∂yAy+∂zAz,curl→⁡A→=∇→∧A→=(∂yAz−∂zAy, ∂zAx−∂xAz, ∂xAy−∂yAx),Δf=div⁡grad→⁡f=∂x2f+∂y2f+∂z2f,\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 ΔA→\Delta\vect A is the Laplacian taken on each Cartesian component. Their meaning is independent of coordinates: grad→⁡f⋅ ⁣dl→= ⁣df\operatorname{\vect{grad}}f \cdot\dd\vect l = \dd f is the change of ff along  ⁣dl→\dd\vect l; div⁡A→\operatorname{div} \vect A is the outgoing flux of A→\vect A through the surface of a small volume, per unit volume; curl→⁡A→⋅n→\operatorname{\vect{curl}}\vect A\cdot\vect n is the circulation of A→\vect A around a small loop of normal n→\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.
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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