Del operator ∇

Definition

The del operator, also called nabla, is a compact way to write spatial derivatives.

In Cartesian coordinates:

$$\nabla =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)$$

It looks like a vector, but do not treat it as a normal vector. It is an operator. It does something to whatever comes after it.

Gradient

For a scalar field $f(x,y,z)$,

$$\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$$

Read this as: “which way does $f$ increase fastest?”

The gradient points uphill. Its size tells you how steep that uphill direction is. In ML, gradient descent goes in the negative gradient direction because that is downhill for the loss.

Divergence

For a vector field $\mathbf{F}=(F_x,F_y,F_z)$,

$$\nabla \cdot \mathbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$

Read this as: “is stuff flowing out of this point, into this point, or neither?”

• positive divergence: source-like, field spreads out
• negative divergence: sink-like, field flows in
• zero divergence: no net source or sink

Solenoidal magnetic fields and no magnetic monopoles uses:

$$\nabla \cdot \mathbf{B}=0$$

meaning magnetic field lines do not start or end.

Curl

For a vector field $\mathbf{F}=(F_x,F_y,F_z)$,

$$\nabla \times \mathbf{F}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)$$

Read this as: “would a tiny paddle wheel spin here?”

Curl measures local rotation. The direction of the curl vector is the rotation axis, using the right-hand rule.

Laplacian

$${\nabla }^{2}f=\nabla \cdot (\nabla f)=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$$

Read this as: “how different is this point from its local neighbourhood?”

It shows up in diffusion, waves, potentials, heat, and field equations.