Vector Calculus Examples

Small examples for the four main operators. Nothing fancy, just enough to see what each one is doing.

Gradient

Let

$$f(x,y)=x^2+3y^2$$

Then

$$\nabla f=(2x,6y)$$

At $(1,1)$:

$$\nabla f(1,1)=(2,6)$$

So the function climbs fastest mostly in the $y$ direction.

This is the same basic idea used in ML Equations and definitions: follow the negative gradient to reduce a loss.

Divergence

For $\mathbf{F}=(x,y,z)$,

$$\nabla \cdot \mathbf{F}=1+1+1=3$$

This field points away from the origin. Positive divergence means source-like behaviour: arrows spreading out.

For magnetic fields, Solenoidal magnetic fields and no magnetic monopoles says:

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

so magnetic field lines do not begin or end.

Curl

For $\mathbf{F}=(-y,x,0)$,

$$\nabla \times \mathbf{F}=(0,0,2)$$

This field swirls anticlockwise in the $xy$-plane.

The curl points in the $z$ direction because that is the rotation axis by the right-hand rule.

Laplacian

For $f=x^2+y^2+z^2$,

$${\nabla }^{2}f=2+2+2=6$$

The positive Laplacian says each point is locally below the average of nearby values. Picture a bowl: the centre is lower than what surrounds it.

Physical reading

• gradient: hill slope
• divergence: water appearing from a spring or draining into a sink
• curl: paddle wheel spinning in a flow
• Laplacian: local imbalance with neighbours

When the algebra gets messy, come back to these pictures.