Curl and divergence (∇×, ∇·) in LaTeX
Combining \nabla with \times and \cdot gives curl (\nabla \times \mathbf{F}) and divergence (\nabla \cdot \mathbf{F}) of a vector field, while \nabla^2 gives the Laplacian, the three core second-family operators of vector calculus.
LaTeX command
\nabla \times \mathbf{F}Not the symbol you meant? Draw it to find the command →
Curl, \nabla \times \mathbf{F}, measures the local rotation of a vector field and produces another vector field. Divergence, \nabla \cdot \mathbf{F}, measures net outward flux or source strength at a point and produces a scalar. Keep the operators straight: cross product for curl, dot product for divergence, never the reverse.
The Laplacian, \nabla^2 f (also written \Delta f), is the divergence of the gradient, \nabla \cdot (\nabla f), and is a scalar operator acting on a scalar field. It is central to Laplace's equation \nabla^2 \phi = 0 and to the heat and wave equations.
Two identities are worth memorising because they show up constantly: the curl of any gradient is always zero, and the divergence of any curl is always zero. Both follow from the equality of mixed partial derivatives and are used repeatedly to simplify vector calculus expressions.
Other forms
\nabla \cdot \mathbf{F}divergence\nabla^2LaplacianExamples
$\nabla \times \mathbf{F}$curl of a vector field
$\nabla \cdot \mathbf{F}$divergence of a vector field
$\nabla^2 \phi = 0$Laplace's equation
$\nabla \times (\nabla f) = \mathbf{0}$the curl of any gradient is zero
$\nabla \cdot (\nabla \times \mathbf{F}) = 0$the divergence of any curl is zero
Tips and common mistakes
- Bold the vector field,
\mathbf{F}(or use\vec{F}), so it reads as clearly different from a scalar function like\phi. \nabla \times \mathbf{F}is curl, a vector;\nabla \cdot \mathbf{F}is divergence, a scalar; don't swap\timesand\cdot, they give genuinely different quantities.- The Laplacian can be written either
\nabla^2f or\Deltaf; both are standard, so pick one convention and stay consistent within a document. - Remember
\nabla^2is literally\nabla \cdot \nabla, the divergence of the gradient, a useful way to recall why it is a scalar operator even though\nablaitself behaves like a vector. - Curl and divergence are only defined for a proper vector field (typically three-component); don't apply
\timesto a plain scalar function.
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Convert your maths to LaTeX →FAQ
How do I write curl in LaTeX?
\nabla \times \mathbf{F}, using \times for the cross product between del and the vector field.
How do I write divergence in LaTeX?
\nabla \cdot \mathbf{F}, using \cdot for the dot product between del and the vector field.
What is the LaTeX code for the Laplacian?
\nabla^2 f, or equivalently \Delta f.
What's the difference between \nabla f, \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F}?
\nabla f is the gradient of a scalar field (a vector output). \nabla \cdot \mathbf{F} is the divergence of a vector field (a scalar output). \nabla \times \mathbf{F} is the curl of a vector field (a vector output).