Nabla Symbol (∇): Meaning, Uses and How to Type It

The nabla symbol ∇ is an upside-down triangle that stands for the del operator in vector calculus. Applied to a function it gives the gradient ∇f, combined with a dot or a cross it gives the divergence ∇ · F and the curl ∇ × F, and squared it gives the Laplacian ∇². It also appears in every machine-learning paper as the gradient of a loss function.

Below: the character to copy, the codes for every platform, what ∇ means in each formula, and how to write it in LaTeX.

Nabla Symbol (∇): Copy and Paste

Copy the symbol from the box below, or take the code you need from the table:

1∇
Copy

Do not substitute the triangle shapes ▽ (U+25BD) or ∆ (U+2206, increment). They are different characters, and screen readers and search engines will not recognise them as nabla.

Nabla Definition

In three-dimensional Cartesian coordinates, ∇ is the vector of partial derivative operators:

∇ = (∂/∂x, ∂/∂y, ∂/∂z)

On its own it does nothing; it acts on whatever follows it. William Rowan Hamilton introduced the operator in the 1830s and 1840s, and Peter Guthrie Tait developed it. The name comes from the Greek word for a harp of Phoenician origin, which the inverted triangle resembles. "Del" is the more common spoken name in English-language physics.

Uses of the Nabla Symbol

• Gradient, ∇f: for a scalar field f(x, y, z), ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) is a vector pointing in the direction of steepest increase. In machine learning, gradient descent updates parameters with θ ← θ − η ∇L(θ).
• Divergence, ∇ · F: the dot product of ∇ with a vector field measures how much the field spreads out from a point. Gauss's law in electromagnetism reads ∇ · E = ρ/ε₀.
• Curl, ∇ × F: the cross product of ∇ with a vector field measures rotation. Faraday's law reads ∇ × E = −∂B/∂t.
• Laplacian, ∇² = ∇ · ∇: the sum of second partial derivatives, ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z². It appears in the heat, wave and Schrödinger equations, and is also written Δ (see delta symbols). In differential geometry, ∇ also denotes a connection or covariant derivative, as in ∇_X Y.

How to Type the Nabla Symbol

• Windows: there is no Alt code. In Word, type 2207 and press Alt + X. Elsewhere, copy ∇ from Character Map or from this page.
• Mac: open the Character Viewer (Control + Command + Space) and search "nabla," or enable the Unicode Hex Input source and hold Option while typing 2207.
• Word equations: press Alt + =, type \nabla and a space.
• Google Docs: Insert → Special characters and search "nabla," or type \nabla in an equation.
• HTML: ∇ or ∇. In MathML it is <mo>&nabla;</mo>.
• Phones: no standard keyboard includes ∇; copy it or create a text replacement.

Nabla in LaTeX

The command \nabla works in math mode with no package. Vectors are usually bold, and some authors also make ∇ bold to stress that it is a vector operator.

1\documentclass{article}
2\usepackage{amsmath}
3\usepackage{bm}
4\begin{document}
5Gradient: $\nabla f = \left( \frac{\partial f}{\partial x},
6  \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$
7
8Divergence and curl:
9\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad
10   \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
11
12Laplacian: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$
13
14Bold nabla: $\bm{\nabla} \cdot \mathbf{F}$
15
16Gradient descent: $\theta \leftarrow \theta - \eta \, \nabla_{\theta} L(\theta)$
17\end{document}
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• \mathbf{\nabla} does not make ∇ bold, because it is a symbol rather than a letter; use \bm{\nabla} from the bm package or \boldsymbol{\nabla} from amsmath.
• The physics package adds \grad, \div, \curl and \laplacian. Note that it redefines \div, which normally prints ÷.
• A subscript names the variable: \nabla_{\theta} L is the gradient with respect to θ.

For more on writing derivatives and operators, see the mathematical expressions lesson, the partial derivative symbol, and the full calculus symbol list with LaTeX commands.

Related symbols and guides

• Nabla symbol page — copy the symbol and its LaTeX command
• Math symbols reference — every symbol in the category with its LaTeX command
• Derivative rules — how to typeset these symbols in a document
• Partial derivative symbol (∂)
• Delta symbols (Δ δ)

Nabla Symbol FAQ

What does ∇ mean?

∇ is the del operator, the vector of partial derivatives (∂/∂x, ∂/∂y, ∂/∂z). ∇f is the gradient of f, ∇ · F the divergence, ∇ × F the curl and ∇² the Laplacian.

How do you pronounce ∇?

Usually "del," as in "del f" for ∇f, or "nabla." "Grad f" is also common for the gradient.

What is the difference between ∇ and Δ?

∇ (nabla) is the vector differential operator. Δ (capital delta) means a change in a quantity, as in Δx, and in vector calculus it is also used for the Laplacian, so Δf = ∇²f.

How do I type nabla in LaTeX?

Write \nabla in math mode, for example $\nabla f$ or $\nabla \cdot \mathbf{F}$. For a bold nabla use \bm{\nabla}.

Is there an Alt code for ∇?

No. ∇ is not in the old Windows code pages. In Word type 2207 then Alt + X, or copy the symbol from this page.