Gradient, Divergence, Curl, and Laplacian
Definitions, Physical Meaning, and Identities
The four basic operations of vector calculus — gradient (grad), divergence (div), curl, and the Laplacian — organized around the nabla operator $\nabla$, with definitions, physical meaning, and figures.
Contents
1. The Nabla Operator
Definition: The nabla operator $\nabla$
In a three-dimensional Cartesian coordinate system, the nabla operator is the vector differential operator defined by
$$\nabla = \left( \dfrac{\partial}{\partial x},\; \dfrac{\partial}{\partial y},\; \dfrac{\partial}{\partial z} \right)$$$\nabla$ is an operator built by lining up partial derivatives like the components of a vector; it is not a vector in itself. Yet treating it formally as a vector lets us describe the gradient, divergence, and curl in a unified way. Acting on a scalar field it produces a vector field (the gradient); its dot product with a vector field gives the divergence; and its cross product with a vector field gives the curl.
$\nabla$ is an operator and acts on whatever stands to its right. The expressions $\nabla f$ and $f \nabla$ have different meanings, so the order matters.
2. Gradient (grad)
Definition: gradient
The gradient of a scalar field $f(x, y, z)$ is the vector field
$$\operatorname{grad} f = \nabla f = \left( \dfrac{\partial f}{\partial x},\; \dfrac{\partial f}{\partial y},\; \dfrac{\partial f}{\partial z} \right)$$Physical meaning
- Direction: $\nabla f$ points in the direction in which $f$ increases most rapidly.
- Magnitude: $|\nabla f|$ equals the maximum rate of change in that direction.
- On a level surface $f = \text{const}$, $\nabla f$ is everywhere normal to the surface.
Relation to the total differential
The total differential of $f$ is the dot product of the gradient with the infinitesimal displacement $d\mathbf{r} = (dx, dy, dz)$:
$$df = \nabla f \cdot d\mathbf{r} = \dfrac{\partial f}{\partial x} dx + \dfrac{\partial f}{\partial y} dy + \dfrac{\partial f}{\partial z} dz$$Example: gravitational potential
For a point mass $M$ located at the origin, the gravitational potential is
$$\varphi = -\dfrac{GM}{r}, \quad r = \sqrt{x^2 + y^2 + z^2}$$The gravitational field (the force per unit mass) is $\mathbf{g} = -\nabla \varphi$, which points in the direction of decreasing potential (i.e., toward the mass).
$$\mathbf{g} = -\nabla \varphi = -\dfrac{GM}{r^3} \mathbf{r}$$The force on a body of mass $m$ is then $\mathbf{F} = m\mathbf{g}$.
Property: the curl of a gradient is zero
For any $C^2$ scalar field $f$,
$$\nabla \times (\nabla f) = \mathbf{0}$$That is, every gradient field is irrotational.
Proof
Compute each component of the curl of $\nabla f = \left(\dfrac{\partial f}{\partial x},\, \dfrac{\partial f}{\partial y},\, \dfrac{\partial f}{\partial z}\right)$.
The $x$-component:
$$\dfrac{\partial}{\partial y}\!\left(\dfrac{\partial f}{\partial z}\right) - \dfrac{\partial}{\partial z}\!\left(\dfrac{\partial f}{\partial y}\right) = \dfrac{\partial^2 f}{\partial y\,\partial z} - \dfrac{\partial^2 f}{\partial z\,\partial y}$$Since $f$ is of class $C^2$, Schwarz's theorem (equality of mixed partial derivatives) gives
$$\dfrac{\partial^2 f}{\partial y\,\partial z} = \dfrac{\partial^2 f}{\partial z\,\partial y}$$so the $x$-component vanishes. The $y$- and $z$-components vanish in the same way:
$$y\text{-component}: \quad \dfrac{\partial^2 f}{\partial z\,\partial x} - \dfrac{\partial^2 f}{\partial x\,\partial z} = 0$$ $$z\text{-component}: \quad \dfrac{\partial^2 f}{\partial x\,\partial y} - \dfrac{\partial^2 f}{\partial y\,\partial x} = 0$$Hence $\nabla \times (\nabla f) = \mathbf{0}$. $\blacksquare$
3. Divergence (div)
Definition: divergence
The divergence of a vector field $\mathbf{A} = (A_x, A_y, A_z)$ is the scalar field
$$\operatorname{div} \mathbf{A} = \nabla \cdot \mathbf{A} = \dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}$$Physical meaning
- $\nabla \cdot \mathbf{A} > 0$: the point is a source (net outflow).
- $\nabla \cdot \mathbf{A} < 0$: the point is a sink (net inflow).
- $\nabla \cdot \mathbf{A} = 0$: the field is solenoidal (incompressible); inflow and outflow balance.
Example: divergence of the position vector
The divergence of the position vector $\mathbf{r} = (x, y, z)$ is
$$\nabla \cdot \mathbf{r} = \dfrac{\partial x}{\partial x} + \dfrac{\partial y}{\partial y} + \dfrac{\partial z}{\partial z} = 3$$and, away from the origin ($r \neq 0$),
$$\nabla \cdot \left( \dfrac{\mathbf{r}}{r^3} \right) = 0$$This expresses the fact that an inverse-square field (such as the gravitational or electrostatic field of a point source) has no divergence except at the source itself. The origin is a singularity: in the sense of distributions, $\nabla \cdot \left(\dfrac{\mathbf{r}}{r^3}\right) = 4\pi\,\delta(\mathbf{r})$, corresponding to the point source (mass or charge).
Example: the continuity equation
The conservation law for a density $\rho$ and flux density $\mathbf{j}$ is
$$\dfrac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0$$This continuity equation appears throughout physics, including conservation of charge and conservation of mass.
Property: the divergence of a curl is zero
For any $C^2$ vector field $\mathbf{A}$,
$$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$That is, every curl field is solenoidal (source-free).
Proof
The components of $\nabla \times \mathbf{A}$ are
$$\nabla \times \mathbf{A} = \left( \dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z},\;\; \dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x},\;\; \dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y} \right)$$Take the divergence:
$$\nabla \cdot (\nabla \times \mathbf{A}) = \dfrac{\partial}{\partial x}\!\left(\dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z}\right) + \dfrac{\partial}{\partial y}\!\left(\dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x}\right) + \dfrac{\partial}{\partial z}\!\left(\dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y}\right)$$Expanding gives six terms:
$$= \dfrac{\partial^2 A_z}{\partial x\,\partial y} - \dfrac{\partial^2 A_y}{\partial x\,\partial z} + \dfrac{\partial^2 A_x}{\partial y\,\partial z} - \dfrac{\partial^2 A_z}{\partial y\,\partial x} + \dfrac{\partial^2 A_y}{\partial z\,\partial x} - \dfrac{\partial^2 A_x}{\partial z\,\partial y}$$Since each component of $\mathbf{A}$ is $C^2$, Schwarz's theorem lets us interchange the order of partial differentiation. Pairing terms with the same component yields
$$= \underbrace{\left(\dfrac{\partial^2 A_z}{\partial x\,\partial y} - \dfrac{\partial^2 A_z}{\partial y\,\partial x}\right)}_{=\,0} + \underbrace{\left(\dfrac{\partial^2 A_x}{\partial y\,\partial z} - \dfrac{\partial^2 A_x}{\partial z\,\partial y}\right)}_{=\,0} + \underbrace{\left(\dfrac{\partial^2 A_y}{\partial z\,\partial x} - \dfrac{\partial^2 A_y}{\partial x\,\partial z}\right)}_{=\,0}$$Hence $\nabla \cdot (\nabla \times \mathbf{A}) = 0$. $\blacksquare$
4. Curl (rot)
Definition: curl (also written rot)
The curl of a vector field $\mathbf{A} = (A_x, A_y, A_z)$ is the vector field
$$\operatorname{curl} \mathbf{A} = \nabla \times \mathbf{A} = \begin{vmatrix} \mathbf{e}_x & \mathbf{e}_y & \mathbf{e}_z \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}$$Component form
Expanding the determinant gives the components
$$\nabla \times \mathbf{A} = \left( \dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z},\; \dfrac{\partial A_x}{\partial z} - \dfrac{\partial A_z}{\partial x},\; \dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y} \right)$$Physical meaning
- $\nabla \times \mathbf{A}$ measures the local circulation density (vorticity) of the vector field.
- Direction: the axis of rotation of the local swirl (right-hand rule).
- Magnitude: the strength of the swirl.
Note that the curl does not measure how curved the streamlines are, but the local rigid-body rotation (vorticity) at each point. Streamlines can be curved while the curl is still zero (a free vortex; see Figures 3–5).
Example: rigid-body rotation
The velocity field of a rigid body rotating with angular velocity $\boldsymbol{\Omega}$ is
$$\mathbf{A} = \boldsymbol{\Omega} \times \mathbf{r}$$Its curl is
$$\nabla \times \mathbf{A} = 2\boldsymbol{\Omega}$$so the vorticity is twice the angular velocity.
Property: the curl of a gradient is zero (recap)
For any $C^2$ scalar field $f$,
$$\nabla \times (\nabla f) = \mathbf{0}$$Conversely, on a simply connected domain, if $\nabla \times \mathbf{A} = \mathbf{0}$ then there exists a scalar potential $f$ with $\mathbf{A} = \nabla f$.
An everyday example: the draining vortex — divergence and curl together
Water spiraling into a drain is a familiar example in which divergence (a sink) and curl (a vortex) appear at the same time. The velocity field splits into a radial part $v_r$ pointing toward the center and a tangential part $v_\theta$ circling the axis.
The sink. For an axisymmetric flow,
$$\nabla \cdot \mathbf{v} = \dfrac{1}{r}\dfrac{d(r\,v_r)}{dr}$$and for the inflow $v_r = -\dfrac{Q}{2\pi r}$ the product $r\,v_r$ is constant, so $\nabla \cdot \mathbf{v} = 0$ — as expected for an incompressible fluid. So throughout the swirling water (including along the funnel's sides) the divergence is $0$, and the sink (negative divergence) is concentrated at the drain: whatever flows in flows out through the hole.
The vortex. A purely radial inflow is irrotational ($\nabla \times \mathbf{v} = \mathbf{0}$). Real drains swirl because the tiny rotation the water already had is amplified near the center by conservation of angular momentum:
$$v_\theta\, r \approx \text{const} \;\Longrightarrow\; v_\theta \sim \dfrac{1}{r}$$so it spins faster as it approaches the center (just as a figure skater spins up by pulling in their arms). The Coriolis force does influence the direction in principle, but on the scale of a sink or funnel it is utterly dwarfed by the water's residual circulation (the "hemisphere decides the direction" claim does not hold at everyday scales).
A vortex with zero curl? The ideal free vortex $v_\theta = \dfrac{C}{r}$ has, away from the center,
$$\nabla \times \mathbf{v} = \dfrac{1}{r}\dfrac{d(r\,v_\theta)}{dr} = 0$$so its vorticity is zero: a small paddle wheel floating there merely orbits the center and does not spin about its own axis — the same at any off-center position (Figure 5). Just as the divergence was concentrated at the drain, the vorticity is concentrated in the core, and where the core rotates as a rigid body $v_\theta = \omega r$ we get $\nabla \times \mathbf{v} = 2\omega$ — the same as the rigid-rotation result $\nabla \times \mathbf{A} = 2\boldsymbol{\Omega}$ above. This rigid core plus free-vortex exterior is the Rankine vortex.
5. The Laplacian
Definition: scalar Laplacian
The Laplacian of a scalar field $f$ is defined as the divergence of the gradient:
$$\nabla^2 f = \operatorname{div}(\operatorname{grad} f) = \nabla \cdot (\nabla f) = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$$Definition: vector Laplacian
The Laplacian of a vector field $\mathbf{A} = (A_x, A_y, A_z)$ is obtained by applying the scalar Laplacian to each component:
$$\nabla^2 \mathbf{A} = (\nabla^2 A_x,\; \nabla^2 A_y,\; \nabla^2 A_z)$$Note: $\nabla^2 = \nabla \cdot \nabla = \operatorname{div} \circ \operatorname{grad}$. The Laplacian is the divergence of the gradient, not the "gradient of the divergence."
The Laplacian in physics
Laplace's equation
A function whose Laplacian vanishes is called a harmonic function.
$$\nabla^2 f = 0$$The electrostatic potential in a charge-free region and a steady-state temperature distribution both satisfy this equation.
Heat equation
The time evolution of a temperature $u$, with thermal diffusivity $\alpha$, obeys
$$\dfrac{\partial u}{\partial t} = \alpha \nabla^2 u$$The Laplacian represents the deviation from the local average: at a point that is colder than its surroundings, $\nabla^2 u > 0$ and the temperature rises.
Wave equation
For a wave with propagation speed $c$,
$$\dfrac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$$This describes a wide range of wave phenomena, including sound waves, electromagnetic waves, and the vibrations of a string.
6. Directional Derivative
Definition: directional derivative
The directional derivative of a scalar field $f$ along a unit vector $\hat{\mathbf{n}}$ is
$$D_{\hat{\mathbf{n}}} f = \nabla f \cdot \hat{\mathbf{n}} = |\nabla f| \cos\theta$$where $\theta$ is the angle between $\nabla f$ and $\hat{\mathbf{n}}$.
The directional derivative attains its maximum value $|\nabla f|$ when $\hat{\mathbf{n}}$ is aligned with $\nabla f$, and its minimum value $-|\nabla f|$ in the opposite direction. It is $0$ in any direction orthogonal to $\nabla f$ (i.e., along the level surface).
The operator $(\mathbf{A} \cdot \nabla)$ on a vector field
Acting on a vector field $\mathbf{B}$, the operator $(\mathbf{A} \cdot \nabla)$ differentiates each component of $\mathbf{B}$ along the direction of $\mathbf{A}$:
$$(\mathbf{A} \cdot \nabla) \mathbf{B} = \left( \mathbf{A} \cdot \nabla B_x,\; \mathbf{A} \cdot \nabla B_y,\; \mathbf{A} \cdot \nabla B_z \right)$$This advection operator plays a central role in the convective term $(\mathbf{v} \cdot \nabla)\mathbf{v}$ of fluid mechanics and the Navier-Stokes equations.
Summary
The four basic operations of vector calculus are summarized below.
| Operation | Notation | Input → Output | Physical meaning |
|---|---|---|---|
| Gradient (grad) | $\nabla f$ | scalar → vector | direction of steepest ascent |
| Divergence (div) | $\nabla \cdot \mathbf{A}$ | vector → scalar | strength of source/sink |
| Curl | $\nabla \times \mathbf{A}$ | vector → vector | strength and axis of rotation |
| Laplacian | $\nabla^2 f$ | scalar → scalar | deviation from local average |
Fundamental identities
- $\nabla \times (\nabla f) = \mathbf{0}$ (every gradient field is irrotational)
- $\nabla \cdot (\nabla \times \mathbf{A}) = 0$ (every curl field is solenoidal)
- $\nabla^2 f = \nabla \cdot (\nabla f)$ (Laplacian = divergence of the gradient)