Gradient, Divergence, and Curl

These three differential operators describe how scalar and vector fields change in space.

Gradient — Acts on a scalar field $\phi(x,y,z)$, producing a vector:

$$\nabla\phi = \frac{\partial\phi}{\partial x}\hat{i} + \frac{\partial\phi}{\partial y}\hat{j} + \frac{\partial\phi}{\partial z}\hat{k}$$

$\nabla\phi$ points in the direction of steepest increase and its magnitude is the rate of change.

Divergence — Acts on a vector field $\vec{F}$, producing a scalar:

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

Measures the net outward flux per unit volume — sources ($>0$) and sinks ($<0$).

Curl — Acts on a vector field $\vec{F}$, producing a vector:

$$\nabla\times\vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ F_x & F_y & F_z \end{vmatrix}$$

Measures the local rotation of the field.

Key identities

• $\nabla\times(\nabla\phi) = \vec{0}$ (curl of gradient is zero)
• $\nabla\cdot(\nabla\times\vec{F}) = 0$ (divergence of curl is zero)
• $\nabla^{2}\phi = \nabla\cdot(\nabla\phi)$ (the Laplacian)

Physics link: $\vec{E} = -\nabla V$ (electric field is the negative gradient of potential) and $\nabla\cdot\vec{E} = \rho/\varepsilon_0$ (Gauss's law in differential form).

Common pitfall: Gradient, divergence and curl take different objects in and out: grad turns scalars into vectors; div turns vectors into scalars; curl turns vectors into vectors. Type-check before computing — "div of a scalar" is meaningless.

Vector Fields and the Gradient

A vector field assigns a vector $\vec{F}(x,y)$ to every point in a region.

The gradient of a scalar field $f$ is:

$$\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j}$$

It points in the direction of steepest increase; its magnitude is the rate of change.

A field like $\vec{F} = (-y, x)$ is rotational: streamlines form closed loops, with curl $\nabla \times \vec{F} = 2\hat{k}$.

Gradient, divergence, and curl are the building blocks of vector calculus.