Gradient, Divergence, and Curl

Spatial Operators Overview

These three differential operators map how scalar and vector fields change across space. They form the core building blocks of vector calculus.

Common pitfall: Always type-check your inputs and outputs. A phrase like div of a scalar is mathematically meaningless since divergence only accepts vector fields.

Formulas and Identities

The gradient, divergence, and curl use distinct partial derivative structures:

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

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

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

Key identities connect these operations:

• $\nabla\times(\nabla\phi) = \vec{0}$ (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: $\vec{E} = -\nabla V$ defines electric field as the negative gradient of potential, while $\nabla\cdot\vec{E} = \rho/\varepsilon_0$ is Gauss's law.