TeXipedia

nabla

Represents the gradient, divergence, or curl operator in vector calculus and mathematical physics.

Overview

Essential in multivariable calculus and physics for describing spatial rates of change and vector field operations.

  • Used to calculate gradients of scalar fields (∇f)
  • Forms divergence when applied to vector fields (∇·F)
  • Creates curl operations in three dimensions (∇×F)
  • Appears frequently in electromagnetic theory, fluid dynamics, and quantum mechanics
  • Common in expressing partial differential equations and conservation laws

Examples

Gradient of a scalar field in three dimensions.

∇f=∂f∂xi+∂f∂yj+∂f∂zk\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} 
\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}

Divergence of a vector field.

∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} 
\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Laplacian operator in two dimensions.

∇2f=∂2f∂x2+∂2f∂y2\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} 
\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}