section_path: In calculus/Gradient, divergence and curl
section_name: Gradient, divergence and curl
paragraph_text: Gradient, divergence and curl
  
  In a rectangular coordinate system, the gradient of a (differentiable) function f: R 3 → R {\displaystyle f:\mathbb {R} ^{3}\rightarrow \mathbb {R} } is given by
  
  and in index notation is written
  
  The divergence of a (differentiable) vector field F = U i + V j + W k, that is, a function F: R 3 → R 3 {\displaystyle \mathbf {F} :\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}, is equal to the scalar -valued function:
  
  In index notation, with Einstein summation convention this is
  
  Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of:
  
  where i, j, and k are the unit vectors for the x -, y -, and z -axes, respectively. This expands as follows:
  
  In index notation, with Einstein summation convention this is
  
  where ϵ i j k {\displaystyle \epsilon _{ijk}} is the totally antisymmetric symbol, the Levi-Civita symbol.