Gradient & Divergence Calculator

Scalar Function f(x,y,z)

Understanding Vector Calculus

Vector Operators

Gradient (∇f)

∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Divergence (∇·F)

∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Curl (∇×F)

∇×F = (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k

Physical Interpretations

Gradient

  • Direction of steepest increase
  • Rate of change in all directions
  • Force field from potential energy

Divergence

  • Flux density at a point
  • Source/sink strength
  • Rate of expansion/contraction

Curl

  • Rotation intensity
  • Circulation density
  • Vorticity in fluid flow

Applications in Physics

Electromagnetism

  • Electric field from potential (∇V)
  • Gauss's law (∇·E)
  • Magnetic field circulation (∇×B)

Fluid Dynamics

  • Pressure gradients
  • Flow convergence/divergence
  • Vorticity analysis

Heat Transfer

  • Temperature gradients
  • Heat flux divergence
  • Thermal conductivity