nabla

The nabla is a vector differential operator denoted by the symbol ∇, used in vector calculus to express gradient, divergence, curl, and the Laplacian. It is commonly referred to as the nabla operator, and the symbol ∇ is an inverted delta; the term del operator is also used in many contexts.

In Cartesian coordinates, ∇ is represented as the triple (∂/∂x, ∂/∂y, ∂/∂z). The operator acts on scalar or vector fields. For a scalar field f(x, y, z), the gradient is ∇f, a vector field whose components are the partial derivatives ∂f/∂x, ∂f/∂y, and ∂f/∂z. For a vector field F = (F1, F2, F3), the divergence is ∇·F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z, and the curl is ∇×F = (∂F3/∂y − ∂F2/∂z, ∂F1/∂z − ∂F3/∂x, ∂F2/∂x − ∂F1/∂y). The Laplacian, applied to a scalar field, is ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z².

Beyond Cartesian coordinates, ∇ generalizes to curvilinear systems (cylindrical, spherical, etc.), where its components involve coordinate scale factors and metric terms. This leads to more complex expressions that still capture the same geometric intuition: the rate and direction of change of fields in space.

Applications are widespread in physics, engineering, and mathematics, including electromagnetism, fluid dynamics, thermodynamics, and computer graphics. The nabla operator provides compact, coordinate-aware formulations of fundamental differential relationships and field equations.