Nablaoperator

Nablaoperator, commonly called the nabla operator or del operator, is a vector differential operator used in vector calculus. It is denoted by the symbol ∇ and, in Cartesian coordinates, defined as ∇ = (∂/∂x, ∂/∂y, ∂/∂z). Applied to a scalar field f, it yields the gradient ∇f, a vector pointing in the direction of greatest increase of f with magnitude equal to the rate of increase.

Applied to a vector field F = (F1, F2, F3), the operator yields the divergence ∇·F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z, a scalar describing net outflow per unit volume. The curl ∇×F, a vector, measures rotation: ∇×F = (∂F3/∂y - ∂F2/∂z, ∂F1/∂z - ∂F3/∂x, ∂F2/∂x - ∂F1/∂y). The Laplacian ∇²f is the divergence of the gradient: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²; for a vector field, apply componentwise.

Several identities relate these operations, such as ∇×(∇f) = 0 and ∇·(∇×F) = 0, and the vector identity ∇(∇·F) - ∇×(∇×F) = ∇²F.

In applications, the nabla operator expresses physical laws in physics and engineering, including electrostatics, magnetostatics, fluid flow, heat diffusion, and computer graphics for lighting and surface normals. In non-Cartesian coordinates, expressions involve scale factors, but the principal ideas remain the same.