gradiente

Gradiente, a term used in Spanish and Portuguese, refers to the gradient in multivariable calculus and related disciplines. For a differentiable scalar field f: R^n → R, the gradient of f is the vector ∇f = (∂f/∂x1, ..., ∂f/∂xn). It points in the direction of the steepest ascent of f, and its magnitude |∇f| represents the rate of increase in that direction.

Geometrically, the gradient is perpendicular to the level sets of f, the surfaces where f is constant. The directional derivative of f in a unit direction u is given by D_u f = ∇f · u. When the gradient vanishes at a point, the function has a critical point, indicating no instantaneous increase in any direction.

The gradient is a central tool in many fields. In optimization, gradient descent moves opposite to ∇f to locate minima. In physics, forces often arise from gradients of potential energy, with F = -∇V. In image processing and computer vision, gradients help detect edges by highlighting regions with rapid changes in intensity. Gradients are also used to describe fluxes in meteorology, geophysics, and engineering, such as temperature or pressure differences across a domain.

Computationally, the gradient is obtained from analytic differentiation when f is known explicitly. If f is not available in closed form, finite difference methods or automatic differentiation are commonly employed. The gradient concept extends to vector-valued fields via Jacobians, but the gradient itself refers to scalar fields. The term gradiente is the customary label in Spanish and Portuguese for this mathematical construct.