Gradienteet

Gradienteet (Finnish for gradients) denote the gradient vectors of scalar fields in several variables. For a function f: R^n → R, the gradient is a vector field that at each point points in the direction of the greatest increase of f, and its length equals the rate of increase in that direction. The gradient is denoted ∇f.

Definition: If f is differentiable, ∇f(x) = (∂f/∂x1, ..., ∂f/∂xn). Example: f(x,y) = x^2 + y^2, ∇f(x,y) = (2x, 2y). The gradient is orthogonal to the level sets f = c, meaning its direction is perpendicular to curves or surfaces where f takes a constant value.

Properties: The gradient operator maps a scalar field to a vector field. For smooth f, the curl of a gradient is zero: ∇ × (∇f) = 0, and gradient fields are conservative, meaning line integrals depend only on endpoints. The directional derivative of f in the direction of a unit vector u is D_u f = ∇f · u. For a scalar function, the gradient is related to the Jacobian as the transpose of the Jacobian row vector.

Applications: Gradienteet play a central role in optimization, where gradient descent uses the negative gradient to move toward minima. In physics, the gradient of potential energy gives the force via F = -∇U. In machine learning, gradients are computed to adjust parameters through backpropagation. They also describe spatial variation in fields such as temperature, pressure, or elevation in geography and meteorology.

See also: gradient, Jacobian, Hessian, directional derivative, level set.