Gradien

Gradien, or gradient, is a central concept in multivariable calculus and physics. For a scalar function f that maps n-dimensional space to the real numbers, the gradient of f at a point x is the vector ∇f(x) whose components are the first-order partial derivatives: ∇f(x) = (∂f/∂x1, ..., ∂f/∂xn). The gradient depends on the chosen coordinate system, but its directional meaning is intrinsic to the space.

Interpretation and basic properties. The gradient points in the direction of the steepest ascent of the function. Its magnitude equals the maximum rate of increase of f per unit distance. The gradient is orthogonal to level sets (or level surfaces) of f, meaning it is normal to contours of equal value.

Directional derivative and optimization. The rate of change of f in any unit direction u is given by the directional derivative, D_u f(x) = ∇f(x) · u. In optimization, gradient-based methods use this vector to move toward local minima: x_{new} = x - η ∇f(x), where η is a step size.

Applications and examples. Gradient concepts appear across disciplines: in physics, the gradient of a potential function relates to force via F = -∇V; in machine learning, gradient descent and backpropagation rely on computing gradients of loss functions; in image processing, gradients help detect edges.

Example. For f(x, y) = x^2 + y^2, the gradient is ∇f(x, y) = (2x, 2y). The magnitude increases with distance from the origin, and its direction points away from the origin.

Notes. The gradient is defined for differentiable scalar fields. For vector fields, related concepts include divergence and curl. In discrete settings, gradients can be approximated by finite differences or finite difference operators.