gradyan

Gradyan is the mathematical concept of the gradient of a scalar field, commonly used in Turkish-language mathematics to refer to the gradient. For a differentiable function f: R^n → R, the gradient ∇f is the n-dimensional vector of partial derivatives (∂f/∂x1, ..., ∂f/∂xn). It points in the direction of the steepest increase of f, and its magnitude equals the rate of the fastest increase.

Geometrically, the gradient is normal to the level sets of f, the surfaces where f(x) is constant. The operator ∇, also called the del operator, is a central tool in vector calculus and is used to compute gradients, divergences, and curls.

Example: for f(x, y) = x^2 + y^2, the gradient is ∇f = (2x, 2y). At the point (1, 2), ∇f = (2, 4). The directional derivative of f in the direction of a unit vector u is the dot product ∇f · u, illustrating how the gradient encodes rate of change in every direction.

Properties include linearity and the chain rule. The magnitude |∇f| gives the maximum rate of increase, and the gradient direction provides the path of steepest ascent. Negative gradient directions yield steepest descent, a principle exploited in optimization algorithms such as gradient descent.

Applications span optimization, machine learning, physics, and engineering. In machine learning, gradients are propagated in backpropagation to update model parameters; in physics, forces are often described as the negative gradient of potential energy. In numerical analysis, gradients are approximated via finite differences when analytic derivatives are unavailable. In Turkish contexts, gradyan is the standard term used for the gradient.