Gradient

A gradient is a mathematical concept that represents the rate of change of a function at a given point. It's a vector composed of the partial derivatives of the function with respect to each of its variables. In simpler terms, the gradient gives the direction and magnitude of the steepest ascent of a function.

Interpretation

Geometrically, the gradient points towards the direction where the function increases the most rapidly. For example, in a 2-dimensional space, the gradient of a function will be a 2-dimensional vector that points in the direction of the steepest slope on the function's graph.

Uses

Gradients are widely used in various fields such as physics, engineering, computer science, and machine learning. In physics, gradients are used to represent the direction and magnitude of the force acting on an object. In machine learning, gradients are used in optimization algorithms like gradient descent to minimize a function and find the optimal parameters of a model.

Calculation

To calculate the gradient of a function, one needs to find the partial derivatives of the function with respect to each variable and then combine them into a vector. In calculus, the gradient of a scalar function is represented by the symbol ∇ (nabla).