Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f {\displaystyle f} . If the gradient of a function is non-zero at a point p {\displaystyle p} , the direction of the gradient is the direction in which the function increases most quickly from p {\displaystyle p} , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, machine learning, and artificial intelligence, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f ( r ) {\displaystyle f(\mathbf {r} )} may be defined by: d f = ∇ f ⋅ d r {\displaystyle df=\nabla f\cdot d\mathbf {r} } where d f {\displaystyle df} is the total infinitesimal change in f {\displaystyle f} for an infinitesimal displacement d r {\displaystyle d\mathbf {r} } , and is seen to be maximal when d r {\displaystyle d\mathbf {r} } is in the direction of the gradient ∇ f {\displaystyle \nabla f} . The nabla symbol ∇ {\displaystyle \nabla } , written as an upside-down triangle and pronounced "del", denotes the vector differential operator. When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector whose components are the partial derivatives of f {\displaystyle f}...