In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) whose value at a point is the vector whose components are the partial derivatives of at .[excessive citations] That is, for , its gradient is defined at the point in n-dimensional space as the vector:
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The gradient vector can be interpreted as the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative.[excessive citations] Further, the gradient is the zero vector at a point if and only if it is a stationary point (where the derivative vanishes). The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent.
The gradient is dual to the total derivative : the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear function on vectors. They are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v; that is, . The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.