Gradient
In vector calculus, the gradient is a multi-variable generalization of the derivative. Whereas the ordinary derivative of a function of a single variable is a scalar-valued function, the gradient of a function of several variables is a vector-valued function. Specifically, the gradient of a differentiable function
f
{\displaystyle f}
of several variables, at a point
P
{\displaystyle P}
, is the vector whose components are the partial derivatives of
f
{\displaystyle f}
at
P
{\displaystyle P}
.Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, if at a point
P
{\displaystyle P}
, the gradient of a function of several variables is not the zero vector, it has the direction of greatest increase of the function at
P
{\displaystyle P}
, and its magnitude is the rate of increase in that direction.The magnitude and direction of the gradient vector are independent of the particular coordinate representation.The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.
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