= Gradient =

A '''gradient''' is a vector of [[Calculus/PartialDerivative|partial derivatives]]. It describes the direction of steepest ascent for a [[Calculus/Derivative|differentiable]] function.

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== Notation ==

The gradient of function ''f'' is notated as ''∇f''. In terms of [[Calculus/PartialDerivative|partial derivatives]], the gradient of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is:

{{attachment:gradient.svg}}

At a given point ''p'', as long as the function ''f'' is differentiable at ''p'', the gradient vector is:

{{attachment:gradientvector.svg}}

Note the assumption; it is not negligible. For example, ''(xy)/(x^2^ + y^2^)'' is partially derivable but is itself not totally derivable at point ''p = [0 0]''. Furthermore, it is not derivable if rotated; the [[LinearAlgebra/Basis|basis]] must be [[LinearAlgebra/Orthogonality|orthonormal]].

Also note that, technically, this is the [[LinearAlgebra/Transposition|transpose]] of the gradient; vectors are by default columns.



=== Relation to Jacobian Matrices ===

A [[Calculus/JacobianMatricesAndDeterminants|Jacobian matrix]] is a square matrix of partial derivatives. Consider a vector-valued function, such as a [[Calculus/ParametricEquation|parameterized]] function, but for simplicity let ''F(X) = <f,,1,,(X), f,,2,,(X)>'' where ''X'' is a vector as ''X = <x,,1,,``, x,,2,,``, ... x,,n,,>''. A gradient can be calculated for each of the component function.

{{attachment:jacobian.svg}}

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== Usage ==

By setting a gradient to 0, critical points (local minima, local maxima, and inflections) can be calculated.

More generally, [[Calculus/GradientDescent|gradient descent]] can be used to estimate minima.



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