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| = Gradient Vector = | = Gradient = |
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| A '''gradient vector''' describes the direction of steepest ascent for a differentiable function. | 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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| In terms of [[Calculus/PartialDerivatives|partial derivatives]], the gradient vector of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is ''[∂f/∂x,,1,, ∂f/∂x,,2,, ... ∂f/∂x,,n,,]''. The gradient is notated as ''∇f''. | 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}} |
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| {{attachment:gradient.svg}} | {{attachment:gradientvector.svg}} |
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| 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/Orthonormalization|orthonormal]]. | 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]]. ---- == 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. |
Gradient
A gradient is a vector of partial derivatives. It describes the direction of steepest ascent for a differentiable function.
Notation
The gradient of function f is notated as ∇f. In terms of partial derivatives, the gradient of f(x1, x2, ... xn) is:
At a given point p, as long as the function f is differentiable at p, the gradient vector is:
Note the assumption; it is not negligible. For example, (xy)/(x2 + y2) is partially derivable but is itself not totally derivable at point p = [0 0]. Furthermore, it is not derivable if rotated; the basis must be orthonormal.
Usage
By setting a gradient to 0, critical points (local minima, local maxima, and inflections) can be calculated.
More generally, gradient descent can be used to estimate minima.