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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 partial derivatives. It describes the direction of steepest ascent for a 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/PartialDerivatives|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}} |
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.