／var／log marcus chiu

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Mathematical Analysis

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Multivariable／Multi-Variable／Multivariate Calculus／Analysis

Gradient (𝛻 ｜ 𝛻𝑓)

Created on Dec 16, 2023 · Last Modified on Sep 08, 2024

Gradient (𝛻 | 𝛻𝑓)

the gradient is a linear operator that transforms a scalar-valued differentiable function 𝑓 of several variables is the vector-valued function 𝛻𝑓 whose value at a point 𝑝 gives the direction and the rate of the fastest increase
is a vector of partial derivatives of a scalar-valued differentiable function 𝑓 of several variables

Gradient - Definition

The gradient of a scalar-valued function 𝑓 with respect to vector input 𝐱 is defined as:

$x = x_{1} x_{2} ⋮ x_{n}$

$\nabla_{x} f = \frac{𝜕 f}{𝜕 x}$
$\nabla_{x} f = [\frac{𝜕 f}{𝜕 x _{1}} \frac{𝜕 f}{𝜕 x _{2}} ... \frac{𝜕 f}{𝜕 x _{n}}]$
$\nabla_{x} f (x) = \frac{𝜕 f ( x )}{𝜕 x}$
$\nabla_{x} f (x) = [\frac{𝜕 f ( x )}{𝜕 x _{1}} \frac{𝜕 f ( x )}{𝜕 x _{2}} ... \frac{𝜕 f ( x )}{𝜕 x _{n}}]$

Gradient - Linearity

The gradient operator 𝛻 is a linear operator such that it follows for any scalar-valued functions 𝑓₁ and 𝑓₂, and scalar constants 𝛼 and 𝛽:

$\nabla (𝛼 f_{1} + 𝛽 f_{2}) = 𝛼\nabla f_{1} + 𝛽\nabla f_{2}$
$\nabla_{x} (𝛼 f_{1} + 𝛽 f_{2}) (x) = 𝛼 \nabla_{x} f_{1} (x) + 𝛽 \nabla_{x} f_{2} (x)$

Gradient - Use Cases

can be used to compute the directional derivative

Resources

Steve Brunton’s The Gradient Operator in Vector Calculus: Directions of Fastest Change & the Directional Derivative