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:

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:

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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