Directional Derivatives: Concepts, Definitions, and
Calculations
Introduction
The material provides a detailed overview of directional derivatives, starting from
their significance as an extension of partial derivatives to their strict definition and
methods for calculation. It covers both binary and ternary functions, geometric
interpretations, and the relationship between directional derivatives and gradients.
The ultimate goal is to provide a comprehensive understanding of directional
derivatives and their applications.
Definition of Directional Derivatives for Binary Functions
Core Idea: The directional derivative for binary functions extends the concept of a
derivative to measure the rate of change of a function in a specific direction.
Definition of the directional derivative:
Let z = f(x, y) be a function defined in a neighborhood of a point P₀(a, b).
Let u = (cosθ, sinθ) be a unit vector in the plane.
The directional derivative Duf(a, b) is defined as the limit (as h approaches 0) of [f(a +
h•cosθ, b + h•sinθ) - f(a, b)] / h, if this limit exists.
Significance:
It measures the instantaneous rate of change of f(x, y) at the point P₀(a, b) in the
direction of the unit vector u.
Directional Derivative of a Ternary Function
Core Idea: Extending the concept to three dimensions, the directional derivative
measures the rate of change of a ternary function in a specific spatial direction.
Definition for ternary functions:
For a function w = f(x, y, z), let u = (cosα, cosβ, cosγ) be a unit vector in space.
The directional derivative Duf(x₀, y₀, z₀) is defined as the limit (as h approaches 0) of
[f(x₀ + h•cosα, y₀ + h•cosβ, z₀ + h•cosγ) - f(x₀, y₀, z₀)] / h, if this limit exists.
Geometric Meaning of the Definition
Core Idea: Directional derivatives can be visualized geometrically as the slope of a
tangent line to a space curve on a surface.
Geometric interpretation:
Calculations
Introduction
The material provides a detailed overview of directional derivatives, starting from
their significance as an extension of partial derivatives to their strict definition and
methods for calculation. It covers both binary and ternary functions, geometric
interpretations, and the relationship between directional derivatives and gradients.
The ultimate goal is to provide a comprehensive understanding of directional
derivatives and their applications.
Definition of Directional Derivatives for Binary Functions
Core Idea: The directional derivative for binary functions extends the concept of a
derivative to measure the rate of change of a function in a specific direction.
Definition of the directional derivative:
Let z = f(x, y) be a function defined in a neighborhood of a point P₀(a, b).
Let u = (cosθ, sinθ) be a unit vector in the plane.
The directional derivative Duf(a, b) is defined as the limit (as h approaches 0) of [f(a +
h•cosθ, b + h•sinθ) - f(a, b)] / h, if this limit exists.
Significance:
It measures the instantaneous rate of change of f(x, y) at the point P₀(a, b) in the
direction of the unit vector u.
Directional Derivative of a Ternary Function
Core Idea: Extending the concept to three dimensions, the directional derivative
measures the rate of change of a ternary function in a specific spatial direction.
Definition for ternary functions:
For a function w = f(x, y, z), let u = (cosα, cosβ, cosγ) be a unit vector in space.
The directional derivative Duf(x₀, y₀, z₀) is defined as the limit (as h approaches 0) of
[f(x₀ + h•cosα, y₀ + h•cosβ, z₀ + h•cosγ) - f(x₀, y₀, z₀)] / h, if this limit exists.
Geometric Meaning of the Definition
Core Idea: Directional derivatives can be visualized geometrically as the slope of a
tangent line to a space curve on a surface.
Geometric interpretation:
