Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Directional Derivatives

Description

This lecture covers the concept of directional derivatives in multivariable calculus, focusing on the definition and calculation of unilateral and Dini directional derivatives. It also discusses the properties of directional derivatives and their applications. The lecture emphasizes the chain rule for directional derivatives and provides examples to illustrate the concept.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (27)

Generalizations of the derivative

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. The Fréchet derivative defines the derivative for general normed vector spaces . Briefly, a function , an open subset of , is called Fréchet differentiable at if there exists a bounded linear operator such that Functions are defined as being differentiable in some open neighbourhood of , rather than at individual points, as not doing so tends to lead to many pathological counterexamples.

Chain rule

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces.

Directional derivative

A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. The directional derivative of a scalar function f with respect to a vector v at a point (e.g.

Related lectures (31)

Derivability Conditions

Explores conditions for non-differentiability in multivariable calculus and the implications of the differentiability theorem.

Chain Rules

Explores the extension of the chain rule to higher dimensions and compositions of functions.

Limits of Functions in R^n

Explores limits of functions in R^n and the importance of understanding derivatives in multiple dimensions.

Functions with Multiple Variables

Covers functions with values in R^m, discussing limits, partial derivatives, and differentiability.