**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Tangent Planes and Derivatives

Description

This lecture covers the calculation of directional derivatives and the equation of a tangent plane to the surface z=f(x,y). Topics include determining differentiability, finding normal vectors to the graph of z=sin(xy), and solving for the equation of the tangent plane at a given point.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

Instructor

MATH-106(e): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs
variables.

Related concepts (32)

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.

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.

Semi-differentiability

In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.

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.

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.

Related lectures (39)

Derivatives and Continuity in Multivariable Functions

Covers derivatives and continuity in multivariable functions, emphasizing the importance of partial derivatives.

Tangent Planes and Implicit Functions

Explores tangent planes, implicit functions, and directional derivatives in relation to graphs of functions.

Tangent to Graph of a FunctionMATH-101(c): Analysis I

Explores finding the equation of the tangent line to a function's graph at a point.

Differentiability and Tangent PlanesMATH-106(e): Analysis II

Explores differentiability in functions and tangent planes to surfaces.

Derivatives and Tangent Planes

Covers derivatives, differentiability, and tangent planes for functions of one and two variables.