Optimization in vector spaces | Calculus of variations

Calculus of variations

The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action. Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology. (Wikipedia).

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Calculus 1 Lecture 3.1: Increasing/Decreasing and Concavity of Functions

Calculus 1 Lecture 3.1: Discussion of Increasing and Decreasing Intervals. Discussion of Concavity of functions.

From playlist Calculus 1 (Full Length Videos)

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Differential Equations | Variation of Parameters.

We derive the general form for a solution to a differential equation using variation of parameters. http://www.michael-penn.net

From playlist Differential Equations

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Statement of Calculus of Variations (6.1)

In this video, I state the calculus of variations problem, and describe how to solve it.

From playlist Intermediate Classical Mechanics

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A16 The method of variation of parameters

Starting the derivation for the equation that is used to find the particular solution of a set of differential equations by means of the variation of parameters.

From playlist A Second Course in Differential Equations

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3_6_3 Alternating Series

The alternating series. Solved problems. Estimating error and partial sum estimation for a set maximum error.

From playlist Advanced Calculus / Multivariable Calculus

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3_6_2 Alternating Series

The alternating series. Test for convergence.

From playlist Advanced Calculus / Multivariable Calculus

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Infinite Limits With Equal Exponents (Calculus)

#Calculus #Math #Engineering #tiktok #NicholasGKK #shorts

From playlist Calculus

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2_1 Sequences

Convergence and sequences.

From playlist Advanced Calculus / Multivariable Calculus

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2_2 Sequences

Convergences of sequences.

From playlist Advanced Calculus / Multivariable Calculus

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Calculus of Variations ft. Flammable Maths

Flammable Maths: https://www.youtube.com/channel/UCtAIs1VCQrymlAnw3mGonhw Leibnitz Rule: https://www.youtube.com/watch?v=wkh1Y7R1sOw This video is an introduction to the calculus of variations. We go over what variational calculus is trying to solve, and derive the Euler-Lagrange equatio

From playlist Analysis

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The Palais-Smale Theorem and the Solution of Hilbert’s 23 Problem - Karen Uhlenbeck

Members' Seminar Topic: The Palais-Smale Theorem and the Solution of Hilbert’s 23 Problem Speaker: Karen Uhlenbeck Affiliation: The University of Texas at Austin; Distinguished Visiting Professor, School of Mathematics Date: April 6, 2020 For more video please visit http://video.ias.edu

From playlist Mathematics

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Lec 23 | MIT 18.086 Mathematical Methods for Engineers II

Calculus of Variations / Weak Form View the complete course at: http://ocw.mit.edu/18-086S06 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 18.086 Mathematical Methods for Engineers II, Spring '06

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Soviet Era Math Book for Beginners and Mathematical Experts

This book states that it is written with beginners in mind but it can also be of use to expert mathematicians. The first half of the book is on Differential Equations and the second half is on the Calculus of Variations. In this video I will show you this book and I will also solve a diffe

From playlist Book Reviews

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A Taste of Calculus of Variations

Dirichlet's Principle In this video, I give you a taste of calculus of variations by illustrating Dirichlet's principle, which says that a function u is a minimizer of a certain Dirichlet energy (kinetic + potential energy) if and only if u solves Poisson's equation. This is a neat way of

From playlist Partial Differential Equations

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Karen Uhlenbeck: Some Thoughts on the Calculus of Variations

Abstract: I will talk about some of the classic problems in the calculus of variations, and describe some of the mathematics which was developed to solve them. We will begin with the Greeks and end with some of the tantalizing problems of today. This lecture was given by the 2019 Abel Lau

From playlist Karen K. Uhlenbeck

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Shortest Distance Between Two Points (6.2)

In this video, I use the calculus of variations to prove that the shortest distance between two points is along a straight line.

From playlist Intermediate Classical Mechanics

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Your Daily Equation #19 : At the Core of Fundamental Physics: The Principle of Least Action

Episode 19 #YourDailyEquation: All fundamental laws of physics share a reliance on a single principle: The Principle of Least Action. In this episode of Your Daily Equation, Brian Greene explains this principle in the simplest example and shows how it yields the basic laws of motion introd

From playlist Your Daily Equation with Brian Greene

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11_4_2 The Derivative of the Composition of Functions

A further look at the derivative of the composition of a multivariable function and a vector function. There are two methods to calculate such a derivative.

From playlist Advanced Calculus / Multivariable Calculus

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Options (Lecture 1) by Shashi Jain

Program Summer Research Program on Dynamics of Complex Systems ORGANIZERS: Amit Apte, Soumitro Banerjee, Pranay Goel, Partha Guha, Neelima Gupte, Govindan Rangarajan and Somdatta Sinha DATE : 15 May 2019 to 12 July 2019 VENUE : Madhava hall for Summer School & Ramanujan hall f

From playlist Summer Research Program On Dynamics Of Complex Systems 2019

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