Definitions for calculus of variations
cal·cu·lus of vari·a·tions

Princeton's WordNetRate this definition:0.0 / 0 votes

1. calculus of variationsnoun

   the calculus of maxima and minima of definite integrals

WiktionaryRate this definition:0.0 / 0 votes

1. calculus of variationsnoun

   The form of calculus that deals with the maxima and minima of definite integrals of functions of many variables

WikipediaRate this definition:0.0 / 0 votes

1. Calculus of variations

   The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and 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.

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1. calculus of variations

   The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals, subjected to certain constraints. Essentially, it studies how to find an optimal solution that maximizes or minimizes a certain functional (a mapping from a set of functions to the real numbers). It has applications in various fields such as physics, economics, and engineering.

WikidataRate this definition:0.0 / 0 votes

1. Calculus of variations

   Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero. 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 obviously 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, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least action. Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet 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 a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

Matched Categories

1. • Calculus
   • Mathematics

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How to say calculus of variations in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of calculus of variations in Chaldean Numerology is: 2

2. Pythagorean Numerology

   The numerical value of calculus of variations in Pythagorean Numerology is: 7

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=calculus of variations
2. ^ Wiktionary
   https://en.wiktionary.org/wiki/Calculus_of_Variations
3. ^ Wikipedia
   https://en.wikipedia.org/wiki/Calculus_of_Variations
4. ^ ChatGPT
   https://chat.openai.com
5. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=calculus of variations