What is meant by calculus of variation?

Definition of calculus of variations : a branch of mathematics concerned with applying the methods of calculus to finding the maxima and minima of a function which depends for its values on another function or a curve.

Why is the calculus of variations important?

The calculus of variations is a powerful technique to solve some dynamic problems that are not intuitive to solve otherwise. It is the precursor to optimal control theory as it allows us to solve non-complex control systems.

What is calculus of variations and what are its applications?

The calculus of variations is a field of mathematics about solving optimization problems. This is done by minimizing and maximizing functionals. The methods of calculus of variations to solve optimization problems are very useful in mathematics, physics and engineering.

Who invented calculus of variations?

It was in his 1744 book, though, that Euler transformed a set of special cases into a systematic approach to general problems: the calculus of variations was born.

What is the fundamental result of the calculus of variations?

In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.

What is the difference between variation and differentiation?

variation (delta) is simply the change in a dependent variable due to a change in an independent variable (=delta y) while differentiation is the variation divided by a the change in the independent variable in a small range (=dy/dx).

What is a functional in calculus of variations?

A typical problem in the calculus of variations involve finding a particular function y(x) to maximize or minimize the integral I(y) subject to boundary conditions y(a)=A and y(b)=B. The integral I(y) is an example of a functional, which (more generally) is a mapping from a set of allowable functions to the reals.

What are variation describe various bends of variation?

variation, in biology, any difference between cells, individual organisms, or groups of organisms of any species caused either by genetic differences (genotypic variation) or by the effect of environmental factors on the expression of the genetic potentials (phenotypic variation).

What is extremal in calculus of variations?

A solution of the Euler-Lagrange equation is called an extremal of the functional. By considering y+g, where y is the solution from exercise 1 and g(x) is a variation in y(x) satisfying g(0)=g(1)=0, and then considering I(y+g), show explicitly that y(x) minimizes I(y) in Exercise 1 above.