What is 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 does convex mean in calculus?

In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.

Is calculus of variations functional analysis?

Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis.

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 variation of a functional?

A variation of a functional is the small change in a functional’s value due to a small change in the functional’s input. It’s the analogous concept to a differential for regular calculus.

Where is calculus of variations used?

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: mappings from a set of functions to the real numbers.

How do you know if it is convex or concave?

To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.

What is the example of concave?

The front side of a spoon is curved inwards. Such a surface is called concave. The inside part of a bowl is also an example of a concave surface. Concave mirrors are used in various medical practices.

What is the difference between calculus and functions?

The word function in calculus refers to something like f(x)=x2+2×3 or f(x)=sin(x) etc…. In linear algebra, the word function is used like- A linear transformation is a function from V→W. And the functions of calculus like f(x)=x2+2×3 or f(x)=sin(x) etc.

Why do we need 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: mappings from a set of functions to the real numbers. Many important problems involve functions of several variables.

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 meant by calculus of variation?

calculus of variations, branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible.

How are functions expressed in the calculus of variations?

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.

Is the twice differentiable convex function strictly convex?

Visually, a twice differentiable convex function “curves up”, without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold.

When is a convex function called a quasiconvex function?

A strictly convex function will have at most one global minimum. the sublevel sets { x | f ( x) < a } and { x | f ( x) ≤ a } with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function. .

Which is easier to work with convex or strongly convex functions?

Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets. {\\displaystyle \\phi } is a function that is non-negative and vanishes only at 0.