Calculus of variation. Minimum surface

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Thread starter knockout_artist
Start date Jul 2, 2017
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Calculus Calculus of variation Minimum Surface Variation

In summary, Calculus of variation is a branch of mathematics that deals with finding the optimal solution to a problem involving a functional. The minimum surface problem is a classic example in this field, where the goal is to find the surface with the smallest area connecting a given set of points. This problem has real-world applications in physics, engineering, and materials science. To solve it, a functional representing the surface area is defined and the Euler-Lagrange equation is used to find the function that minimizes it. Calculus of variation has various other applications in fields such as physics, engineering, economics, and is used to solve optimization problems involving moving objects, hanging chains, energy configurations, and more. However, it has limitations such as not always

Jul 2, 2017

knockout_artist

so df/dy' is yy'/ √(1+y'^2)

then we are supposed to do
y' . [ yy'/ √(1+y'^2) ] - y√(1+y'^2)

how does this bring equation 2 in the image ?

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Jul 2, 2017

knockout_artist

imaged rotated :)

Jul 2, 2017

formodular

Multiply ##f = y \sqrt{1+y'^2}## by ##1 = \frac{\sqrt{1+y'^2}}{\sqrt{1+y'^2}}## then subtract from ##y' f_y'##.

Related to Calculus of variation. Minimum surface

What is calculus of variation?

Calculus of variation is a branch of mathematics that deals with finding the optimal solution to a problem involving a functional. A functional is a mathematical expression that takes in a function as an input and outputs a real number. In calculus of variation, we look for the function that minimizes or maximizes the value of the functional.

What is the minimum surface problem?

The minimum surface problem is a classic example in calculus of variation. It involves finding the surface with the smallest area that connects a given set of points. This problem has real-world applications in fields such as physics, engineering, and materials science.

How is the minimum surface problem solved using calculus of variation?

In order to solve the minimum surface problem using calculus of variation, we first define a functional that represents the surface area. Then, we use the Euler-Lagrange equation to find the function that minimizes this functional. This function represents the surface with the smallest area that connects the given points.

What are some other applications of calculus of variation?

Calculus of variation has many applications in physics, engineering, economics, and other fields. It is used to find the optimal path for a moving object, the shape of a hanging chain, the minimal energy configuration of a system, and many other problems involving optimization.

What are the limitations of using calculus of variation?

While calculus of variation is a powerful tool for solving optimization problems, it has some limitations. It may not always provide a unique solution, and it requires a certain level of mathematical expertise to apply effectively. Additionally, it may not be suitable for certain types of problems, such as those with discontinuous or non-differentiable functions.