Minimize surface area - calculus of variations

In summary, the concept of "minimize surface area" in calculus of variations involves using calculus to find the shape or surface with the smallest possible area based on a set of constraints. This has various applications, such as determining the shape of soap bubbles or a hanging chain. The Euler-Lagrange equation is used to find the critical points of the functional and ultimately the shape that minimizes the surface area. However, there are limitations to using this method, such as only being applicable to continuous surfaces and being challenging for complex shapes. It can also be applied to three-dimensional objects by using a three-dimensional functional.
  • #1
Shackleford
1,656
2
I can't get rearrange the last equation into a nice form to integrate with respect to x to minimize the surface area.

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-17122502.jpg?t=1287336976

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-17123734.jpg?t=1287339019

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-17122444.jpg?t=1287336976
 
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  • #2
The photos cannot be viewed
 
  • #3
I can't get them either. He may have figured it out by now and taken them down.
 

1. What is the concept of "minimize surface area" in calculus of variations?

The concept of "minimize surface area" in calculus of variations involves finding the shape or surface that has the smallest possible area based on a set of constraints. This is achieved by using calculus to optimize a functional, which represents the surface area, and finding the critical points where the functional is minimized.

2. What are the applications of "minimize surface area" in calculus of variations?

The applications of "minimize surface area" in calculus of variations are numerous, with some examples including finding the shape of soap bubbles, minimizing the energy of a soap film, optimizing the shape of a hanging chain, and determining the shape of a membrane under tension.

3. What is the Euler-Lagrange equation and how is it used in "minimize surface area"?

The Euler-Lagrange equation is a necessary condition for finding the critical points of a functional. In the context of "minimize surface area" in calculus of variations, it is used to find the shape that minimizes the surface area by setting the first variation of the functional to zero and solving the resulting differential equation.

4. Are there any limitations to using calculus of variations to "minimize surface area"?

One limitation of using calculus of variations to "minimize surface area" is that it is only applicable to continuous surfaces and may not accurately represent surfaces with discontinuities or sharp edges. Additionally, it may be challenging to find an analytical solution for complex shapes and numerical methods may need to be used instead.

5. Can "minimize surface area" be applied to three-dimensional objects?

Yes, "minimize surface area" can be applied to three-dimensional objects by using a three-dimensional functional that represents the surface area. The same principles and techniques of calculus of variations can be applied to find the shape that minimizes the surface area in this case.

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