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persia7
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How do you prove that for a given volume, sphere has the minimum surface area?
persia7 said:i tried to find proofs but i don't find please show me proofs
can you show me a proof in internet?jackmell said:How about first doing it for a surface of revolution, say for y(x) between -1 and 1. Then the surface is:
[tex]S=2\pi \int_{-1}^1 \sqrt{1+(y')^2}dx[/tex]
and the volume of this surface of revolution is:
[tex]V=\pi \int_{-1}^1 y^2 dx[/tex]
via discs.
Now, in the language of variation, we wish to minimize S while keeping V constant. The Euler condition for this problem is:
[tex]\frac{\partial H}{\partial y}-\frac{d}{dx}\frac{\partial H}{\partial y'}=0[/tex]
with
[tex]H=V+\lambda S[/tex]
Now, I've never solved this particular variational problem before ok, but if it were mine, I'd first see if I can get a circle out of Euler's equation thus showing me that a sphere has the minimum surface area for a given volume.
persia7 said:can you show me a proof in internet?
jackmell said:How about first doing it for a surface of revolution, say for y(x) between -1 and 1. Then the surface is:
[tex]S=2\pi \int_{-1}^1 \sqrt{1+(y')^2}dx[/tex]
The minimum surface area of a sphere is the smallest possible surface area that a sphere can have, while still maintaining a spherical shape.
The minimum surface area of a sphere is calculated using the formula A = 4πr², where A is the surface area and r is the radius of the sphere.
The minimum surface area of a sphere is important because it represents the most efficient use of surface area for a given volume. This has practical applications in fields such as packaging, architecture, and biology.
No, the minimum surface area of a sphere cannot be exceeded. It is the most compact and efficient shape for a given volume.
The minimum surface area of a sphere is often considered to be the most efficient among all shapes, as it has the smallest surface area for a given volume. Other shapes, such as cubes or cylinders, may have larger surface areas for the same volume.