A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area.
2.
A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.
3.
There are only two minimal surfaces of revolution ( surfaces of revolution which are also minimal surfaces ) : the plane and the catenoid.
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For a given constraint there may also exist several minimal surfaces with different areas ( for example, see minimal surface of revolution ) : the standard definitions only relate to a local optimum, not a global optimum.
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