8Minimal Surface and Membrane Wave Equations

8.1 Derivation of Minimal Surface Equation

To derive the minimal surface equation, suppose images is the height of a surface with images on the boundary images of images, which minimizes the surface area (derived in most Calculus III texts):

images

Then if images is any smooth function with images on the boundary, images for any α; thus

images

should have a minimum at images and so images should be zero. But

so

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