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I'm currently writing a project on minimal surfaces and have come across the First Variation of Area formula, however I'm finding it difficult to understand it's significance when understanding Minimal Surfaces.

Would someone be able to explain/clarify this?

Many thanks for your time

Sarah

Sarah Jayne
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  • I'm not sure if I understood question. When searching minimum of an ordinary 1D function one observes its first derivative, right? The first variation plays the same role for functors. –  Feb 02 '14 at 12:57
  • It would help if you supplied more details on what formula you're confused about and what context it appears in. – E.P. Feb 03 '14 at 00:53

1 Answers1

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Consider and choose a bounded domain D and differentiable function on $h:D→R$. The variation on the normal of h is the map given by $φ:D×(-ε,ε)→R^3$ definided $φ(u, v, t)=x(u, v)+th(u, v)N(u, v)$ . Minimize to funtional

$$A(t)=\int_D\sqrt{E^tG^t−(F^t)^2}dudv,$$

this is $$A'(0)=-\int_D2hH\sqrt{EG-F^2}dudv.$$

For other properties of minimal surfaces, to be convenient to introduce, for any surface regul ar parameterization, the average curvature vector defined by H $=HN$. The geometric significado on direction of H can be obtained from the equation, choosing $h=H$ we have, for this variation on particular

$$A'(0)=-2\int_D\langle H,H\rangle \sqrt{EG-F^2}dudv<0$$.

This means that if we deform $x(D)$ at the direction on the vector H, the area initially decreases.

jimbo
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