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{wiki=First_variation_of_area_formula}

With $A(X,Y)=(\overline\nabla_XY)^\perp$ and $\mathbf H=\operatorname{tr}A$, a variation with velocity $V$ satisfies
$$
\left.\frac d{dt}\operatorname{Area}(M_t)\right|_{t=0}
=-\int_M\langle\mathbf H,V\rangle,d\mu
+\int_{\partial M}\langle V,\eta\rangle,d\sigma.
$$
Thus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.


First variation of area

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 4 / a / Solution

First variation of area

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First variation of area

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