Shape derivative

Summary

The shape derivative measures how a quantity that depends on a domain changes when the domain is deformed a little. One moves each point $x$ of the domain to $x + \theta(x)$ for a small vector field $\theta$. The shape derivative is the derivative of the quantity with respect to $\theta$. For smooth domains it usually depends only on the normal component of $\theta$ on the boundary. This fact is known as the Hadamard structure theorem. It is the gradient that drives numerical methods of shape optimization.