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We recover the gradient of a scalar conductivity defined on a smooth boundedopen set in $\mathbb{R}^d$ from the Dirichlet to Neumann map arising from the$p$-Laplace equation. For any boundary point we recover the gradient usingDirichlet data supported on an arbitrarily small neighbourhood of the boundarypoint. We use a Rellich-type identity in the proof. Our results are new when $p\neq 2$. In the $p = 2$ case boundary determination plays a role in severalmethods for recovering the conductivity in the interior.

Calder\'on problem for the p-Laplacian: First order derivative of conductivity on the boundary