This paper, originally published by A. P. Calderón in 1980, discusses an inverse boundary value problem. The problem involves determining a bounded, measurable function \(\gamma\) with a positive lower bound in a bounded domain \(D\) in \(\mathbb{R}^n\) from the quadratic form \(Q_{\gamma}(\phi)\), where \(\phi\) is a function in \(H^1(\mathbb{R}^n)\) with zero boundary conditions on \(\partial D\). The differential operator \(L_{\gamma}(w) = \nabla \cdot (\gamma \nabla w)\) is defined on functions in \(H^1(D)\). The main goal is to decide whether \(\gamma\) is uniquely determined by \(Q_{\gamma}\) and to calculate \(\gamma\) if it is. This problem arises in electrical prospecting, where \(\gamma\) represents the electrical conductivity of an inhomogeneous body and needs to be determined from surface measurements. Calderón introduces norms for functions \(\gamma\) and quadratic forms \(Q(\phi)\) and shows that the mapping \(\Phi: \gamma \rightarrow Q_{\gamma}\) is bounded and analytic. He proves that the linearized problem, \(d\Phi|_{\gamma=\text{const}}\), is injective, meaning that if \(\gamma\) is constant, it is nearly determined by \(Q_{\gamma}\). However, the full injectivity of \(\Phi\) remains an open problem. The paper also provides an expression for the solution to the equation \(L_{\gamma}(W) = \nabla \cdot (\gamma \nabla W)\) and calculates the derivative \(d\phi|_{\gamma=1}\). It shows that \(dQ_{\gamma}(\phi)|_{\gamma=1}\) is injective, and under certain conditions, \(\gamma\) can be approximated with an error much smaller than \(\|\delta\|_{L^{\infty}}\). Finally, the paper discusses the case where \(\gamma\) is assumed to be in \(C^m\) and provides a method to approximate \(\gamma\) with an error that decreases as \(\|\delta\|_{L^\infty}\) decreases.This paper, originally published by A. P. Calderón in 1980, discusses an inverse boundary value problem. The problem involves determining a bounded, measurable function \(\gamma\) with a positive lower bound in a bounded domain \(D\) in \(\mathbb{R}^n\) from the quadratic form \(Q_{\gamma}(\phi)\), where \(\phi\) is a function in \(H^1(\mathbb{R}^n)\) with zero boundary conditions on \(\partial D\). The differential operator \(L_{\gamma}(w) = \nabla \cdot (\gamma \nabla w)\) is defined on functions in \(H^1(D)\). The main goal is to decide whether \(\gamma\) is uniquely determined by \(Q_{\gamma}\) and to calculate \(\gamma\) if it is. This problem arises in electrical prospecting, where \(\gamma\) represents the electrical conductivity of an inhomogeneous body and needs to be determined from surface measurements. Calderón introduces norms for functions \(\gamma\) and quadratic forms \(Q(\phi)\) and shows that the mapping \(\Phi: \gamma \rightarrow Q_{\gamma}\) is bounded and analytic. He proves that the linearized problem, \(d\Phi|_{\gamma=\text{const}}\), is injective, meaning that if \(\gamma\) is constant, it is nearly determined by \(Q_{\gamma}\). However, the full injectivity of \(\Phi\) remains an open problem. The paper also provides an expression for the solution to the equation \(L_{\gamma}(W) = \nabla \cdot (\gamma \nabla W)\) and calculates the derivative \(d\phi|_{\gamma=1}\). It shows that \(dQ_{\gamma}(\phi)|_{\gamma=1}\) is injective, and under certain conditions, \(\gamma\) can be approximated with an error much smaller than \(\|\delta\|_{L^{\infty}}\). Finally, the paper discusses the case where \(\gamma\) is assumed to be in \(C^m\) and provides a method to approximate \(\gamma\) with an error that decreases as \(\|\delta\|_{L^\infty}\) decreases.