sense) Composition with a function More generally, the delta distribution may be composed with a smooth function g(x) in such a way that the familiar change of variables formula holds, that provided that g is a continuously differentiable function with g′ nowhere zero. That is, there is a unique way to assign meaning to the distribution so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies if g is nowhere zero, and otherwise if g has a real root at x0, then It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by where the sum extends over all roots (i.e., all the different ones) of g(x), which are assumed to be simple. Thus, for example In the integral form, the generalized scaling property may be written as Properties in n dimensions The delta distribution in an n-dimensional space satisfies the following scaling property instead, so that δ is a homogeneous distribution of degree −n. Under any reflection or rotation ρ, the delta function is invariant, As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function uniquely so that the identity for all compactly supported functions f. Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function such that the gradient of g is nowhere zero, the following identity holds where the integral on the right is over g−1(0), the -dimensional surface defined by with respect to the Minkowski content measure. This is known as a simple layer integral. More generally, if S is a smooth hypersurface of Rn, then we can associate to S the distribution that integrates any compactly supported smooth function g over S: where σ is the hypersurface measure associated to S. This generalization is associated with the potential theory of simple layer potentials on S. If D is a domain in Rn with smooth boundary S, then δS is equal to the normal derivative of the indicator function of D in the distribution sense, where n is the outward normal. For a proof, see e.g. the article