structural:scholar_edition
Properties in n dimensions

The delta distribution in an n -dimensional space satisfies the following scaling property instead, δ (α x) = | α | − n δ (x) , {\displaystyle \delta (\alpha {\boldsymbol {x}})=|\alpha |^{-n}\delta ({\boldsymbol {x}})~,} so that δ is a homogeneous distribution of degree − n.

Under any reflection or rotation ρ, the delta function is invariant, δ (ρ x) = δ (x). {\displaystyle \delta (\rho {\boldsymbol {x}})=\delta ({\boldsymbol {x}})~.}

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function g: R → R uniquely so that the following holds ∫ R n δ (g (x)) f (g (x)) | det g ′ (x) | d x = ∫ g (R n) δ (u) f (u) d u {\displaystyle \int _{\mathbb {R} ^{n}}\delta (g({\boldsymbol {x}}))\,f(g({\boldsymbol {x}}))\left|\det g'({\boldsymbol {x}})\right|d{\boldsymbol {x}}=\int _{g(\mathbb {R} ^{n})}\delta ({\boldsymbol {u}})f({\boldsymbol {u}})\,d{\boldsymbol {u}}} 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 g : R → R such that the gradient of g is nowhere zero, the following identity holds ∫ R n f (x) δ (g (x)) d x = ∫ g − 1 (0) f (x) | ∇ g | d σ (x) {\displaystyle \int _{\mathbb {R} ^{n}}f({\boldsymbol {x}})\,\delta (g({\boldsymbol {x}}))\,d{\boldsymbol {x}}=\int _{g^{-1}(0)}{\frac {f({\boldsymbol {x}})}{|{\boldsymbol {\nabla }}g|}}\,d\sigma ({\boldsymbol {x}})} where the integral on the right is over g (0), the (n − 1) -dimensional surface defined by g (x) = 0 with respect to the Minkowski content measure. This is known as a simple layer integral.

More generally, if S is a smooth hypersurface of R, then we can associate to S the distribution that integrates any compactly supported smooth function g over S: δ S = ∫ S g (s) d σ (s) {\displaystyle \delta _{S}=\int _{S}g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {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 R with smooth boundary S, then δ is equal to the normal derivative of the indicator function of D in the distribution sense,

− ∫ R n g (x) ∂ 1 D (x) ∂ n d x = ∫ S g (s) d σ (s), {\displaystyle -\int _{\mathbb {R} ^{n}}g({\boldsymbol {x}})\,{\frac {\partial 1_{D}({\boldsymbol {x}})}{\partial n}}\,d{\boldsymbol {x}}=\int _{S}\,g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}}),}

where n is the outward normal. For a proof, see e.g. the article on the surface delta function.

In three dimensions, the delta function is represented in spherical coordinates by:

δ (r − r 0) = { 1 r 2 sin θ δ (r − r 0) δ (θ − θ 0) δ (ϕ − ϕ 0) x 0, y 0, z 0 ≠ 0 1 2 π r 2 sin θ δ (r − r 0) δ (θ − θ 0) x 0 = y 0 = 0, z 0 ≠ 0 1 4 π r 2 δ (r − r 0) x 0 = y 0 = z 0 = 0 {\displaystyle \delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{0})={\begin{cases}\displaystyle {\frac {1}{r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})\delta (\phi -\phi _{0})&x_{0},y_{0},z_{0}\neq 0\\\displaystyle {\frac {1}{2\pi r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})&x_{0}=y_{0}=0,\ z_{0}\neq 0\\\displaystyle {\frac {1}{4\pi r^{2}}}\delta (r-r_{0})&x_{0}=y_{0}=z_{0}=0\end{cases}}}