For multi-dimensional functions, the [[Fourier transform]] is simply repeated for each variable such that ^8348fd $\widetilde{f}(k_1,...,k_d)=\int dx^1\,e^{-ix^1k_1}...\int dx^d \, e^{-ix^{d}k_d}f(x^1,...,x^d)$ ^546b78 where here by convention we have vectors $(k_1,...,k_d)^T$, which is a [covariant](Covariant.md) vector as indicated by the use of subscript indices and $(x_1,...,x^j)$ is a [contravariant](Contravariant.md) vector as indicated by the use of superscript. The the Fourier transform of a multidimensional function is written as: $\mathcal{F}[f(\mathbf{x})]_\mathbf{k} = \widetilde{f}(\mathbf{k}) = \int dx^1... dx^d e^{-i\mathbf{k}\cdot\mathbf{x}}f(\mathbf{x})$ where the inverse is $\mathcal{F}^{-1}[f(\mathbf{k})]_\mathbf{x} = f(\mathbf{x}) = \int \frac{dk_1}{2\pi}... \frac{dk_d}{2\pi} e^{i\mathbf{k}\cdot\mathbf{x}}\widetilde{f}(\mathbf{x})$ and $\mathbf{k}\cdot\mathbf{x}=\sum_j k_j x^j$, meaning that the exponent is a coordinate invariant scalar. # Multidimensional Completeness relation In multiple dimensions, the completeness relation involves the [[Multi-dimensional Dirac delta function]] such that, $\delta(\mathbf{x}) = \int \frac{dk_1}{2\pi}... \frac{dk_d}{2\pi} e^{i\mathbf{k}\cdot\mathbf{x}}$ #MathematicalFoundations/Analysis/FourierAnalysis/Integrals #MathematicalFoundations/Analysis/Functions/Functionals/Integrals/IntegralTransforms