the multidimensional Fourier transform

Real problems live in space, not just on a line: heat flows through a three-dimensional block, waves spread across a plane, a potential fills a room. The multidimensional Fourier transform is the same idea as the one-dimensional one, but applied to all the space variables at once. Instead of decomposing a function of x into waves of one frequency xi, you decompose a function of a position vector into plane waves, each labelled by a frequency vector that points in some direction and has some wavelength.

For a function f(x) of a vector x in n-dimensional space, the transform is f-hat(xi) = integral over all of R^n of f(x) e^(-i xi·x) dx, where xi is also an n-vector and xi·x is the dot product. A single ingredient e^(i xi·x) is a plane wave: it oscillates along the direction of xi with wavelength set by the length |xi|, and is constant along the perpendicular directions. The differentiation rule generalizes cleanly: a derivative in the j-th direction multiplies f-hat by i*xi_j, and — the key fact for PDEs — the Laplacian (the sum of all the second derivatives, u_xx + u_yy + u_zz) transforms to multiplication by -|xi|^2, the negative squared length of the frequency vector. The whole Laplacian, the central operator of the subject, becomes a single number for each xi.

This is what lets the transform crack PDEs in higher dimensions. The heat equation u_t = k Laplacian u becomes u-hat_t = -k|xi|^2 u-hat for each frequency vector, an ODE again, and inverting recovers the n-dimensional Gaussian heat kernel. More generally, for any constant-coefficient operator the number that multiplies f-hat, viewed as a function of xi, is the symbol of the operator — and a great deal of modern PDE theory is the study of operators through their symbols. The honest note: the bookkeeping of 2*pi factors and conventions multiplies by the dimension, and rotational problems are often handled more efficiently by the Hankel transform than by the full n-dimensional Fourier integral.