I am reading here Section 3.4 equation(3.73). It says that Fourier transform is given by: $$U(k,t|r_0,t_0)=\int_{-\infty}^{\infty} dr ~.r ~u(r,t|r_0,t_0) ~~e^{-ikr}$$
I am an engineer and have just studied $f(t)$ transform and not $f(r,t)$ Fourier transform, I want to know from where does this come from? Further is it possible to have Fourier transform of a function like $f(r,z,t)$ if $r$ is the radius and $z$ is height in cylindrical coordinate system. If yes what will be the transform equation for this case? I need it to solve it PDE to a question given here
The Fourier Transform is given by the formula $$\mathcal F(f) (k) = \int_{-\infty}^\infty f(x)e^{-ixk}dx.$$
By the Fourier inversion theorem (https://en.wikipedia.org/wiki/Fourier_inversion_theorem) we can write the inverse of $\mathcal F$ as
$$\mathcal F^{-1}(g)(x) = (2\pi)^{-1}\int_{-\infty}^\infty g(k) e^{ixk}dk.$$
In the case $f$ has two variables $(x_1,x_2)$ we can write the transform with respect to one of the variables as:
$$\mathcal F_{x_1}(f) (k_1,x_2) = \int_{-\infty}^\infty f(x_1,x_2)e^{-ix_1k_1}dx_1.$$
which is just the usual transform keeping one of the variables fixed.
The inverse of $\mathcal F_{x_1}$ is just
$$\mathcal F_{x_1}^{-1}(g)(x_1,x_2) = (2\pi)^{-1}\int_{-\infty}^\infty g(k_1,x_2) e^{ix_1 k_1}dk_1.$$
You can perform the partial Fourier transform described with as many variables as you wish because you are just keeping the other variables fixed.