I want to understand a solution from an exercise where we should find a solution of the heat equation:
$$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ with $u(x,0)=f(x)\in L^1$.
The first step is to use the Fouriertransformation on both sides. But in order to get a ODE we must have the partial $\partial$ outside of the Fouriertransformation. How we can take it out?
First you'll want to show that for $g: \mathbb{R} \to \mathbb{R}$ $$ \mathscr{F} \left ( \frac{ d g }{dx} \right ) = i \xi \mathscr{F}( g ) $$ where the Fourier transform takes $\mathscr{F} : u(x) \to \hat u( \xi)$. Apply this to $\Delta u$ now to see $$ \mathscr{F} ( \Delta u ) = - || \xi ||^2 \hat u $$ Since you'll take this as a spacial transform (not time), the time derivative will pass through the transform. i.e. $$ \mathscr{F} (u_t ) = \hat u_t$$ Putting all the pieces together shows that the Fourier transform applied to the PDE gives $$ \Delta u = u_t \to - | \xi|^2 \hat u = \hat u_t $$ i.e. an ODE in $t$.