I'm given the following stationary heat equation in 2D:
$$\nabla\cdot q=f, \quad q=-k\nabla u, \tag1$$
where $u$ is the temperature, $q$ is the heat flow, $k$ is the conduction coefficient and $f$ a given production term.
Normally, the heat equation in 2D is given as
$$u_t = -k(u_{xx} + u_{yy}), \tag2$$
but how do I go from (1) to (2)? My first thought was that i can multiply (1) with $\nabla$ and then get
$$\nabla\cdot q=-k\nabla^2u \implies f=-k\nabla^2u=-k(u_{xx}+u_{yy}).$$
Is $f=u_t$ then? If $q$ is the heat flow, then $f=\nabla\cdot q$ is the change in heat flow, and not the change in temperature $u_t$ as I want it.
What am I missing?
You're not really missing anything.
You are given a specific stationary differential equation.
Since it represents a heat equation, we can conclude that $u_t=f$, where $f$ would be independent of $t$ since the heat distribution is supposed to be stationary.
That is, you're supposed to match the given equation to the heat equation and draw conclusions.