Is heat flux density and heat flux the same thing? Heat flux and heat flux density is the same thing, while electric flux density and electric flux is not the same thing?
It makes me confused since we compare Fourier's law with Ohm's law.
Here is a statement from Wikipedia.

To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small.

Is heat flux defined at a point or on a surface? I have never been found any defintion of heat flux or heat flux density.
As a mathematical concept, flux is represented by the surface integral of a vector field,
$$\Phi_F = \iint_A \mathbf{F}\cdot \mathrm{d}\mathbf{A}$$
where $\mathbf{F}$ is a vector field, and $\mathrm{d}\mathbf{A}$ is the vector area of the surface $A$, directed as the surface normal.
Heat is often denoted $\vec{\phi_q}$ and we integrated the heat flux density $\vec{\phi_q}$ over the surface of the system to have the heat rate but we integrated the $\mathbf{E}$-filed to get the electric flux?
Thanks.
 A: If we take the definition of heat flux given here seriously, then heat flux is defined as a vector field $\vec\phi$ with units of energy per unit time, per unit area.  At every point $\vec x$ in space, the vetor $\vec\phi(\vec x)$ tells you the direction and magnitude of heat flow in a neighborhood of that point.  In particular, if we consider some two-dimensional surface $d\vec A$ containing $\vec x$, then
$$
  \vec\phi(\vec x) \cdot d\vec A
$$
will tell us the amount of energy per unit time flowing through that surface.  In particular, notice that here flux is being using to describe a vector field, not a scalar as in electric flux in EM.  Perhaps this is rather bad terminology for this reason.
A: As I understand it:
Electromagnetism and mathematics defines "flux" as the current or flow rate (dimensions of quantity/time), which is equal to the surface integral of a "flux density" (dimensions of quantity/time/area).
In the study of transport phenomena (including thermal transport and mass diffusion), "flux" refers to integrand (quantity/time/area), and the integrated result (quantity/time) is called a flow or transport rate.
However, I have found the field of heat transport to have the most inconsistency. As OP noted, often the same symbol will be given to the two quantities (flow rate and flux by Transport convention, or flux and flux density, by E&M/math convention). To further complicate the matter, in heat transport one may see both conventions used, so "flux" could refer to either quantity/time/area or quantity/time. "Flux density" and "flow rate" are unambiguous, but are based on different conventions. Personally I prefer the Transport convention for heat transport, defining flux as the area derivative of flow rate.
