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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.

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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.

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Do we have infinity many choice of two-dimensional surface $\mathrm{d}\vec{A}$ containing $\vec{x}$. –  Brooks Jun 6 '13 at 19:45
    
Would any different choice leads a different result? –  Brooks Jun 6 '13 at 20:00
    
@Brooks Yes you can choose any $d\vec A$ you'd like, and the discussion still applies. Different choices would indeed lead to different results in general. For example, if the direction of the heat flow were parallel to a given choice of $d\vec A$, then one would have $\vec\phi(\vec x)\cdot d\vec A = 0$. –  joshphysics Jun 6 '13 at 21:41
    
Would you mind to help me with these two probelm?physics.stackexchange.com/questions/67033/… . physics.stackexchange.com/questions/67307/… –  Brooks Jun 6 '13 at 22:04
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