[Physics] How could flux can be a vector and a scalar

Vector Fields

Here is the General mathematical definition of Flux on Wikipedia:

The frequent symbol is $j$, and a definition for scalar flux of physical quantity $q$ is the limit:
$$j=\lim\limits_{A\to 0}\cfrac{I}{A}=\cfrac{\mathrm{d}I}{\mathrm{d}A}$$
where:
$$I=\lim\limits_{\Delta t\to 0}\cfrac{\Delta q}{\Delta t}=\cfrac{\mathrm{d}q}{\mathrm{d}t}$$
is the flow of quantity $q$ per unit time $t$, and $A$ is the area through which theh quantity flows.For vector flux, the surface intergal of $j$ over surface S, followed by an integral over the time duration $t_1$ to $t_2$, gives the total amount of the property flowing through the surface in that time $(t_2 − t_1)$:
$$q=\int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\hat{\mathbf{n}}\, \mathrm{d}A\mathrm{d}t$$

Although I have already know the relationship between $\mathrm{d}A$ and $\mathrm{d}\mathbf{A}$. However, what's the relationship between $j$ and $\mathbf{j}$? What's the definition of $\mathbf{j}$?

I ask this question because of this: problem.

I don't solve my problem from the two answers. I am not sure about the definition of heat flux. It is necessary for you to write rigorous description to this question.

Best Answer

The relation between the scalar $j$ and the vector $\vec j$ is simply :

$$j = \vec j \cdot \vec n$$

where $\vec n$ is a unit vector normal to the surface $\mathrm{d}A$.

So, you could write:

$$j ~ \mathrm{d}A = \vec j \cdot \vec n ~ \mathrm{d}A = \vec j \cdot \vec {\mathrm{d}A}$$

with the notation $\vec {\mathrm{d}A} = \vec n ~ \mathrm{d}A$

So, choose the notation that you prefer, but it must be clear for you.

Suppose, for instance, that you are working with a conserved quantity $Q$. We may define a quantity density $\rho$, and a quantity flux $\vec j$, so that the local law of the conservation of the quantity is written:

$$\frac{\partial \rho}{\partial t} + \operatorname{div} \vec j = 0$$

Considering now a volume $V$, with boundary surface $A$. The variation of the quantity $Q$, in the volume V, between 2 times $t_1$ and $t_2$ is:

$$\Delta Q = \int_{t_1}^{t_2} \int_V \frac{\partial \rho(\vec x,t)}{\partial t} \mathrm{d}^3x = - \int_{t_1}^{t_2}\int_V \operatorname{div} \vec j \mathrm{d}^3x = - \int_{t_1}^{t_2} \int_A \vec j.\vec {\mathrm{d}A}$$

The last equality comes from Stokes Theorem.

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