Surface Current Density I'm trying to derive the boundary conditions for magnetic field and if I try to derive the condition for the tangential component of magnetic field.

From the figure and Maxwell's law for line integral of  magnetic field:
$$\oint \vec H \cdot \vec {dl} = \oint_A \vec H \cdot \vec {dl}+\oint_B \vec H \cdot \vec {dl}+\oint_C \vec H \cdot \vec {dl}+\oint_D \vec H \cdot \vec {dl} = I_{enclosed}$$
As l and w become small $\oint_B \vec H \cdot \vec {dl}+\oint_D \vec H \cdot \vec {dl} \rightarrow 0$
which implies $\oint_A \vec H \cdot \vec {dl}+\oint_C \vec H \cdot \vec {dl}=I_{enclosed}$
All the manipulations till this point are clear to me and after this he goes ahead and replaces $I_{enclosed}$ by $J_s \cdot (\Delta l \hat t \times \hat n)$ where $J_s$ is the surface current density and $\Delta l$ is the length of A and C.
First and foremost is surface current density same as current density J and because w reduces to almost close to zero current density becomes like line current density of some sort?
Is $(\Delta l \hat t \times \hat n)$ added to include the area element to the current density?
But then again $J_s$ is the surface current density so why is the area element added instead of some line element like $\vec {dl}$?
Impressive and subtle as the rest of the derivation the question above is bugging me?
Can anybody help?
 A: "Current density" (without any other adjectives) means "charge passing a plane per unit time, per unit area of the plane". "Surface current density" means "amount of charge in a thin sheet of current passing a given place, per unit time, per unit length across the wide dimension of the sheet of current".
To be clear, let's consider a sheet of current in the shape of a rectangle with widths $a$ and $b$, with $a \gg b$. Let $I$ be the total current in this sheet. Then the volume current density is
$$
j = \frac{I}{ab}
$$
and if $b$ is small compared to all other significant lengths in the problem under consideration, then we can say this sheet of current can be regarded as a surface current,  with surface current density
$$
k = \frac{I}{a}
$$
In practice the concept of a surface current usually involves approximation, because the sheet of current is not really infinitely thin. For example, the thickness might be a nanometre in some cases, so if you are studying a problem involving waves of wavelength hundreds of nanometres or more, then you can regard the current as a thin sheet. This is what is going on when people refer to surface current in the context of continuity conditions for magnetic fields. The idea is that if we don't care about the details of the distribution of current across the small dimension of the thin sheet, then it is more convenient just to talk about the net current per unit distance along the sheet.
Note that I use different letters for the different ways of discussing current, because they refer to quantities with different physical dimensions (current per unit area in one case, current per unit length in the other). However there is no universal convention on the choice of letter. You sometimes see $\sigma$ used for surface current, but since it is also used for surface charge, I prefer to use another letter such as $k$. That may not always be a great choice, for example if one is dealing with waves then one might want to use $k$ for a wave vector.
