In my physics book the definition of current is
If a charge $∆Q$ crosses an area in time $∆t$, we define the average electric current through the area during this time as
$I=\frac{∆Q}{∆t}$. The current at time $t$ is $I=\lim_{∆t\to0} \frac{∆Q}{∆t}=\frac{dQ}{dt}$
And the definition of current density is
If $∆I$ is the current through the area $∆S$, the (same as the top) ... $J=\frac{dI}{dS}$
My question is what is the difference between the 1st definition area and the 2nd one?
The total current $I$ through a surface is a global quantity (with respect to the surface). The current density $\bf J$ is a vector and it is a function of the point, in general. The integral over the surface of $\bf J \cdot n$ is the total current $I$, $\bf n$ being a unit vector orthogonal to each surface element. If the surface is so small that variations of $\bf J$ over the surface can be neglected, and if the surface is orthogonal to the direction of $\bf J$ there is a relation between the small surface area $\Delta S$, $|{\bf J}|$, and the total current through the same surface: $I=|{\bf J}| \Delta S.$ If the unit vector makes an angle $\theta$ with $\bf J$ the previous formula becomes: $$ I=|{\bf J}| \Delta S\cos \theta.\tag{1} $$ The right-hand side can be interpreted as the product of the modulus of the density current by the cross-sectional area of the surface element.
Summarizing: if $I$ is the total current through a small surface element $\Delta S$ and we want to connect it to an underlying current density ${\bf J}$ the surface element must be the same, in general. However, formula $(1)$ tells us that many surface areas, oriented to give the same cross-sectional area, will give the same total current.
