Producing the Maxwell-Ampere law in differential form

The Maxwell-Ampere law in intergral form:

(1)
Substituting from the relationship:

(2)

(3)

(4)

(5)
Applying Stoke's Thm:

(6)

(7)
Now, expressing the terms with respect to the area S:

(8)

(9)

(10)

(11)

(12)
So,

(13)

(14)
Finally, introducing D by the relation:

(15)
we get the Maxwell-Ampere law in differential form:

(16)