# Usability of the Maxwell-Ampère equation outside the region of the current

If the Maxwell-Ampère equation is written in the form, \begin{align}\nabla \times H = \dfrac{\partial D}{\partial t} + J\end{align} that the both sides of the equation will be zero. So the Maxwell-Ampère equation should be unuseful outside the region of current. If the area does not approach zero when the curl of the magnetic field is calculated, the equation becomes meaningful. But that violates the definition of curl.

The questions:

Is the Maxwell-Ampère equation in differential form is unuseful outside the region of the current?

Is the integral form more general than the differential form? However, the integral form misses the information of the direction of the vectors.

• It takes a lot of guts to question the usefulness of the Maxwell-Ampère equation, to put it mildly. – my2cts Sep 30 '18 at 8:54
• I am sorry. I am poor at English skill. Could you give me an example? – IvanaGyro Oct 1 '18 at 2:05