Physics Equations: Integral vs Vector Derivative form [duplicate]

Question

I have been doing D.J. Griffiths's book on electrodynamics. The new level of mathematical formalism does make things a bit less farmiliar...

So, I'm asking if there is any fundamental difference (advantage, disadvantage) in writing equations in their integral form vs the vector derivative form...

For example, does writing down Maxwell's equations using the curl or divergence make it any easier to solve certain configurations, or are the operators simply there to make things look less tedious... ?

Another example would be Poisson's equation. How would one go about setting up and solving Poisson's equation for some configuration (say, a charged ring)?

Answer 1

The vector calculus form of the physical equations is mostly preferred due to brevity.

In case of Maxwell's equations the set of equations is usually considered together when solving problems and the equations use to be plugged into each other. This is certainly more convenient when using the vector calculus forms. Example: obtaining the wave equations by taking $\nabla \times$ of the curl equation .

The integral forms however help understand the physics of the equations easier. For example, consider the induction law $\nabla \times E = -\frac{\partial B}{\partial t}$. The physical meaning of the left hand side of that equation is not that intuitive. By looking at it in the integral form $\int_{\partial \Sigma} E\cdot dl=-\frac{\rm d}{{\rm d}t}\int\int_{\Sigma} B\cdot dS$ one would rather see the physical meaning: the electric field along a curve which is the boundary for the magnetic field flux surface.