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Maxwell's four equations are enough to specify the fields (electric and magnetic) in a region, once you specified the charge density and current density. Maxwell's four equations basically give you the divergence and curl of the field. If I defined a matrix $A$ for some vector function $V(x,y,z)=(V_x,V_y,V_z)$ that looks like $$ A=\begin{bmatrix} \frac{\partial V_x}{\partial x} & \frac{\partial V_x}{\partial y} & \frac{\partial V_x}{\partial z}\\ \frac{\partial V_y}{\partial x} & \frac{\partial V_y}{\partial y} & \frac{\partial V_y}{\partial z} \\ \frac{\partial V_z}{\partial x} & \frac{\partial V_z}{\partial y} & \frac{\partial V_z}{\partial z} \end{bmatrix}$$

Then divergence and curl are just a combination of a component of this matrix $A$ ( for field vector). I know about this famous theorem Helmholtz's theorem, which says that in rough form

If you give me a region of space and div and curl of a vector field & you specify normal component of curl on the surface this volume element then field is uniquely determinant.

But I don't able to see physical, Why do I specify some special combination of this matrix? Why not some other? How do I know this is enough? So basically I want some intuition behind this.

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  • $\begingroup$ In principle you can define anyhow some operators from these partial derivatives but there is del operator already defined which can act in 3 different ways ( divergence, curl and gradient). And using those operations is really helpful because we are able to express laws using them in a compact form and we know divergence and curl depend on physical quantities such as charge density, current density, etc. So, using them we get a complete formulation of laws. If you can find some other combination of matrix elements that have some meaning and can be used to express laws , then its fine. $\endgroup$ Jan 15, 2021 at 12:43
  • $\begingroup$ For helmholtz theorem, we can say that since there are three unknowns( three components) ,we need three conditions or three equations amongst them to find them uniquely. Also , you can say that only curl or divergence is not enough because they can remain same even after changing the vector on which they are acting because curl of gradient is zero and divergence of curl is zero.So,we do need to specify some sort of boundary condition. For concreteness , you can see it here, $\endgroup$ Jan 15, 2021 at 12:50
  • $\begingroup$ en.wikipedia.org/wiki/…( $\endgroup$ Jan 15, 2021 at 12:50

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