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.
