# Determination of electric field E

Theorem in electromagnetism says if the divergence $\nabla\cdot E$ and the curl $\nabla\times E$ are known, then E is uniquely determined. So my question is, the divergence equation gives 1 constraint and the curl equation gives 3 constraints, which add up to 4 constraints. On the other hand, the electric filed E has only 3 unknown components. Does it mean 4 equations are overdetermined and thus has dependency?

Jano L.
Gold Member
If the system of equations is overdetermined, it means there is no solution.

If the equations are dependent, it means one or more equations are superfluous (do not give new constraint).

Divergence and curl of a field are both necessary to reconstruct the field. See

http://mathworld.wolfram.com/HelmholtzsTheorem.html

Dale
Mentor
2020 Award
Theorem in electromagnetism says if the divergence $\nabla\cdot E$ and the curl $\nabla\times E$ are known, then E is uniquely determined. So my question is, the divergence equation gives 1 constraint and the curl equation gives 3 constraints, which add up to 4 constraints. On the other hand, the electric filed E has only 3 unknown components. Does it mean 4 equations are overdetermined and thus has dependency?
You might like this paper

http://faculty.uml.edu/cbaird/95.657(2013)/Maxwell_Uniqueness.pdf