Mfrhtyj

I was reading Classical Electrodynamics by J.D.Jacskon (section 1.5) where he said:

Perhaps some readers know that a vector field can be specified almost completely if its divergence and curl are given everywhere in space

with a comment

(almost)-Up to the gradient of a scalar function that satisfies the Laplace equation.

why is this so

if we want to know $vec{E}$ in 3-D space we really just want to find out is $E_x,E_y$ and $E_z$ (in cartesian coordinates) which are just 3 scalar functions.
If we know the curl is (let's say) zero, that is $$nabla times vec{E}=0$$

in cartesian coordinates that gives us three equations
$$frac{partial E_z}{partial y}-frac{partial E_y}{partial z}=0$$
$$frac{partial E_x}{partial z}-frac{partial E_z}{partial x}=0$$
$$frac{partial E_y}{partial x}-frac{partial E_x}{partial y}=0$$

these are three equations for 3 unknown quantities, that should give us enough* information about these three functions, it obviously isn't so hence we need to know the divergence as well, my question is why?
is there some theorem that proves this (I suspect there is $:)$ )

this is obviously a mathematical question although it is physics motivated
so I hope it is ok that I post it here.

(*up to maybe a constant because all the equations involve first derivatives, I can live with that)

If

$nabla times vec E = 0, tag 1$

we know from a standard result that

$vec E = nabla phi tag 2$

for some scalar function $phi$.

If the divergence of $vec E$ is also a known function $rho$,

$nabla cdot vec E = rho, tag 3$

then combining (2) and (3) we obtain

$nabla^2 phi = nabla cdot nabla phi = nabla cdot vec E = rho, tag 4$

which can in principle be solved for $phi$ (assuming we admit appropiate boundary conditions on $phi$; $phi(x) to 0$ sufficiently fast as $vert x vert to infty$ is often used; Jackson's book covers this, mos' likely); thus, we may discover $vec E$ from (2).

We observe that such a solution is not unique; indeed, let $psi$ be any harmonic function,

$nabla cdot nabla psi = nabla^2 psi = 0; tag 5$

then

$nabla^2(phi + psi) = nabla^2 phi + nabla^2 psi = rho + 0 = rho, tag 6$

and if

$vec E = nabla (phi + psi), tag 7$

we still obtain

$nabla times vec E = nabla times nabla (phi + psi) = nabla times nabla phi + nabla times nabla times psi = 0, tag 8$

since the curl of any gradient vanishes. Also,

$nabla cdot vec E = nabla cdot (nabla phi + nabla psi) = nabla^2 phi + nabla ^2 psi = nabla^2 phi = rho; tag 9$

these last two equations show that we may transform any solution according to

$phi to phi + psi, tag 9$

$vec E = nabla phi to nabla phi + nabla psi, tag{10}$

$psi$ as in (5), and preserve the divergence and curl of $vec E$; so any solution is not unique; uniqueness may be attained by specifying appropriate boundary conditions on $phi$ and $psi$ which can then become unambiguously determined.

The above discussion addresses the relatively simple case (1), (3); we can, in fact, also address the significant generalization

$nabla times vec E = vec F, ; nabla cdot vec E = rho, tag{11}$

where $vec F$ is a pre-specified vector field; the situation is more complicated since we may no longer assume $vec E$ is a gradient as in (1)-(2). In this case we
instead invoke the vector calculus identity

$nabla times (nabla times vec A) = nabla (nabla cdot vec A) - nabla^2 vec A, tag{12}$

where the Laplacian operator $nabla^2$ occurring on the right-hand side is understood to act component-wise on $vec A$; thus we have

$nabla times vec E = vec F tag{13}$

leading to

$nabla times (nabla times vec E) = nabla times vec F, tag{14}$

which we trasform according to (12):

$nabla (nabla cdot vec E) - nabla^2(vec E) = nabla times vec F; tag{15}$

by virtue of (3) we write this as

$nabla rho - nabla^2 vec E = nabla times vec F, tag{16}$

whence we find

$nabla^2 vec E = nabla rho - nabla times vec F, tag{17}$

which we may in principle solve component-wise for $vec E$; for example,

$nabla^2 E_x = dfrac{partial rho}{partial x} - left ( dfrac{partial F_z}{partial y} - dfrac{partial F_y}{partial z} right) = dfrac{partial rho}{partial x} -
nabla^2 E_x = dfrac{partial rho}{partial x} - dfrac{partial F_z}{partial y} + dfrac{partial F_y}{partial z}; tag{18}$

of course, we need boundary conditions and perhaps certain restrictions on $rho$ and $vec F$; but I'll leave it to folks like David Jackson to explain such matters.

Apparently solutions to (11), (18) are still not unique; it is still true that adjusting $vec E$ by the gradient of a harmonic function $psi$ gives rise to another solution; of course, we may determine such $psi$ uniquely via appropriate boundary conditions, and then a certain uniqueness is attained for $vec E$.