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Divergence and Curl

My notes say that if we know the divergence and curl of a field then that uniquely determines the field.

Can somebody give me an example of how, given only the div and curl of a field, we can deduce the field?

I considered the electric field where we have,
[itex]\nabla \cdot \vec{E}=\frac{\rho}{\epsilon_0}, \nabla \times \vec{E}=0[/itex]
but we can't actually establish E using only vector calculus can we? we need other techniques do we not? perhaps i'm just being silly?
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I think the boundary condition needs to be specified.
The electrostatic case is easy to understand. To find the electric field at a point, you need to integrate the electric field contributions from each infinitesimal charge. Since the local charge density is found from the divergence of the E-field, you are essentially integrating the divergence over the volume of interest.

In practice, the charge density is not always known a priori. Sometimes the electric field or potential is specified along some boundary, which is where Neumann and Dirichlet boundary conditions come into play.
yeah. can you just confirm a couple of things:

(i) the RHS of the divergence equation is easy enough to integrate given a charge density. To integrate the LHS however, I would use divergence theorem and then take it from there using an appropriate Gaussian surface yes?

(ii)In the method you just gave for finding the E field in post 3, you have only used the divergence equation, how would the curl one come into play?

to make sure it's electrostatic
i.e. a non zero curl results in a non continuous tangential component of electric field at a conductor's surface and so the tangential component of electric field outside the conductor isn't necessarily 0 (as is guaranteed if the curl were 0) and so charges are moving on the surface - hence it's no longer electrostatics!

is that what you mean?
em...i would rather say a nonzero curl is resulted from a changing magnetic field, by faraday's law.So it won't be not electrostatics.
Actually electric fields and magnetic fields are the same thing if you consider standing still the same as moving in the time dimension.

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