List all the relations between the various space derivatives of the three components of \(\mathbf{E}\), using Gauss's law and the fact that an electrostatic field is conservative.

Gauss's Law for Electricity is a fundamental principle in electromagnetism that relates the electric field to the charge distribution present. It provides a way to calculate the electric field generated by a given charge distribution. In integral form, the law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. In differential form, which is typically used when dealing with the spatial derivatives, Gauss's Law states that the divergence of the electric field, abla \( \abla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \) . Here, \mathbf{E} \mathbf{E}\mathbf{E} represents the electric field, \rho represents the charge density, and \epsilon_0 is the vacuum permittivity. For an electrostatic field, which is static and does not change with time, the charge density is zero in free space (no free charges). This leads to \abla \abla \cdot \mathbf{E} = 0 \abla \abla \cdot \mathbf{E} = 0\abla \abla \cdot \mathbf{E} = 0, simplifying the evaluation of electric fields in electrostatic situations. Understanding this law is crucial for solving many problems in electromagnetism and is particularly important when working with electrostatic fields.

A conservative electric field is one of the key characteristics of an electrostatic field. This field conserves mechanical energy, meaning that the work done in moving a charge between two points is independent of the path taken.

Mathematically, this is expressed by stating that an electrostatic field can be represented as the gradient of a scalar potential function, \mathbf{E} = -\abla V. Furthermore, the work done by a conservative field around any closed loop is zero, which leads to one of Maxwell's equations for an electrostatic field: \abla \abla \times \mathbf{E} = \mathbf{0}. The implication of this is profound: it allows us to use the potential function \( V \) to describe an entire electrostatic field and guarantees that electric fields are irrotational in the absence of a magnetic field.

Understanding that an electric field is conservative is vital for grasping why certain problems can be greatly simplified by looking at potential rather than the field itself as it provides a tool for predicting behavior without complex calculations.

The spatial derivatives of an electric field describe how the field changes in space. They are essential for understanding the behavior of the field and for predicting how it will interact with charges and currents in space.

When dealing with the spatial derivatives of the components of the electrostatic field, we apply both Gauss's Law and the fact that the field is conservative to list out specific relations. These relations are necessary when attempting to solve for the electric field in a given configuration. We have from Gauss's Law in differential form that the sum of the second partial derivatives of the electric field components must equal zero: \[\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} = 0\]. And from the conservative nature of the field, we establish equalities between the mixed partial derivatives: \[\frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x}, \frac{\partial E_x}{\partial z} = \frac{\partial E_z}{\partial x}, \frac{\partial E_y}{\partial z} = \frac{\partial E_z}{\partial y}\]. These equalities are a direct result of the fact that the order of differentiation is irrelevant for the scalar potential function from which the electric field is derived. Understanding these derivatives is crucial for deeper insights into the electric field structure, enabling more accurate predictions and solutions in the study of electrostatics.