Final answer:
To determine the divergence of a vector field, one calculates the dot product of the del operator with the vector field. A nonzero divergence indicates a net outflow of the vector field from a point or region, while zero divergence indicates a closed field with no net outflow or inflow.
Step-by-step explanation:
To determine whether the divergence of a vector field is complete or incomplete, one must first calculate the divergence using differential calculus. The divergence of a vector field ℝ\vec{E} at any point gives a measure of how much the vector field 'spreads out' from a point. If the divergence is nonzero at a point or in a region, it indicates a net 'outflow' of the vector field from that point or region, hence indicating an incomplete closure. Conversely, if the divergence is zero, it implies no net outflow or inflow, indicating completeness or a closed field.
Divergence is calculated as the dot product of the del operator (nabla operator) with the vector field; in Cartesian coordinates, this is:
div(\vec{E}) = \\abla \cdot \vec{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}.
Understanding this concept can be useful in analyzing one-dimensional and two-dimensional relative motion problems, as well as in various physics applications like calculating electric field vectors, dealing with scalar and vector quantities, and understanding the properties of conservative forces. For instance, the electric field vector at a point due to a charge distribution has components Ex, Ey, and Ez, and computing the divergence can provide insights into the nature of the field's source or sink at that point.