Final answer:
To determine if a line integral in a vector field is complete or incomplete, one needs to assess if the vector field is conservative and pay close attention to the single parameter that describes the path, applying the principle of superposition if necessary.
Step-by-step explanation:
To determine whether a line integral of a vector field is complete or incomplete, we must assess whether the vector field is conservative. If a force field is conservative, the line integral is independent of the path taken, implying that the field work is completely determined by its start and end points. In non-conservative fields, the integral would be path-dependent and potentially incomplete depending on the specified limits and path.
When calculating line integrals, variables not explicitly given need to be eliminated, such as the radial distance (r) and the changes along the path. This is usually done by expressing the variables in terms of a single parameter that can be varied along the path of integration and using dependencies like charge density functions correctly.
Superposition principle is often assumed when evaluating such integrals. When working with components, as in our examples, simplifications often arise. However, choosing the right form (x, y components or magnitudes and angles) is crucial and depends on the particular situation.
In summary, to determine the completeness of a line integral over a vector field, it's essential to understand whether the field is conservative and to apply the principle of superposition and correct expressions for differential elements, like dl, dA, or dV, based on the nature of the vector field.