True.

In a conservative vector field, the field is defined as the gradient of a potential function $f$. This means that the vector field $\vec{F}$ can be expressed as:

$\vec{F} = \nabla f$

The potential function $f$ determines the vector field uniquely, up to an additive constant. This is because any constant added to $f$ does not affect the gradient $\nabla f$, as the derivative of a constant is zero. Therefore, multiple potential functions differing only by a constant will yield the same conservative vector field.

Would you like further details or have any questions?

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Here are five related questions for further exploration:

1. What properties must a vector field satisfy to be conservative?
2. How can one determine if a given vector field is conservative?
3. What role does the path independence of line integrals play in conservative vector fields?
4. How does the Fundamental Theorem for Line Integrals relate to conservative vector fields?
5. Can a conservative vector field exist in a region with holes or singularities?

Tip: For a vector field to be conservative, it must be defined in a simply connected region where all closed path integrals equal zero.