The statement presented is:

True or false? If $\vec{F}$ is a conservative vector field and $f$ is a potential function for $\vec{F}$, then $\int \vec{F} \cdot d\vec{r} = f + C.$

Let's analyze this:

1. A conservative vector field $\vec{F}$ implies that there exists a potential function $f$ such that $\vec{F} = \nabla f$.
2. In conservative vector fields, the line integral of $\vec{F}$ over a path from a point $A$ to a point $B$ is given by $\int_A^B \vec{F} \cdot d\vec{r} = f(B) - f(A).$
3. This result follows from the Fundamental Theorem of Line Integrals for conservative vector fields.

However, the statement $\int \vec{F} \cdot d\vec{r} = f + C$ is a bit misleading. If the integral is evaluated between two points, the result should be $f(B) - f(A)$, not simply $f + C$. For an indefinite integral (without specified bounds), it would be more appropriate to represent it as $f(\vec{r}) + C$, where $C$ is a constant of integration.

Conclusion: The statement is True if we interpret it as an indefinite line integral resulting in $f + C$, which aligns with finding a potential function up to an additive constant. However, if interpreted as a definite integral, it would be False because that result would depend on the end points.

Thus, the correct answer is True for the indefinite interpretation.

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Related Questions

1. What is a conservative vector field, and how is it identified?
2. How does the Fundamental Theorem of Line Integrals apply to conservative fields?
3. What distinguishes an indefinite integral from a definite integral in vector calculus?
4. What role does the constant $C$ play in potential functions?
5. How can potential functions be used to simplify calculations in vector fields?

Tip

For a vector field to be conservative, its curl must be zero in simply connected regions.