https://file.aiquickdraw.com/mathbot/file/1730719524429vntrlkma.jpg

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. 

---

Would you like a detailed breakdown or have any questions?

### 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.

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

Analyze the relationship between a conservative vector field and its potential function in vector calculus. This problem explores whether \( \int \vec{F} \cdot d\vec{r} = f + C \) is true and examines the Fundamental Theorem of Line Integrals. Suitable for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation