https://file.aiquickdraw.com/mathbot/file/1728893143604r3s0fe9p.jpg

Let's analyze the vector fields and paths in the provided image to determine whether the line integrals of each vector field along the oriented path are positive, negative, or zero. Here's the breakdown:

1. **Top-left (Square path)**:
   - The vector field is uniform and appears to be radially outward or constant across the field, and the path is a closed square loop.
   - Since the vectors along the path are mostly perpendicular to the path direction (no net circulation), the integral is **zero**.

2. **Top-middle (Half-circle path)**:
   - The vector field shows radial symmetry with arrows pointing outward from the center, but the path is along a half-circle.
   - Since the field is symmetric and perpendicular to the path, there is no net flow or circulation along the path. The integral is **zero**.

3. **Top-right (Straight path)**:
   - The vector field is also outward from the center and the path is a straight line.
   - As the vectors along the path are perpendicular, there is no work done along the path. The integral is **zero**.

4. **Bottom-left (Triangular path)**:
   - The vector field is pointing outward, and the path is a triangle.
   - The path moves in such a way that the vector field will do positive work along some segments (where the vectors are aligned with the path). Hence, the integral is **positive**.

5. **Bottom-middle (Spiral path)**:
   - This vector field has a circular pattern, and the path is a spiral. Here, the orientation of the path and the direction of the vector field are mostly opposing each other.
   - This leads to a negative contribution to the integral. Therefore, the integral is **negative**.

6. **Bottom-right (Half-circle path)**:
   - The vector field shows a uniform direction, and the path is a half-circle.
   - The vector field and path are generally aligned, contributing positively to the integral. Hence, the result is **positive**.

### Correct Answers:
1. **Zero**  
2. **Zero**  
3. **Zero**  
4. **Positive**  
5. **Negative**  
6. **Positive**

Would you like more details on any of the steps or clarifications?

### Further Questions:
1. How is the concept of vector fields related to circulation?
2. What are the conditions for a line integral of a vector field to be zero?
3. Can a vector field with no divergence still have a non-zero line integral?
4. How does the orientation of the path affect the result of a line integral?
5. What role does the symmetry of a vector field play in calculating line integrals?

### Tip:
Always check if the vector field aligns or opposes the direction of the path when determining if the line integral will be positive, negative, or zero.

Determine whether the line integral of each vector field (in blue) along the oriented path (in red) is positive, negative, or zero.

Understand how to determine the sign of line integrals for various vector fields along oriented paths. Explore the application of Stokes' Theorem and the fundamental concepts of vector fields and circulation. Suitable for students in Grades 11-12.

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