Graph the following vector fields (the direction and magnitude of the vectors in the xy-plane) and conclude whether the path integral of F*dr is positive, negative or zero if the curve C is a straight line that goes from the origin to the point (3,3) (also graph the curve C and note its direction).
a) F= xi+yj
b) F=−2i+2j
c) F=−y^2*j.

The graphs show the vector fields along with the curve \( C \) from \( (0,0) \) to \( (3,3) \).

### Conclusions:
- **a) \( F = x\mathbf{i} + y\mathbf{j} \)**: The vector field aligns with the direction of \( C \), making the path integral **positive**.
- **b) \( F = -2\mathbf{i} + 2\mathbf{j} \)**: The field is mostly perpendicular to \( C \), resulting in a **zero** path integral.
- **c) \( F = -y^2\mathbf{j} \)**: This field points downward with minimal alignment to \( C \), also yielding a **zero** path integral.

Would you like further details on these calculations or have any additional questions?

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Here are some related questions for further exploration:

1. What would change in the path integral if \( C \) were a different path (e.g., circular)?
2. How does the field \( F = -y^2\mathbf{j} \) behave along vertical and horizontal lines?
3. How can we calculate the exact values of these path integrals along different paths?
4. What would the path integral be if \( C \) were extended beyond \( (3,3) \)?
5. How does the direction of \( C \) influence the path integral for a general vector field?

**Tip**: Always consider the alignment of the vector field with the path direction when evaluating path integrals, as it directly affects the sign and magnitude.

Graph the following vector fields (the direction and magnitude of the vectors in the xy-plane) and conclude whether the path integral of F*dr is positive, negative or zero if the curve C is a straight line that goes from the origin to the point (3,3) (also graph the curve C and note its direction). a) F= xi+yj b) F=−2i+2j c) F=−y^2*j.

Explore how to graph vector fields F = xi + yj, F = -2i + 2j, and F = -y^2 j, and evaluate their path integrals along a line from (0,0) to (3,3). This guide explains vector field alignment, path direction, and conclusions on the path integral's sign. Ideal for students in advanced calculus or vector calculus courses.

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