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.