To match the given vector fields $F(x, y)$ with their corresponding graphs, we analyze how the components of the vector field affect the direction and magnitude of the vectors. Let's break down each option and compare it to the graphs provided.

1. Analysis of Vector Fields:

a. $F(x, y) = x \, \hat{i} - y \, \hat{j}$

• The $x$-component increases linearly along the $x$-axis.
• The $y$-component is negative and increases linearly with $y$, pointing downward for positive $y$ and upward for negative $y$.
  Behavior: Radial pattern with vectors pointing outward along the $x$-axis and inward along the $y$-axis. Likely matches Graph I.

b. $F(x, y) = y \, \hat{i} + (x - y) \, \hat{j}$

• The $x$-component is proportional to $y$.
• The $y$-component is determined by $x - y$, leading to a combination of increasing $x$-direction and decreasing $y$-direction.
  Behavior: Slanted pattern depending on the combination of $x$ and $y$. Likely matches Graph IV.

c. $F(x, y) = y \, \hat{i} + (y + 2) \, \hat{j}$

• The $x$-component is proportional to $y$.
• The $y$-component increases linearly with $y$ and always points upward.
  Behavior: Vectors move upward with a slight tilt depending on $y$. Likely matches Graph II.

d. $F(x, y) = \cos(x + y) \, \hat{i} + x \, \hat{j}$

• The $x$-component oscillates due to the cosine function of $x + y$.
• The $y$-component increases linearly with $x$.
  Behavior: Complex field with oscillations in the $x$-direction and consistent increase in $y$-direction. Likely matches Graph III.

2. Matching Results:

• a matches Graph I.
• b matches Graph IV.
• c matches Graph II.
• d matches Graph III.

Would you like more details or further explanation? Here are 5 follow-up questions to expand on this:

1. What is a vector field, and how can you interpret its components?
2. How do the properties of trigonometric functions (e.g., cosine) affect vector fields?
3. What are the typical patterns for radial and oscillatory vector fields?
4. How can you graph vector fields from scratch?
5. How does scaling the components of a vector field affect its visualization?

Tip: When analyzing vector fields, always observe how the direction and magnitude of vectors change with respect to $x$ and $y$. This helps identify patterns like rotation, oscillation, or divergence.