To match the vector fields given with the plots, let's analyze each vector field and see how it corresponds to the visual patterns in each plot.

Analysis of the Vector Fields

1. $(x, -y)$:
   This vector field will have vectors pointing away from or toward the origin, but with opposite signs in the $x$ and $y$ directions. We can expect the field to display symmetry across the axes, with vectors pointing in the same direction on opposite sides of the $x$-axis but opposite on the $y$-axis.

2. $(y, x - y)$:
   This is a more complex field, where the $x$-component depends on $y$, and the $y$-component depends on both $x$ and $y$. It likely results in a rotational or swirling pattern, as each component influences the other in a non-linear way.

3. $(y, y + 2)$:
   Here, the $x$-component of the vector field is $y$, and the $y$-component is $y + 2$. This vector field suggests a linear pattern with a consistent shift due to the $+2$ in the $y$-component.

4. $(\cos(x + y), x)$:
   This vector field has a cosine function, meaning it will exhibit oscillatory behavior in the $x$-component. The $y$-component depends on $x$, which could result in a repeating pattern along certain axes.

Matching the Plots

Now let’s examine each plot:

• Plot (a): It shows vectors pointing in a consistent direction but with a mild oscillation horizontally. This might match an oscillatory component, such as $\cos(x + y)$ in the $x$-direction.

• Plot (b): This plot shows vectors with a swirling or rotational pattern, indicating a complex interaction between the $x$ and $y$ components, likely matching $(y, x - y)$.

• Plot (c): The vectors are mostly vertical and consistent with a slight upward shift, likely matching $(y, y + 2)$ due to the $+2$ in the $y$-component.

• Plot (d): This plot has a symmetrical pattern across the axes with vectors pointing towards or away from the origin. This symmetry suggests it matches $(x, -y)$.

Solution

Based on the analysis, here’s the matching of vector fields to plots:

1. $(x, -y)$ — Plot (d)
2. $(y, x - y)$ — Plot (b)
3. $(y, y + 2)$ — Plot (c)
4. $(\cos(x + y), x)$ — Plot (a)

Would you like further explanations on any of these matches, or do you have any questions?

Related Questions

1. How do vector fields illustrate symmetry properties in mathematical functions?
2. What are some real-world applications of oscillatory vector fields like $(\cos(x + y), x)$?
3. How can rotational patterns in vector fields, such as $(y, x - y)$, be identified in more complex fields?
4. What impact does a constant shift, such as the $+2$ in $(y, y + 2)$, have on a vector field’s appearance?
5. How do changes in vector components along different axes affect the overall pattern?

Tip

When analyzing vector fields, focusing on individual components along each axis can simplify understanding of the overall pattern.