The image contains three vector field diagrams, labeled d, e, and f.### Diagram d:- **Description:** The vector field exhibits a saddle point pattern. The field lines extend outward along the vertical and horizontal axes, while they curve inward from the diagonal corners toward the center. This creates a structure reminiscent of hyperbolas intersecting at the center. The arrows indicate the direction of flow, showing vectors moving away from the center along the axes and towards it along the diagonals.### Diagram e:- **Description:** This diagram illustrates a counterclockwise vortex. The vector field lines form a spiral pattern, rotating around a central point. The lines start wider at the perimeter and become tighter as they approach the center, indicating increasing field intensity. The arrows on the lines point towards the center, marking the inward spiral motion.### Diagram f:- **Description:** This vector field shows a checkerboard-like pattern with a central point where the lines curve inward from all sides, meeting at the center and then extending outward. The lines create square-like regions characterized by the lines opposing their direction diagonally. The arrows indicate flow direction, suggesting a rotational or twisting motion that diverges and converges in quadrants.These diagrams are useful in studying different types of vector fields, such as electromagnetic or fluid flow fields, demonstrating how lines of force or flow can be represented graphically. # Eigenvalues and Phase Portraits of Two-Dimensional SystemsIn this lesson, match pairs of eigenvalues to the phase portraits of two-dimensional systems. Understanding how eigenvalues relate to phase portraits helps in analyzing the behavior of dynamical systems.**Match the pairs of eigenvalues:**1. \(\lambda_1 = 1\), \(\lambda_2 = 2\)2. \(\lambda_1 = -1\), \(\lambda_2 = -2\)3. \(\lambda_1 = 1 + i\sqrt{5}\), \(\lambda_2 = 1 - i\sqrt{5}\)4. \(\lambda_1 = -2 + 3i\), \(\lambda_2 = -2 - 3i\)5. \(\lambda_1 = 5\), \(\lambda_2 = -3\)6. \(\lambda_1 = 2i\), \(\lambda_2 = -2i\)**Phase Portraits:**- **Portrait a:** This diagram features spiraling curves indicating a center or spiral point. The lines gradually approach or move away from the equilibrium point.- **Portrait b:** This diagram shows hyperbolic trajectories, characteristic of a saddle point. The lines appear to diverge from one axis and converge towards the other.- **Portrait c:** This diagram depicts concentric circular patterns around a central point suggesting a center or a focus, usually associated with purely imaginary eigenvalues.Analyze these properties to determine which pair of eigenvalues corresponds to each phase portrait. Examining the real parts and imaginary parts of the eigenvalues is key in recognizing the types of nodes, spirals, centers, or saddles.

To match the phase portraits of the given eigen values:1). If both the eigen values are of same sign and positive then it's phase portrait is unstable node. So, the required portrait is option (d).2). If both the eigen values are of same sign and negative then it's phase portrait is stable node. So, the required portrait is option (b).3). If the eigen values are pair of imaginary roots with positive real part then it's phase portrait is unstable spiral. So, the required portrait is option (a).4). If the eigen values are pair of imaginary roots with negative real part then it's phase portrait is stable spiral. So, the required portrait is option (e).5). If the eigen values are real, unequal and of opposite sign then it's phase portrait is unstable saddle point. So, the required portrait is option (f).6). If the eigen values are pair of pure imaginary roots then it's phase portrait is center. So, the required portrait is option (c).The required solution is:-1)-(d).2)-(b).3)-(a).4)-(e).5)-(f).6)-(c).