We begin this lab by developing some graphical tool to better visualise to solutions to dif-...

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We begin this lab by developing some graphical tool to better visualise to solutions to dif- ferential equations. We begin with Quiver plots: A quiver plot is used to display a two di- mensional vector field. It is called in python with plt.quiver. This function requires four arguments, all of which must be arrays of the same size. The first two arguments define a grid of (x, y) points. The grid can be created using the meshgrid function in numpy: |X, Y = np.meshgrid(np.arange(0, 2 * np.pi, .2), np.arange(0, 2 * np.pi, 2)) This creates a 2-dimensional grid of points in the range 0 to 2 in the x and y directions. The next arguments specify the components of a 2-dimensional vector, V(x, y) = (Vx(x, y), Vy(x, y)), at the corresponding grid points. Python will draw an arrow at each grid point to represent the vector field V(x, y). As an example; define the vector field, V(Vx, Vy), as: Vx = np.cos (X) Vy = np. sin (Y) Implement and run the following code illustrating how to make a quiver plot, as well as affect figure size, geometry and labelling: import matplotlib.pyplot as plt import numpy as np np.meshgrid(np. arange (0, 2 np.pi, .2), np.arange (0, 2* np.pi, .2)) plt.close() X, Y = Vx = np.cos(X) Vy np.sin(Y) = plt.figure (figsize=(6,6)) plt.gca ().set_aspect ('equal', adjustable='box') #Make plot box square plt.xlabel('x') plt.ylabel('y') plt.title('Example of a quiver plot') plt.quiver (X, Y, Vx, Vy, pivot='mid', label='$V_x$ plt.legend () plt.show() = cos ($x$), $V_y$ = sin ($y$)') 6 5 4 31 2 1 Example of a quiver plot | V₁ = cos(x), V, = sin(y) > 3 0- Quiver plot with reduced density V = cos(x), Vy= sin(y) 4 (a) (b) Figure 1: Examples of quiver plots for the vector field defined by (cos(x), sin(y)) for different grid densities and plot options The code should produce a plot as shown in Fig. 1(a) Note that at each gridpoint the arrow is proportional to the magnitude and direction of the vector V at that point. Vary the grid density and explain what you are observing: It is possible to add a legend for the arrow lengths on to the quiver plot as follows: Q-plt.quiver (X, Y, U, V, pivot='mid', label='$V_x$ = cos($x$), $V_y$ = sin($y$)') plt.quiverkey (Q, 0.9, 0.9, 2, r'$2\frac{m} {s}$', labelpos='E', coordinates='figure') One can produces a less dense quiver plot without redefining the original array by omit- ting points in the original array. On can use an addition of a scatter plot to add a red dot at the grid points. The result of these modifications is shown in Fig. 1(b). Q-plt.quiver (X[::3, ::3], Y[::3, ::3], Vx[::3, 1:3], Vy[::3, ::3], pivot='mid', label='$V_x$ = cos ($x$), $V_y$ = sin($y$)') plt.quiverkey (Q, 0.9, 0.9, 2, r'$2\frac{m} {s}$', labelpos='E', coordinates='figure') plt.scatter (X[::3, ::3], Y[::3, ::3], color='r', s=10) A detailed description of the quiver command can be found with the help command help (plt.quiver), or online. We begin this lab by developing some graphical tool to better visualise to solutions to dif- ferential equations. We begin with Quiver plots: A quiver plot is used to display a two di- mensional vector field. It is called in python with plt.quiver. This function requires four arguments, all of which must be arrays of the same size. The first two arguments define a grid of (x, y) points. The grid can be created using the meshgrid function in numpy: |X, Y = np.meshgrid(np.arange(0, 2 * np.pi, .2), np.arange(0, 2 * np.pi, 2)) This creates a 2-dimensional grid of points in the range 0 to 2 in the x and y directions. The next arguments specify the components of a 2-dimensional vector, V(x, y) = (Vx(x, y), Vy(x, y)), at the corresponding grid points. Python will draw an arrow at each grid point to represent the vector field V(x, y). As an example; define the vector field, V(Vx, Vy), as: Vx = np.cos (X) Vy = np. sin (Y) Implement and run the following code illustrating how to make a quiver plot, as well as affect figure size, geometry and labelling: import matplotlib.pyplot as plt import numpy as np np.meshgrid(np. arange (0, 2 np.pi, .2), np.arange (0, 2* np.pi, .2)) plt.close() X, Y = Vx = np.cos(X) Vy np.sin(Y) = plt.figure (figsize=(6,6)) plt.gca ().set_aspect ('equal', adjustable='box') #Make plot box square plt.xlabel('x') plt.ylabel('y') plt.title('Example of a quiver plot') plt.quiver (X, Y, Vx, Vy, pivot='mid', label='$V_x$ plt.legend () plt.show() = cos ($x$), $V_y$ = sin ($y$)') 6 5 4 31 2 1 Example of a quiver plot | V₁ = cos(x), V, = sin(y) > 3 0- Quiver plot with reduced density V = cos(x), Vy= sin(y) 4 (a) (b) Figure 1: Examples of quiver plots for the vector field defined by (cos(x), sin(y)) for different grid densities and plot options The code should produce a plot as shown in Fig. 1(a) Note that at each gridpoint the arrow is proportional to the magnitude and direction of the vector V at that point. Vary the grid density and explain what you are observing: It is possible to add a legend for the arrow lengths on to the quiver plot as follows: Q-plt.quiver (X, Y, U, V, pivot='mid', label='$V_x$ = cos($x$), $V_y$ = sin($y$)') plt.quiverkey (Q, 0.9, 0.9, 2, r'$2\frac{m} {s}$', labelpos='E', coordinates='figure') One can produces a less dense quiver plot without redefining the original array by omit- ting points in the original array. On can use an addition of a scatter plot to add a red dot at the grid points. The result of these modifications is shown in Fig. 1(b). Q-plt.quiver (X[::3, ::3], Y[::3, ::3], Vx[::3, 1:3], Vy[::3, ::3], pivot='mid', label='$V_x$ = cos ($x$), $V_y$ = sin($y$)') plt.quiverkey (Q, 0.9, 0.9, 2, r'$2\frac{m} {s}$', labelpos='E', coordinates='figure') plt.scatter (X[::3, ::3], Y[::3, ::3], color='r', s=10) A detailed description of the quiver command can be found with the help command help (plt.quiver), or online.

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To vary the grid density in the quiver plot and observe the differences you can adjust the slicing i... View the full answer