Lab 19: Plotting Vector Valued Functions¶
In this lab you will create visualizations to help you understand vector fields. Before beginning the lab, please review how to create quiver plots in python.
http://problemsolvingwithpython.com/06-Plotting-with-Matplotlib/06.15-Quiver-and-Stream-Plots/
A quiver plot is a plot that shows vectors as arrows and is especially useful when talking about vector fields.
To practice, plot the vector field \(\mathbf F(x,y) = (y^2 - 2xy)\mathbf i + (3xy-6x^2)\mathbf j\). This won’t be graded.
Task 1a¶
Write a function, plot_vfield_2d(F, xa, xb, ya, yb, dx, dy, title), that plots the vector field for a vector-valued function, F, on the domain [xa, xb] X [ya, yb] with step sizes dx and dy. Recall that the tails of your vectors are the inputs to F and the tips of your vectors are the outputs from F. Be sure to set the aspect ratio of the plot to "equal" and to set the display bounds to [xa - dx, xb + dx] X [ya - dy, yb + dy] using plt.axis([xa - dx, xb + dx, ya - dy, yb + dy]) and to set the title to title if provided. Do NOT change any other properties of the plot.
Task 1b¶
Use your code from the previous exercise to visualize \(\mathbf F(x) = (r^2 - 2r)\mathbf x\), where \(\mathbf x = \langle x,y \rangle\) and \(r = |\mathbf x|\). In NumPy, this might be written as F = lambda x: np.linalg.norm(x, axis=0) * (np.linalg.norm(x, axis=0) - 2) * x. Visualize this function on [-1,1] X [-1,1] with step sizes of 0.1 and 0.1.
Task 1c¶
Use your code from the previous exercise to visualize F = lambda x: np.linalg.norm(x, axis=0) * (np.linalg.norm(x, axis=0) - 2) * x on [-5,5] X [-5,5] with step sizes of 0.5 and 0.5.
Gradient Vector Fields¶
In the next few tasks, we will plot the gradient vector field of \(f\) together with a contour map of \(f\) for:
\(f(x,y) = \ln(1+x^2+2y^2)\)
\(f(x,y) = \cos x - 2 \sin y\)
Task 2a¶
Write a function, plot_sfield_2d(f, Df, xa, xb, ya, yb, dx, dy, title), that plots the scalar field for a two-dimensional, scalar-valued function, f, using a contour plot and plots the vector field of its gradient, Df, on top on the domain [xa, xb] X [ya, yb] with step sizes dx and dy. Be sure to set the aspect ratio of the plot to "equal" and to set the display bounds to [xa - dx, xb + dx] X [ya - dy, yb + dy] using plt.axis([xa - dx, xb + dx, ya - dy, yb + dy]) and to set the title to title if provided and display the colorbar (using plt.colorbar(ax)). Do NOT change any other properties of the plot.
Task 2b¶
Use your code from the previous exercise to visualize f = lambda x: np.log(1 + x[0] ** 2 + 2 * x[1] ** 2) and Df = lambda x: 2 * np.array([x[0], 2 * x[1]]) / (1 + x[0] ** 2 + 2 * x[1] ** 2) on [-5,5] X [-5,5] with step sizes of 0.5 and 0.5 and 16 levels.
Task 2c¶
Use your code from the previous exercise to visualize f = lambda x: np.cos(x[0]) - 2 * np.sin(x[1]) and Df = lambda x: -np.array([np.sin(x[0]), 2 * np.cos(x[1])]) on [-5,5] X [-5,5] with step sizes of 0.5 and 0.5 and 16 levels.
Vector Fields in Higher Dimensions¶
Of course there are also vector fields in higher dimensions. See http://matplotlib.org/3.1.1/gallery/mplot3d/quiver3d.html
In the following tasks, we will plot these vector fields:
\(\mathbf F(x,y,z) = \mathbf i + 2\mathbf j + 3 \mathbf k\)
\(\mathbf F(x,y,z) = \mathbf i + 2\mathbf j + z \mathbf k\)
\(\mathbf F(x,y,z) = x\mathbf i + y\mathbf j + z \mathbf k\)
\(\mathbf F(x,y,z) = \sin y\mathbf i + (x\cos y + \cos z)\mathbf j - y \sin z \mathbf k\)
Task 3a¶
Write a function, plot_vfield_3d(F, xa, xb, ya, yb, za, zb, dx, dy, dz, length, title), that plots the vector field for a vector-valued function, F, on the domain [xa, xb] X [ya, yb] X [za, zb] with step sizes dx, dy, and dz and a forced arrow length, length (set normalize=True). Recall that the tails of your vectors are the inputs to F and the tips of your vectors are the outputs from F. Be sure to set the aspect ratio of the plot to "equal" and to set the display bounds to [xa - dx, xb + dx] X [ya - dy, yb + dy] X [za - dz, zb + dz] using plt.axis([xa - dx, xb + dx, ya - dy, yb + dy, za - dz, zb + dz]) and to set the title to title if provided. Use plt.subplot(111, projection="3d") to create a 3d plot in CodeBuddy. Do NOT change any other properties of the plot.
Task 3b¶
Use your code from the previous exercise to visualize F = lambda x: np.array([1, 2, 3]) on [-1,1] X [-1,1] X [-1, 1] with step sizes of 0.4, 0.4, and 0.4 and an arrow length of 0.2.
Task 3c¶
Use your code from the previous exercise to visualize F = lambda x: np.array([np.ones_like(x[0]), 2 * np.ones_like(x[1]), x[2]]) on [-1,1] X [-1,1] X [-1, 1] with step sizes of 0.4, 0.4, and 0.4 and an arrow length of 0.2.
Task 3d¶
Use your code from the previous exercise to visualize F = lambda x: x on [-1,1] X [-1,1] X [-1, 1] with step sizes of 0.4, 0.4, and 0.4 and an arrow length of 0.2.
Task 3e¶
Use your code from the previous exercise to visualize F = lambda x: np.array([np.sin(x[1]), x[0] * np.cos(x[1]) + np.cos(x[2]), -x[1] * np.sin(x[2])]) with step sizes of 0.4, 0.4, and 0.4 and an arrow length of 0.2.