The equation $y^2 + z^2 = 1$ represents a cylinder in 3D space. The cross-section in the $yz$-plane is a circle with radius 1, and the cylinder extends infinitely along the $x$-axis. To plot this in 3D, you need to parametrize the circular cross-section and sweep it along the $x$-axis.

Steps to plot $y^2 + z^2 = 1$ in 3D:

1. Parametrize the equation:

   • In cylindrical coordinates, $y$ and $z$ form a circle of radius 1. So, you can parametrize it as: $y = \cos(\theta), \quad z = \sin(\theta)$ where $\theta$ runs from $0$ to $2\pi$.

2. Extend the circle along the $x$-axis:

   • Since $y^2 + z^2 = 1$ holds for all $x$, $x$ can take any value.

3. Plotting:

   • Use a 3D plotting library (like Python's matplotlib) to create a 3D plot.

Here’s some Python code using matplotlib to visualize this:

pythonCopy code
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Create the theta values
theta = np.linspace(0, 2 * np.pi, 100)

# Parametric equations for y and z
y = np.cos(theta)
z = np.sin(theta)

# Create a range of x values
x = np.linspace(-5, 5, 100)  # Example x values from -5 to 5

# Create the 3D plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Plot the surface by repeating the circle for different x values
for xi in x:
    ax.plot(np.full_like(theta, xi), y, z, color='b')

# Set labels
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')

plt.show()

This will generate a 3D plot of a cylinder where $y^2 + z^2 = 1$, and it stretches along the $x$-axis.

Would you like a deeper explanation of any step, or details on how this works?

Related Questions:

1. How can this cylindrical equation be converted to spherical coordinates?
2. What does the surface look like for other values of constants, such as $y^2 + z^2 = r^2$?
3. How would the plot differ if we restricted the range of $x$?
4. How would this change if we added an $x$-dependent term, like $y^2 + z^2 = x$?
5. How can you visualize this cylinder using software other than Python?

Tip:

In 3D plotting, varying the resolution (like increasing the number of $\theta$ or $x$ values) can give you a smoother surface, improving the visual quality of your plots.