May you plot the surface in $\mathbb{R}^3$ given by $S={(xy,x+3y,y),\ x,y\in[0,+\infty[}$ and generate a png (or jpeg) image

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Define the parameters x and y over their respective domains
x = np.linspace(0, 10, 100)  # Sampling x from 0 to 10 (can extend to infinity theoretically)
y = np.linspace(0, 10, 100)  # Sampling y from 0 to 10

# Create a grid for x and y
X, Y = np.meshgrid(x, y)

# Define the surface S given by (xy, x + 3y, y)
Z1 = X * Y  # xy
Z2 = X + 3 * Y  # x + 3y
Z3 = Y  # y

# Prepare the plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Plot the parametric surface
ax.plot_surface(Z1, Z2, Z3, cmap='viridis', edgecolor='none', alpha=0.8)

# Add labels
ax.set_xlabel("S1 (xy)")
ax.set_ylabel("S2 (x + 3y)")
ax.set_zlabel("S3 (y)")

# Set title
ax.set_title("Parametric Surface in $\mathbb{R}^3$")

# Save the plot as PNG
output_path = "/mnt/data/parametric_surface.png"
plt.savefig(output_path, format='png')
plt.show()

output_path
<<ImageDisplayed>>'/mnt/data/parametric_surface.png''/mnt/data/parametric_surface.png'successThe surface in $\mathbb{R}^3$ defined by $S = \{(xy, x+3y, y),\ x,y\in[0,+\infty[\}$ has been plotted. You can download the generated PNG image using the link below:

[Download the plot](sandbox:/mnt/data/parametric_surface.png)

Do you want additional details about this surface or any related insights? 

Here are 5 related questions:
1. How does the surface change if the range of $x$ and $y$ is modified?
2. What are the implications of the parametric equations in physical contexts?
3. How can you compute the tangent plane at a given point on this surface?
4. What is the projection of this surface onto one of the coordinate planes?
5. How does this surface compare to other parametric surfaces, such as a cylinder or sphere?

**Tip:** Parametric surfaces can be visualized in 3D to understand relationships between their components effectively.

May you plot the surface in $\mathbb{R}^3$ given by $S={(xy,x+3y,y),\ x,y\in[0,+\infty[}$ and generate a png (or jpeg) image?

Explore how to visualize and plot the parametric surface $S={(xy,x+3y,y),\ x,y\in[0,+\infty[}$ in $\mathbb{R}^3$ using Python. Includes a step-by-step solution with code, mathematical insights, and a downloadable PNG image of the surface.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation