Problem 51 - University Calculus: Early Transcendentals

Question

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. $$z=x^{2}+y^{2}, \quad-3 \leq x \leq 3, \quad-3 \leq y \leq 3$$

Step-by-Step Solution

Answer

Plot a 3D paraboloid \( z = x^2 + y^2 \) over \(-3 \leq x, y \leq 3\). Rotate for different views.

Step 1: Identify the Equation Type

The given equation is \( z = x^2 + y^2 \). This is a well-known equation for a paraboloid. Understanding the shape helps in predicting how it will look when plotted.

Step 2: Define the Domain

The given domain for both variables, \(x\) and \(y\), is \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\). This defines a square region in the \(xy\)-plane where the surface will be plotted.

Step 3: Set Up the Grid for Plotting

Create a grid of \(x\) and \(y\) values within the domain. For example, you can use a grid spacing of 0.1 to 0.5 to get a smooth surface. In Python, this can be done using the `numpy.meshgrid` function.

Step 4: Calculate \(z\) Values

For each pair of \((x, y)\) in the grid, calculate the corresponding \(z\) value using the equation \(z = x^2 + y^2\). This gives a matrix of \(z\) values that can be plotted to create the surface.

Step 5: Plot the Surface

Use a plotting library such as Matplotlib in Python. The function `plot_surface` in the `Axes3D` toolkit is useful for plotting 3D surfaces. It requires the grid of \(x\), \(y\), and calculated \(z\) values.

Step 6: Adjust Viewing Angle

To rotate and view the surface from different angles, modify the azimuth and elevation angles in the plot. This can be done using the `view_init` function in Matplotlib, setting the desired angles to view the paraboloid differently.

Key Concepts

Paraboloid

A paraboloid is a three-dimensional surface that has a distinct curved shape. It can be thought of as an extension of a parabola, which is a two-dimensional curve. In mathematical terms, a paraboloid is described by equations of the form \(z = x^2 + y^2\) or \(z = -x^2 - y^2\). The equation given in the exercise, \(z = x^2 + y^2\), represents a standard paraboloid. This type has a bowl-like shape that opens upwards.
Paraboloids have an axis of symmetry and are examples of quadratic surfaces. The vertices of these shapes are either the maximum or minimum points, depending on whether they face up or down. For an upward-facing paraboloid like \(z = x^2 + y^2\), the vertex, located at the origin, is the minimum point of the surface.
Understanding the nature of the paraboloid is crucial for visualizing and predicting its representation when plotted in 3D space, especially concerning its symmetry and openness.

Plotting Grids

Plotting grids are essential for mapping surfaces in three-dimensional space. The concept involves creating a mesh or grid of values over a defined domain, which facilitates the calculation of surface points. For the paraboloid equation \(z = x^2 + y^2\), the domain is given as \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\). This means you would want to create a grid covering these ranges.
To set up a grid:

• Divide the domain into evenly spaced intervals. Smaller intervals result in smoother surface representations.
• Utilize tools like `numpy.meshgrid` in Python to efficiently create arrays of \(x\) and \(y\) values that cover the entire square region.
• For each grid point, calculate the corresponding \(z\) value using the surface's equation. This produces a matrix of points that represent the surface in 3D space.

A properly defined grid ensures that the plotted surface accurately represents the geometric concept it aims to depict, allowing for better understanding and visualization.

3D Plotting with Matplotlib

3D plotting with Matplotlib allows us to visualize mathematical surfaces in a three-dimensional space. It renders the concept more tangible and easy to grasp through direct visualization.
Matplotlib's `plot_surface` method is particularly useful for plotting the surface points calculated from the paraboloid equation. This function requires three main inputs: the grid of \(x\), \(y\), and the computed \(z\) values. Setting up the plot involves:

• Importing `matplotlib.pyplot` and `mpl_toolkits.mplot3d` which provide the necessary functions for 3D plotting.
• Passing the grid matrices to the `plot_surface` function to visualize the surface.
• Utilizing `view_init` to change the viewing angles, allowing the surface to be seen from different perspectives. Adjusting the azimuth and elevation angles can provide a comprehensive view of the surface's structure.

Through these steps, Matplotlib helps present complex mathematical concepts in an accessible and engaging manner, aiding learners in developing an intuitive understanding of 3D geometry.