14.1 - Multivariable functions

Definition: Function of 2 Variables f(x,y)

• $f(x,y)$ assigns every ordered pair $(x,y)$ in the domain $\mathcal{D}$ of f

• A unique number $z=f(x,y)$ in the range of $f$

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An Example Multivariable Function

Volume of a Cylinder:

$v(h,r)=\pi r^{2}h$

Filled out:

$v(2,3)=\pi 3^{2}2=18\pi$

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Contour Maps

Simple Python MatplotLib contour map:

Python

import micropip await micropip.install("numpy") await micropip.install("matplotlib") import matplotlib.pyplot as plt import numpy as np from mpl_toolkits.mplot3d import Axes3D # Create grid x = np.linspace(-5, 5, 100) y = np.linspace(-5, 5, 100) x, y = np.meshgrid(x, y) # Bowl shape (z = x^2 + y^2) z = x**2 + y**2 # Create figure fig = plt.figure() ax = fig.add_subplot(111, projection="3d") # Plot contour map contour = ax.contour(x, y, z, levels=[5, 10, 15, 20, 25], cmap="viridis", extend3d=True) ax.clabel(contour, inline=True, fontsize=10) # Set labels ax.set_xlabel("X") ax.set_ylabel("Y") ax.set_zlabel("Z") # Add title ax.set_title("3D Contour Map of Bowl Shape") # Show plot plt.show()

Output

Now, projecting this onto a 2d plane:

Python

import micropip await micropip.install("numpy") await micropip.install("matplotlib") import matplotlib.pyplot as plt import numpy as np # Create grid x = np.linspace(-5, 5, 100) y = np.linspace(-5, 5, 100) x, y = np.meshgrid(x, y) # Bowl shape (z = x^2 + y^2) z = x**2 + y**2 # Create figure fig, ax = plt.subplots() # Plot 2D contour map with specified levels contour = ax.contour(x, y, z, levels=[5, 10, 15, 20, 25], cmap='viridis') # Label contours ax.clabel(contour, inline=True, fontsize=10) # Set labels ax.set_xlabel('X') ax.set_ylabel('Y') # Add title ax.set_title('2D Contour Map of Bowl Shape') # Show plot plt.show()

Output

You might notice that the “tighter” the contours, the “steeper” the surface.

How do you go back and forth between the projection and the 3d graph?

2D Vs. 3D Contour Maps: Projection and Lifting

In a 3D contour map, we’re “lifting” the contours off the surface, giving each curve a specific height (or z value). Imagine slowly pulling up a flexible surface to see its shape in space. Each contour line represents a different height level, like a slice of the shape.

When we “squish” this back down into a 2D contour map, we flatten the surface while still showing where those height changes occur. The heights aren’t lost, but represented through lines on the flat plane, much like a topographical map of terrain. Each line corresponds to a different elevation, giving us a view of the shape without the 3D depth.

NOTE

This is similar to how topographical maps show mountains and valleys. Instead of showing peaks, we use contour lines to indicate changes in elevation.

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Example

Let’s make a contour map for the following equation:

$\frac{x^2}{4}-\frac{y^2}{9}=z$

In the plane $z=1$:

$\frac{x^2}{4}-\frac{y^2}{9}=1$

Plotting this with MatPlotLib yields:

Python

import micropip await micropip.install("numpy") await micropip.install("matplotlib") import numpy as np import matplotlib.pyplot as plt # Create grid x = np.linspace(-10, 10, 400) y = np.linspace(-10, 10, 400) x, y = np.meshgrid(x, y) # Equation for z = 1: x^2 / 4 - y^2 / 9 = 1 z = (x**2 / 4) - (y**2 / 9) # Create figure fig, ax = plt.subplots() # Plot the contour where z = 1 contour = ax.contour(x, y, z, levels=[1], colors="blue") # plot the contour where z = 2 contour = ax.contour(x, y, z, levels=[2], colors="red") # Plot the contour where z = 3 contour = ax.contour(x, y, z, levels=[3], colors="green") # Label contours ax.clabel(contour, inline=True, fontsize=10) # Set labels ax.set_xlabel("X") ax.set_ylabel("Y") # Add title ax.set_title(r"Contour map for $\frac{x^2}{4} - \frac{y^2}{9} = 1$") # Show plot plt.show()

Output

Contour Density and Steepness

In a contour map, the spacing between lines reveals the steepness of the surface. When the contours are far apart, the elevation is changing gradually. But when the contours get closer together, the surface becomes steeper, showing a rapid change in elevation.

In the case of the hyperbolic contour map for $\frac{x^2}{4}-\frac{y^2}{9}=1$, as the contours tighten around certain regions, it indicates that the surface is becoming much steeper there. This is similar to how, on a topographical map, closely packed lines represent a sharp incline on a mountain.

NOTE

The tighter the contours, the faster the change in height, just like on steep slopes in nature!

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