Objectives
- Create a vector valued function representing a section or contour for a multivariable function
- Understand the relationship between a contour plot and a multivariable function
Section 6.3 Sections on MVF and Contour Plots
Subsection 6.3.1 Sections on Graphs of Multivariable Functions
Activity 6.3.1. Using Sections to Draw MVF graphs.
Let \(f(x,y)=yx^2\) with domain restricted to \(\{(x,y)|-1\geq x \geq 1, -1\geq y \geq 1\}\text{.}\) We will use the steps below to try to make a good plot of \(z=f(x,y)\text{.}\)
(a)
Draw the section corresponding to \(y=-1\) on the image provided above.
(b)
Draw the section corresponding to \(y=1\) on the image provided above.
(c)
Draw the section corresponding to \(x=-1\) on the image provided above.
(d)
Draw the section corresponding to \(x=1\) on the image provided above.
(e)
Use other sections as appropriate to complete the graph of \(z=f(x,y)\text{.}\)
Subsection 6.3.2 Contours and Level Curves
We have explored means for studying multivariable functions using contour plots. To find a specific contour for a function \(z=f(x,y)\) we substitute a value \(c\) for \(z\) and find the resulting two dimensional graph of
\begin{equation*}
c=f(x,y)\text{.}
\end{equation*}
In this way, the value of \(c\) can be thought of as the elevation we are interested index and the graph of \(c=f(x,y)\) represents multivariable surface at that elevation. Often times it is useful to describe these graphs with one or more parametric equations.
Activity 6.3.2.
Consider the ellipsoid given by
\begin{equation}
z=4x^2 + y^2.\tag{6.3.1}
\end{equation}
(a)
Form implicit equation for the contours at \(c=0,1,2,4\) and graph the equations.
(b)
Find parametric equations \((x(t), y(t))\) for the contours from the previous part. You may want to refer to Section 1.3 if you need assistance.
(c)
Confirm the parametric and implicit equations agree by using the tool below
(d)
Now we would like to move our contours onto the graph of (6.3.1). Form parametric equations in three dimensions using \(x(t), y(t)\) along with
\begin{equation*}
z(t) = c\text{.}
\end{equation*}
Add your graphs to the ellipsoid below.
Activity 6.3.3. Graphs and Contours.
(a)
Give enough values to either \(x\) or \(y\) until you can imagine how the graph of \(z=f(x,y)\) looks in space. Draw the graph as carefully as you can and include a verbal description for your surface.
- \(\displaystyle f(x,y)=x-y^2\)
- \(\displaystyle f(x,y)=(x+y)^2\)
- \(\displaystyle f(x,y)=\sqrt{x-y}\)
- \(\displaystyle f(x,y)=\sqrt{y-x^2}\)
(b)
Draw a contour diagram for the function \(z=f(x,y)\) as carefully as you can and verify that your diagram is consistent with the graph you drew in the previous part. You must draw enough contours so that the behavior of the function in all of its domain may be deduced. Show all your work.
- \(\displaystyle f(x,y)=x-y^2\)
- \(\displaystyle f(x,y)=(x+y)^2\)
- \(\displaystyle f(x,y)=\sqrt{x-y}\)
- \(\displaystyle f(x,y)=\sqrt{y-x^2}\)
Activity 6.3.4.
The following is a contour diagram of a plane.
(a)
Find \(m_x\text{.}\) (Remember that the plane is in 3D and that therefore both \(x\) and \(y\) refer to “horizontal”, while “vertical” refers to \(z\)).
(b)
Find \(m_y\text{.}\) (Remember that the plane is in 3D and that therefore both \(x\) and \(y\) refer to “horizontal”, while “vertical” refers to \(z\)).
(c)
Express vertical change \(dz\) as a function of the horizontal change in the \(x\) direction and the horizontal change in the \(y\) direction (that is, in terms of \(dx\) and \(dy\)).
Exercises 6.3.3 Exercises
For the following sketch a contour map of the surface using level curves for the given \(c\)-values.
1.
\(f(x,y) = 6-2x-3y, \qquad c=0,2,4,6\)
2.
\(f(x,y) = \sqrt{9-x^2-y^2}, \qquad c=0,1,2,3\)
3.
\(f(x,y) = 2^{1-x^2-y^2}, \qquad c=1,2,4,8\)
Hint.
Take the logarithm of both sides.
