Active Calculus - Multivariable

In single variable calculus, we learn that a differentiable function is locally linear. In other words, if we zoom in on the graph of a differentiable function around a point, the graph will look like the tangent line to the function at that point. In Figure 9.5.1 you can look at the graph of a curve and zoom in around a point on that curve to see that eventually the graph of the function will look linear. You can use the sliders at the top of change how zoomed in the plot is displayed or change the location on the curve that you are examining. You should visually verify at a several points that this function is differentiable because at any point shown, the graph is locally linear.

Recall that linear functions have the form \(f(x)=ax+b\text{.}\) These functions play important roles in single variable calculus and are useful in approximating differentiable functions, in approximating roots of functions (Newton’s Method), and approximating solutions to first order differential equations (Euler’s Method). In multivariable calculus, we will study curves in space and we will see that connection between local linearity and differentiability exists for curves in three (or more) dimensions. As we extend our study of functions to include those of two variables in Chapter 11, we will see that a function is locally linear at a point if the surface defined by the function looks like a plane (the tangent plane) as we zoom in on the graph.

Because of their important role in understanding curves and surfaces in \(\R^3\text{,}\) it is important for us to understand both lines and planes in space, as can both be thought of as generalizations of lines in two dimensions to three-dimensional space. In your study of one-variable functions, you probably worked with slope, direction, and measuring change using lines several times before you generalized those ideas for a curve in general. Similarly, we will want to understand how we talk about direction and measuring change along flat objects in \(\R^3\) (lines and planes) before we start generalizing to curved objects. In this section, we discuss how we can use vectors to describe lines in space. The next section will be dedicated to understanding planes.

The way most people draw a line is more connected to the ideas in Preview Activity 9.5.1 than slope or algebraic forms. Take a minute and draw a plot of the line described by \(x+y-1=0\text{.}\) You should not continue reading until you have made a plot (by hand) of \(x+y-1=0\) that you would be proud to share with classmates.

Figure 9.5.7 shows the plot of a line in two-space. The vector \(\overrightarrow{OP}\) is illustrated in blue and points from the origin to our starting point \(P\text{.}\) The direction vector \(t \vv\) as in Equation (9.5.1) is shown in red with its initial point at point \(P\text{.}\) The line is traced out by the terminal points of the green vector \(\overrightarrow{OP}+t\vv\text{.}\) In other words, the tips (terminal points) of the green vectors trace out the line as \(t\) changes. You can use the slider to change the value of \(t\) for this example line. You should pay attention to how \(\overrightarrow{OP} + \vv t\) changes as you vary \(t\text{.}\) Be sure to see what happens with \(t=0\) as well as for some negative values of \(t\text{.}\)

To summarize, the terminal points of the vectors of the form in (9.5.1) define a linear function \(\vr\) in space of the following form, which is valid in any dimension and gives some geometric insight into lines in higher dimensions.

We formalize the discussion in the last part of the previous activity by giving names to the two nonintersecting configurations that lines in \(\R^n\) for \(n\geq 3\) can have.

Notice that there are many different parametric equations for the same line. For example, choosing another point \(P\) on the line or another (parallel) direction vector \(\vv\) produces another set of parametric equations. In many physical applications, it is useful to think of \(t\) as a time parameter and the parametric equations as telling us where we are on the line at each time. In this way, the parametric equations describe a particular way to walk along the line. There are, of course, many possible ways to walk along the same line, including in opposite directions on at different speeds. In mathematical terms, we say that the parameterization of a line is not unique.

Before we move on to our discussion of planes in the next section, we will talk about some terminology that may helpful in describing features and measurements later. A line is a one-dimensional, flat graph in three (or \(n\)) dimensions. We say the line is a one-dimensional graph because there is only one direction (forward/backward) to go while staying on the graph. We say that a line is flat because the direction you are allowed to go doesn’t change as you move along the graph. Note that flat does not mean horizontal. A circle would be another example of a one-dimensional graph that can be drawn in two or more dimensional space (because there is only one direction to move along the circle). However, a circle is not flat since the direction you go to stay on the figure changes as you move along the graph. As we will see in the next section, a plane in three dimensional space is an example of a flat, two-dimensional graph. If you were an ant on a plane, you would have two directions (forward/backward or left/right) to move in while staying on the graph. However, the directions that the graph extends do not change when you change locations on the plane.