In this recitation, we covered functions of several variables and planes.

Special Unit Vectors

Let \(\vec{f}(t)\) be a vector valued function. Recall that the unit tangent vector for \(\hat{f}(t)\) is the function

\[\vec{\tau}(t) := \frac{\vec{f}'(t)}{\mid \vec{f}(t) \mid}.\]

Since \(\mid \vec{\tau}(t) \mid = 1\), we know from earlier that its derivative will be orthogonal to it. The principal unit normal vector is defined by

\[\vec{n}(t) := \frac{\vec{\tau}'(t)}{\mid \vec{\tau}'(t) \mid}.\]

This is a vector which is orthogonal to the tangent vector and is unit.

Remark: Normal, perpendicular, and orthogonal all mean the same thing when it comes to vectors.

Functions of Several Variables

We’ve gone from scalar valued functions, i.e. functions of the form \(f : \mathbb{R} \rightarrow \mathbb{R}\), to vector valued functions, i.e. functions of the form \(f : \mathbb{R} \rightarrow \mathbb{R}^n\). The next step is to think about functions of the form \(f : \mathbb{R}^n \rightarrow \mathbb{R}\); we call these functions of several variables. These are functions which take ordered tuples of real numbers \((x_1, \ldots, x_n)\) and output a real number. You can also think of these as functions that take vectors and output real numbers by using the correspondence between points in space and vectors.

Let \(f : \mathbb{R}^n \rightarrow \mathbb{R}\) be a function of several variables (although technically the upcoming definition works for any kind of function). The preimage of a point \(t \in \mathbb{R}\) is the collection of all points \((x_1, \ldots, x_n)\) so that \(f(x_1, \ldots, x_n) = t\). We write this as

\[f^{-1}(t) = \{ (x_1, \ldots, x_n) \in \mathbb{R}^n \ | \ f(x_1, \ldots, x_n) = t\}.\]

This is also sometimes referred to as the level set (especially in the context of functions of several variables).

Warning: The \(-1\) is misleading – this is not an inverse. This works for functions that are not even invertible.

Example: Consider the function \(f(x) = x^2\). Then \(f^{-1}(1) = \{-1, 1\}\).

Observation: Recall that a function \(f\) is one-to-one if \(f(x) = f(y)\) implies \(x=y\). If a function is one-to-one, then the preimage of any point is at most one point. So \(f(x) = x^2\) is not one-to-one, since the preimage of \(1\) has two points.

Going back to functions of several variables, let’s consider the function

\[F : \mathbb{R}^2 \rightarrow \mathbb{R}, \ \ F(x,y) := x^2 + y^2.\]

Notice that for \(r > 0\) we have that the level set \(F^{-1}(r^2)\) is a circle of radius \(r\).

If the domain of a function of several variables is a subset of \(\mathbb{R}^2\), then we can visualize it by plotting it in three dimensions, with the third dimension being the output of the function. More precisely, we can look at the graph of the function of several variables by looking at

\[\{(x,y, F(x,y)) \in \mathbb{R}^3 \ | \ (x,y) \text{ are in the domain of } F\}.\]

Planes

We now have the ingredients to describe a plane in \(\mathbb{R}^3\). Given a vector \(\vec{v}\), we can define the function

\[F_{\vec{v}}(x,y,z) := \langle x,y,z \rangle \cdot \vec{v}.\]

This is a function of several variables, and the level set \(F_{\vec{v}}^{-1}(0)\) is the collection of all vectors orthogonal to \(\vec{v}\). Notice that this level set is free in two variables (meaning that once we know two variables, the other one is determined), so this determines a plane in \(\mathbb{R}^3\) which goes through the origin. Suppose we know that the point \(p = (p_1, p_2, p_3)\) lies on the plane. We wish to treat this like the origin \((0,0,0)\). We can consider the function \(\tau_p : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) given by

\[\tau(x,y,z) := (x-p_1, y-p_2, z-p_3).\]

This shifts everything so that \(p\) is now the origin. We can compose our functions now to get

\[F_{\vec{v}, p}(x,y,z) := F_{\vec{v}} \circ \tau_p(x,y,z).\]

By the description above, we see that \(F_{\vec{v}, p}^{-1}(0)\) determines a plane which goes through the point \(p\). We call \(\vec{v}\) the normal vector to the plane.

Given two vectors \(\vec{w}_1, \vec{w}_2\) such that \(F_{\vec{v}, p}(\vec{w}_1) = F_{\vec{v}, p}(\vec{w}_2) = 0\) and \(\vec{w}_1 \times \vec{w}_2 \neq \vec{0}\) (or, even simpler – \(\vec{w}_1\) and \(\vec{w}_2\) are not parallel), we can give a parameterization for the plane by

\[L(s,t) := \vec{p} + s \vec{w}_1 + t \vec{w}_2.\]