Higher Dimensions

This section expands on the idea of level curves from and the interpretations of the gradient from . This section can likely be covered in a single class or split up and incorporated into work for and . Much of this section is centered around describing the generalization of derivatives to functions of three or more variables because students will likely struggle with understanding much of the development of these ideas due to the lack of geometric tools. For this reason, the activities in this section are more limited in scope when compared to the rest of this text.

In this chapter, we have primarily worked with functions of two variables including understanding graphs of the form \(z=f(x,y)\) and measuring change with functions of two variables. In this section, we explore how many of the tools we have developed for understanding two-variable functions can be generalized to functions of three or more variables.

A significant impediment to expanding the ideas in this chapter to functions of three or more variables will be that is it difficult (or impossible) to draw plots of these functions, which means we must rely on simplifications or analogies to understand these measurements geometrically. For this reason, most efforts to understand functions of three or more variables is algebraically-focused.

We start in the Preview Activity by generalizing the idea of level curves.

In this section, we will often use the idea of level sets to explore functions of three or more variables. The topics we explore are not likely to surprise you: domain and range, graphs, limits, and rates of change.

A function is a rule that assigns an single output for each allowed input. For instance, your school likely uses a function that takes your student ID number as an input and outputs your name. While you may not be the only student at your school with your name, this function should not associate more than one name with each student ID number. A function of \(n\) variables can be thought of as taking points in \(\R^n\) as input and outputs a real number for each valid input.

Recall from that the domain of a function is the set of input values for which a function is defined. The range of a function is the set of values actually output by the function.

Given a function \(f\) of several variables the level set of \(f\) at the value \(k\) is the collection of input points \(P\) for which \(f(P)=k\). A level set is a subset of the domain of the function \(f\) because the level set is a collection of input points.

If \(f(x,y)\) is a function of two variables, then a level set is a contour or a level curve of the form \(f(x,y)=k\). Notice that the level set in this case is a set of points in the \(xy\)-plane. As we saw in , we can make a contour plot by graphing a collection of these level curves on the same two-dimensional plot, which will give us an idea of what the surface corresponding to \(z=f(x,y)\) looks like. In other words, we were able to give enough information in a two-dimensional plot to describe a three-dimensional surface.

If \(g(x,y,z)\) is a function of three variables, then a graph of \(g\) function would require four dimensions: three dimensions for the inputs and one dimension for the output. Drawing plots in three dimensions is challenging, but we will use three-dimensional graphs to understand functions of three variables rather than attempting to visualize four-dimensional graphs directly. We will do this by generalizing our approach of using contour plots to express a three-dimensional plot in a two-dimensional setting. Specifically, we will use level sets of \(g(x,y,z)\) to understand a graph of \(w=g(x,y,z)\) and help measure change in the output of \(g\).

A level set of \(g\) is a collection of points in \(xyz\)-space that corresponds to a surface of the form \(g(x,y,z)=k\). As you saw in , the level sets of a function of three variables are often surfaces, which we call level surfaces. For example, if \(g(x,y,z)=x^2+y^2+z^2\), then the level surface corresponding to the value \(1\) is all points that satisfy the equation \(x^2+y^2+z^2=1\). This set of points forms the sphere of radius \(1\) centered at the origin.

These level surfaces will often be a different kind of surface than we have been working with throughout . Surfaces of the form \(z=f(x,y)\) are called explicit surfaces because one of the coordinates can be solved explicitly as a function of the other coordinates. Surfaces such as spheres or hyperboloids are not explicit surfaces because there is no way to solve for one variable as a function of the other coordinates. Two perspectives from which you can recognize that are first that these surfaces fail the vertical line test in \(\R^3\) and algebraically, the \(\pm\) that comes from needing to take square roots prevents this for many quadric surfaces.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Conceptually, limits for functions of three or more variables work the same as for functions of two variables. Given a function \(f\), we say that \(f\) has limit \(L\) as the inputs approach \(P_0\) provided that we can make \(f(P)\) as close to \(L\) as we like by taking \(P\) sufficiently close (but not equal) to \(P_0\). We write \[\begin{aligned}\end{aligned}\]. If the limit of a function \(f\) along every path through an input point \(P_0\) exists and all of those limits are the same value \(L\) then we say the limit of \(f\) at \(P\) is \(L\). There is not much more insight into limits of multivariable functions to be had at this point, so we will move on to measuring the change in output of our multivariable functions of three or more variables.

The partial derivative of a multivariable function measures the rate of change in the output of the function when one variable is changed and all others are held constant. Our definition and notation of partial derivatives given for functions of two variables only needs to be updated to account for three or more input variables.

also generalizes to higher dimensions: if all mixed partials are continuous near an input point of a function of three or more variables, then the mixed partials at that point are equal.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Directional derivatives and gradients for functions of three or more variables are critical concepts, both in this course and in future coursework in math, economics, and the physical sciences. Recall that the directional derivative measures the instantaneous rate of change of a multivariable function when the inputs are changed in a particular direction. None of the arguments in were specific to functions of two variables, but rather than stating these results in terms of an abstract function of \(n\) variables, we will illustrate the various definitions and results in terms of functions of three or four variables: \(f(x,y,z)\) or \(g(x,y,z,w)\).

Let \(f = f(x,y,z)\) be a function of three variables. The derivative of \(f\) at the point \((x,y,z)\) in the direction of the unit vector \(\vu = \langle u_1, u_2 , u_3 \rangle\) is denoted \(D_{\vu}f(x,y,z)\) and is given by \[\begin{aligned}\end{aligned}\] for those values of \(x\), \(y\), and \(z\) for which the limit exists. We can make a similar limit definition for a function of more than three variables because we are able to separate the length of the step in a particular direction (\(t\) in the above statement) and the unit vector in that particular direction (\(\vu\) from above) for vectors with any number of components.

We can calculate the directional derivative in terms of partial derivatives of the function and \(\vu\). This result comes from using the chain rule on a composition of the multivariable function with the line in the direction of \(\vu\). If \(f(x,y,z)\) and \(g(x,y,z,w)\) are functions of three and four variables, respectively, then \[\begin{aligned}\end{aligned}\] and \[\begin{aligned}D_{\vu} g(x,y,z,w)= g_x(x,y,z,w) u_1 \amp+ g_y(x,y,z,w) u_2 \\ \amp +g_z(x,y,z,w) u_3+ g_w(x,y,z,w) u_4\end{aligned}\]. Remember that the direction vector will have as many components as there are inputs to the function because the direction vector corresponds to a change in the inputs of the function.

The formulas above have the same form of a dot product of the gradient and the direction vector, so we more compactly write \[\begin{aligned}\end{aligned}\] and \[\begin{aligned}\end{aligned}\] where \(\nabla f =\langle f_x(x,y,z), f_y(x,y,z) , f_z(x,y,z)\rangle\) and \[\begin{aligned}\end{aligned}\]. This generalization shows that in any dimension, the directional derivative can be calculated as the dot product of the gradient and the direction vector.

This also means that all of our work to understand the meaning of the gradient will generalize to any dimension as well. In particular, we update our summary of the meaning of the gradient below.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.