Section 11.1 Calculus of Several Variables

For much of your study of algebra, precalculus, and calculus you have been focused on working with functions that have a scalar input and a scalar output, like \(y=f(x)\text{.}\) In Chapter 10, we used vector-valued functions of one variable as our first case of multivariable functions. Vector-valued functions of one variable have a single scalar input and an output corresponding to multiple variables that we either organized as a vector output such as \(\vr(t)=\langle x(t),y(t),z(t) \rangle\) or as parametric functions of the form (\(x(t)\text{,}\) \(y(t)\text{,}\) and \(z(t)\)) . Our work in Chapter 10 focused on paths in space and the motion of an object on these paths. This narrow focus was because we had just one direction to move while staying on the path. We worked with vector-valued functions of one variable as our first new class of functions because the calculus of these object was relatively easy. We applied limits, derivatives, and integrals to these functions componentwise, but we saw how useful a combination of vector tools and calculus measurements was for describing many features of vector-valued functions of one variable and their graphs as paths in space.

A wide range of theoretical and applied problems involve a larger space of inputs and outputs. For instance, when studying weather patterns and behavior it is useful to measure temperature or atmospheric pressure. Both temperature and pressure are scalar measurements because they are measured by a single number. However, these measurements vary over three dimensions (location in terms of north/south, east/west, and elevation). Temperature can be given by a function with a location in three dimensions as the input and the temperature at that location, a scalar, as an output.

Wind direction and strength are also very important when working with weather patterns. Because it has both magnitude and direction, wind is measured with a vector that varies by location in a three-dimensional space. Therefore, wind would be given by a function that takes a location in space as its input and outputs a vector. We would call both the temperature and the wind functions functions of several variables or multivariable functions because each of these functions has multiple scalar inputs. We will look at the calculus of multivariable functions with scalar outputs such as temperature and pressure in this chapter and the next. The final chapter 13 of the book studies functions with multivariable inputs and outputs.

In Chapter 10, all of the applications and analogies were centered around an object moving along a curve in space. When working with functions of several variables, we will want to use several different types of functions for our applications and examples because of the possible conceptual relationships between inputs and outputs. For instance, the first type of example we will consider is thinking of the elevation or height of land as being a function of map coordinates. In this case, the input and output have the same units and are connected to make the surface. In other words the input and output are each the same type of measurement and would be easily understood by the plot of a surface.

The example of temperature as a multivariable function as described above gives a different kind of interpretation, one in which the output is a different kind of measurement than the inputs. For instance, think of a function whose input is the \((x,y)\) location on a heated metal plate and the output is the temperature of the metal plate at this location. In this case, the input is thought of as a point or location and the output is a measure of thermal energy at that location. The inputs and outputs are different kinds of measurements, but a plot of the different temperature values across the different \((x,y)\) locations would still visually make sense.

We can also consider a function that describes the horizontal distance traveled by an arrow before the arrow hits the ground as determined by the angle at which the arrow is shot and the initial speed of the arrow. In this case, the inputs to our function are an angle and a speed, while the output is a distance representing horizontal displacement. The units and types of measurements for the inputs and outputs are all different, and a plot of the corresponding surface of displacement, angle, and speed values would be a more abstract visual representation than in the temperature and elevation examples.

In the next few chapters, we will use these types of examples, as well as applications to economics, to motivate our study of functions of several variables.