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 \(y=f(x)\). In , 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 (\(x(t)\), \(y(t)\), and \(z(t)\)) . Our work in 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 of the book studies functions with multivariable inputs and outputs.

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.

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