Why Multivariable?

In your earlier calculus courses, you have explored how to compute a limit, how to measure the rate of change of a function, and how to calculate the area under the graph of a function. You likely also spent several years before calculus getting used to working algebraically and geometrically with a variety of different kinds of functions like polynomials and rational polynomials, as well as sinusoidal, exponential, and logarithmic functions. Nearly all this study involved functions that have one number as an input and one number as an output. In your first semester calculus, you considered how small changes in the input variable would lead to changes in the output variable.

In your previous mathematics and science courses, you probably also saw a few places where the math you were studying was not general enough to describe some reasonable scenarios. For instance, the calculus tools you learned in your earlier courses will not apply to graphs, like a circle or a hyperbola, where \(y\neq f(x)\).

More generally, we will need to work with functions and expressions that depend on more than one input or involve outputs more complicated than a single scalar. For instance, when trying to find the optimal price for a product, you will need to consider fixed costs like a building lease, labor costs, and the varying costs of all the materials that are used in the product. Another example would be trying to measure the atmospheric temperature, which will vary over a three-dimensional space and will vary over time of day as well. If we wanted to use wind to help in modeling weather phenomena, we will need new tools since wind has both a direction and a strength associated with it, and thus cannot be measured by a single number. For each of the cases mentioned above, it is not entirely clear how to relate changes in the input variables to changes in the different kinds of outputs. A long-term goal of our work in this book will be to carefully define the relationship between changes of inputs and outputs for many new types of functions.

With our new expanded view of the world around us, we will also need to describe a variety of new examples of mathematical objects that are algebraically nice and exhibit different geometric features. Most students took several math classes to understand and apply to the calculus of functions with one-variable in and one-variable out; In this text, we will expand our tools in much less time because of the depth and variety of ideas that you have already encountered. Before we look at the calculus of these new kinds of functions, we will need to do a bit of precalculus again.