Integrating Multivariable Vector-Valued Functions

In , we looked at the calculus of functions with one scalar input and multiple outputs, usually organized as a vector. In we looked at measuring changes involving functions with multiple input variables and a single scalar output. In we looked at integration involving scalar valued multivariable functions. In this chapter, we will look at combining all of our multivariable function tools, as well as our vector tools, to look at the calculus of functions with multiple input variables and multiple output variables. This chapter is significantly longer than others in this text because there are many different measurements to investigate, as well as several fundamental thereoms used in applications of these new measurements.

The example of wind has been used several times in this text to describe a measurement that would require multiple inputs and outputs. In particular, wind will change over the points in some region (as well as changing over time). Additionally, wind requires a measurement of strength in each direction which is typically stored as a vector. Thus wind has multiple scalar inputs corresponding to the location of the measurement and multiple scalar outputs that measure the strength in coordinate directions.

Recognizing that not all institutions will cover all the material in the chapter on vector calculus, we have created the following chart for the dependencies of the topics in .