Organization

• Narrative (Mathematics in Context) <--> Mathematical Infrastructure

• Ideals Meet Reality

Narrative (Mathematics in Context) <--> Mathematical Infrastructure

This material is organized around the interplay between mathematics and its applications. There are two sides to this interplay.

• Narrative (Mathematics in Context)

• Mathematical Infrastructure

Narrative (Mathematics in Context)

Our basic philosophy is that mathematics should be learned in the context of real, significant and engaging applications for four reasons.

• Mathematical ideas are more meaningful and more easily understood in the context of the real and tangible things they were developed to study.

• Mathematics is most frequently used to study real things and the best way to learn how to apply mathematics in real situations is by using it in real situations.

• We learn more about our world by using mathematics.

• By learning new mathematical ideas and techniques "just-in-time" we are able to see immediately how they are used and why they are important. In addition, new ideas and techniques are immediately reinforced by use.

For these reasons the main narrative thread of this material is mathematics in context.

Mathematical Infrastructure

The more traditional study of mathematics is organized around the mathematical structure of the material. This same structure underlies much of the main narrative -- although it appears to be "application-driven," the applications were carefully chosen to build up a coherent mathematical infrastructure. This mathematical infrastructure is central for four reasons.

• Mathematics is not just a series of disjoint tools, each developed to solve one particular problem -- it is an entire tool box and a way of thinking.

• The real power of mathematics comes from the way in which the same mathematical tools and ideas can be used in many different settings and the way in which mathematics enables us to make precise analogies among seemingly different situations.

• There is an important logical structure to mathematics. We build up the infrastructure step-by-step in a way that is logically well-founded.

• Because the same mathematical ideas come up in many different settings, it is more efficient to study each idea once in a general setting rather than over and over again in each individual setting.

For these reasons we will explicitly develop the mathematical infrastructure with very general ideas, tools, and even theorems. You will see the following icon throughout this material and in the navigation frame at the top of each module.

This button leads to the table of contents for the mathematical infrastructure side of this interplay.

More important than the two buttons and the two sides of this interplay are the large number of links between the two sides. As you work through the mathematics in context narrative, you will follow links to and build up the mathematical infrastructure side. If you look at a particular page on the mathematical infrastructure side you will see many links back to the mathematics in context side.

Ideals Meet Reality or A Funny Thing Happened on the Way to Press

Multivariable calculus, linear algebra, and differential equations are not three separate subjects. Prepublication reviewers of Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World were unanimous and enthusiastic in their belief that these subjects should be integrated into a single year-long course. Unfortunately, however, they were also unanimous in adding "but here at my university I won't be able to use this book because ..."

• "not every student takes all three courses."

• "we can't require our students to take a year-long course."

• "we only have one section of each course, so we can't offer options."

• "different majors require different combinations of these three courses at different times."

• "our students often interrupt their studies in midyear."

Whatever the perceived reasons there is clearly a problem. We believe that these three courses should be taught as a single subject and we have preserved this basic idea. We have adapted the structure and organization of the book and this web-based material, however, so that they can be used for three individual but highly connected courses and so that students can study Multivariable Calculus or Linear Algebra in either order and then take Differential Equations. The book has four parts -- the first part Prolog: Virtual Reality -- Mathematical Reality begins both Multivariable Calculus and Linear Algebra. It ties the two subjects together and lays the foundation for Differential Equations. Students who take both courses will find the prolog rich enough so that they will gain quite a bit the second time through. After either Multivariable Calculus or Linear Algebra students can take Differential Equations. The figure below illustrates this organization.

In addition to being tied together by the prolog, the three subjects have many links. Although we hope that many students will use the book and this Web material for a year-long integrated course, we believe that it is possible to use them for three separate courses as described above without losing the connections among these subjects.