calculus book - Mathematics and Computer Science

PREFACE xi
To make one last analogy, think of mathematics as terrain. Some
areas are flat and grassy, good for a pleasant stroll; others are hilly,
but with well-worn trails. There are cliffs scaled by narrow, difficult
paths that require ropes, and ravines where it is easy to see how to
cross, but difficult to scramble through the undergrowth. Here and
there, mountain peaks climb far above the plains. The views from the
top are stunning, allowing you to see wide vistas and their connections,
but are achieved only by a long, difficult climb. Still, the trails have
been blazed, and steady work will bring you to the summit. The book
is your tour guide, showing you interesting trails and geographical features
in the part of the map called “real analysis”. The skills you learn
will accumulate, allowing you to tackle more difficult trails. Along the
way, there are springs, waterfalls, and wildflowers as rewards for steady
progress. All the places on the itinerary are well-mapped out, but occasionally
we will pass a cliff that no one has yet scaled, and a few times
you will glimpse the vast, uncharted hinterlands where current research
occurs.