Differentials and Geographical Maps (UMAP)

This module seeks to: 1) strengthen the students' intuition in multidimensional geometry; 2) consolidate their command of differentials; and 3) show important applications that rarely appear in calculus texts. It shows how to use differentials to determine whether a geographical map preserves angles, areas, or neither, and how to draw a loxodrome (a path along which the azimuth remains constant) between any two points on the surface of the Earth. The exercises are at a level intermediate between mechanical problems and abstract proofs and fit in any multivariable calculus course.

Table of Contents:

INTRODUCTION

DIFFERENTIALS OF MAPPINGS BETWEEN EUCLIDEAN SPACES

MAPS THAT PRESERVE ANGLES OR AREAS
Linear Transformations that Preserve Angles or Areas
Differentiable Maps that Preserve Angles or Areas
Changes of Coordinates
No Map Preserves All Distances

OTHER USES OF CONFORMAL MAPS

SAMPLE EXAM PROBLEMS

SOLUTIONS TO THE ODD-NUMBERED EXERCISES

SOLUTIONS TO THE SAMPLE EXAM

REFERENCES

ACKNOWLEDGMENTS

ABOUT THE AUTHOR