• James J. CallahanEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


Approximations are at the heart of calculus. In Chapter 1 we saw that the transformation of differentials dx = φ'(s)ds can be traced back to the linear approximation Δx ≈ φ'(s)Δs (the microscope equation), and that the factor φ'(s) represented a local lengthmultiplier.We also suggested there that the transformationdxdy = rdrdθ of differentialsfrom Cartesian to polar coordinates has the same explanation: the polar coordinate change map has a linearapproximation (a twovariable “microscope” equation) and the factor r is the local area multiplier for that map. In this chapter we construct a variety of useful approximations to nonlinear functions of one or more variables. However, we save for the following chapter a discussion of the most important approximation, the derivative of a map.


Homogeneous Polynomial Remainder Function Taylor Polynomial Binomial Expansion Microscope Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSmith CollegeNorthamptonUSA

Personalised recommendations