Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T23:08:23.936Z Has data issue: false hasContentIssue false

2 - Differentiation

Published online by Cambridge University Press:  11 January 2010

J. J. Duistermaat
Affiliation:
Universiteit Utrecht, The Netherlands
J. A. C. Kolk
Affiliation:
Universiteit Utrecht, The Netherlands
Get access

Summary

Locally, in shrinking neighborhoods of a point, a differentiable mapping can be approximated by an aftine mapping in such a way that the difference between the two mappings vanishes faster than the distance to the point in question. This condition forces the graphs of the original mapping and of the aftine mapping to be tangent at their point of intersection. The behavior of a differentiable mapping is substantially better than that of a merely continuous mapping.

At this stage linear algebra starts to play an important role. We reformulate the definition of differentiability so that our earlier results on continuity can be used to the full, thereby minimizing the need for additional estimates. The relationship between being differentiable in this sense and possessing derivatives with respect to each of the variables individually is discussed next. We develop rules for computing derivatives; among these, the chain rule for the derivative of a composition, in particular, has many applications in subsequent parts of the book. A differentiable function has good properties as long as its derivative does not vanish; critical points, where the derivative vanishes, therefore deserve to be studied in greater detail. Higher-order derivatives and Taylor's formula come next, because of their role in determining the behavior of functions near critical points, as well as in many other applications. The chapter closes with a study of the interaction between the operations of taking a limit, of differentiation and of integration; in particular, we investigate conditions under which they may be interchanged.

Linear mappings

We begin by fixing our notation for linear mappings and matrices because no universally accepted conventions exist.

Type
Chapter
Information
Multidimensional Real Analysis I
Differentiation
, pp. 37 - 86
Publisher: Cambridge University Press
Print publication year: 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×