Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-20T11:46:37.873Z Has data issue: false hasContentIssue false

5 - Applications

Published online by Cambridge University Press:  20 August 2009

Tobin A. Driscoll
Affiliation:
University of Delaware
Lloyd N. Trefethen
Affiliation:
University of Oxford
Get access

Summary

Conformal mapping in general, and Schwarz–Christoffel mapping in particular, are fascinating and beautiful subjects in their own rights. Nevertheless, the history of conformal mapping is driven largely by applications, so it is appropriate to consider when and how SC mapping can be used in practical problems.

It is not our intent in this final chapter to recount every instance in which Schwarz–Christoffel mapping has been brought to bear. Rather, after a brief look at a few areas full of such examples, we describe some situations in which SC ideas can be applied in ways that are computational and perhaps not transparent. The most famous application of conformal mapping is to Laplace's equation, and we devote three sections to it. Beyond this it is clear that Schwarz–Christoffel mapping has a small but important niche in applied mathematics and science.

In applications it is common to pose a physical problem in the z-plane, which maps to a canonical region in the w-plane. This convention runs counter to our discussion in the earlier chapters, in which w was the plane of the polygon. In the following sections we attempt to be consistent with established applications literature where appropriate.

Why use Schwarz–Christoffel maps?

Schwarz–Christoffel mapping is an incomparably effective tool for a very specific sort of problem. The most natural and satisfying application is the solution of Laplace's equation in the plane with piecewise constant (and in the case of derivative conditions, homogeneous) boundary conditions.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2002

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
×