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The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative

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by Shawn W. Walker


2015 / x + 152 pages / Softcover / ISBN 978-1-611973-95-2 / List Price $74.00 / SIAM Member Price $51.80 / Order Code DC28

Keywords: differential geometry, shape derivative, shape optimization, gradient flow, surface PDEs


Many things around us have properties that depend on their shape—for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a “shape variable.” This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts.

Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included.

This book is a convenient reference for various shape derivative formulas and should be of value to anyone interested in surface geometry and shape optimization. Graduate students can use it to quickly get up to speed on the machinery of shape differential calculus. Scientists studying continuum mechanics, fluid mechanics, numerical analysis, and PDEs will find the book helpful for problems in which surface geometry is critical and/or geometry evolves in time. Those who want to learn the basics of shape differentiation will also find it useful.

About the Author
Shawn W. Walker is an assistant professor of mathematics at Louisiana State University (LSU), with a joint appointment in the Center for Computation and Technology (CCT). He held a postdoctoral position at the Courant Institute (New York University) and joined the LSU faculty in 2010 in the computational mathematics group. He is a member of SIAM, AMS, MRS, and APS. His research interests include PDEs for fluids and moving/free boundaries, geometric evolution problems, numerical analysis and finite element methods, mesh generation, and optimal PDE control of shape.

ISBN 9781611973952

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