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

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.

Audience: 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.

Contents: Chapter 1: Introduction; Chapter 2: Surfaces and Differential Geometry; Chapter 3: The Fundamental Forms of Differential Geometry; Chapter 4: Calculus on Surfaces; Chapter 5: Shape Differential Calculus; Chapter 6: Applications; Chapter 7: Willmore Flow; Appendix A: Vectors and Matrices; Appendix B: Derivatives and Integrals