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

Authors: Shawn W. Walker



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.

Table of Contents
        
Front Matter
        
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

Back Matter