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.
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
Shawn 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.
