An Introduction to Differential Geometry with Applications to Elasticity

Focusing on the mathematical foundations of curvilinear coordinates, this work provides a direct proof of the three-dimensional Korn inequality and explores Koiter's equations for modeling thin elastic shells using two-dimensional curvilinear coordinates. The text establishes the existence, uniqueness, and regularity of solutions to these equations, leveraging a specialized Korn inequality and an infinitesimal rigid displacement lemma. Additionally, it introduces differential geometry concepts relevant to boundary value problems in elasticity and shell theory, with some material adapted from previous works.