This chapter provides an overview of the mathematical theory of elasticity, focusing on fundamental equations, plane theory, and torsion and bending. The book, authored by N. I. Muskhelishvili, is a comprehensive treatise on the subject, covering various aspects of elasticity theory and its applications. The content is divided into several parts, including: 1. **Fundamental Equations**: Discusses the basic principles of elasticity, such as stress analysis, strain analysis, and the fundamental law of elasticity. 2. **Plane Theory**: Introduces the plane theory of elasticity, including the stress function, complex representation, and solutions for various plane problems. 3. **Torsion and Bending**: Explores the torsion and bending of homogeneous and compound bars, providing detailed solutions and applications. The book is known for its rigorous mathematical approach and practical examples, making it a valuable resource for both researchers and engineers in the field of elasticity. The author, N. I. Muskhelishvili, is recognized for his significant contributions to the theory of elasticity, and the book has been widely influential in the development of the subject.This chapter provides an overview of the mathematical theory of elasticity, focusing on fundamental equations, plane theory, and torsion and bending. The book, authored by N. I. Muskhelishvili, is a comprehensive treatise on the subject, covering various aspects of elasticity theory and its applications. The content is divided into several parts, including: 1. **Fundamental Equations**: Discusses the basic principles of elasticity, such as stress analysis, strain analysis, and the fundamental law of elasticity. 2. **Plane Theory**: Introduces the plane theory of elasticity, including the stress function, complex representation, and solutions for various plane problems. 3. **Torsion and Bending**: Explores the torsion and bending of homogeneous and compound bars, providing detailed solutions and applications. The book is known for its rigorous mathematical approach and practical examples, making it a valuable resource for both researchers and engineers in the field of elasticity. The author, N. I. Muskhelishvili, is recognized for his significant contributions to the theory of elasticity, and the book has been widely influential in the development of the subject.