This book provides comprehensive coverage of the calculus of variations and its applications, focusing on the integration of partial differential equations with geometrical techniques. It encompasses various methods of variations and presents their practical applications in different fields. The book also discusses the Hyers-Ulam stability of Euler's equation, providing insights into the behavior and robustness of this fundamental equation in the calculus of variations. Furthermore, it presents exact solutions of nonlinear partial differential equations using a new double integral transform combined with an iterative method.The book goes beyond traditional applications and explores unique and unexpected areas where the calculus of variations has been successfully applied. For example, it examines the vibration of a flexible follower in a cam mechanism with time-dependent boundary effects, addressing practical engineering sproblems. It also presents a straightforward sufficiency proof for a nonparametric problem of Bolza, demonstrating the versatility and applicability of variational methods in different mathematical contexts. Surprisingly, the book explores how the calculus of variations has been utilized in fields such as economics, literature, and interior design. Given the wide range of topics covered, the book will be beneficial for researchers and professionals working in the field of mathematics, particularly those interested in the calculus of variations, partial differential equations, and geometrical techniques.