** Exterior Calculus** textbook covers the fundamental requirements of exterior calculus in curricula for third-year students, as it contains the fundamentals related to Geometric algebra or Grassmann algebra oriented to Calculus. Without any doubt, this algebra has important implications in Science and Engineering. Chapters start from Heaviside-Gibbs algebra, and progress to different concepts in Grassmann’s algebra. The final section of the book covers applications of exterior calculus with solutions.

The ebook thoroughly examines the elements of Geometric algebra G over the Real field and these operators: inner product, outer product, and geometric product, their components, and their geometric representation, as well as their properties and the rigid transformations on the plane and in space. It also reviews the differentiation and the integration over Geometric algebra, including the line integral and surface integral. The Green, Stokes and Gauss theorems are also studied in detail and the Theorem of Fundamental Calculus is generalized.

Readers will find a concise and clear study of vector calculus and differential geometry, along with several examples and exercises. The solutions to the exercises are also included at the end of the book.

This is an ideal book for students with a basic background in mathematics who wish to learn about exterior calculus as part of their college curriculum and equip themselves with the knowledge to apply relevant theoretical concepts in practical situations.

**About the Author:**

Dr. Carlos Polanco PhD is an Associate Professor in the Department of Mathematics at the Universidad Nacional Autónoma de México (UNAM), where he has been a faculty member since 2006, and postdoctoral researcher and Head of Department of Electromechanical Instrumentation at the Instituto Nacional de Cardiología Ignacio Chávez since 2021.

Carlos completed his Ph.D. and his undergraduate studies at UNAM. His research interests lie in the area of High-Performance Computing, with a focus on the design of mathematical-computational models relating to Structural Proteomics, and Mathematical Epidemiology using Clustering, and MicroChip solutions.

**Keywords**: Exterior calculus, *Grassmann* algebra, Heaviside-Gibbs algebra, vector calculus, Calculus.

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