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Exterior calculus is a branch of mathematics which involves differential geometry. In Exterior calculus the concept of differentiations is generalized to antisymmetric exterior derivatives and the notions of ordinary integration to differentiable manifolds of arbitrary dimensions. It therefore generalizes the fundamental theorem of calculus to Stokes' theorem. This textbook covers the fundamental requirements of exterior calculus in curricula for college students in mathematics and engineering programs. Chapters start from Heaviside-Gibbs algebra, and progress to different concepts in Grassman algebra. The final section of the book covers applications of exterior calculus with solutions. 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.

Calculus.

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Calculus.

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Markov Chain Process: Theory and Cases is designed for students of natural and formal sciences. It explains the fundamentals related to a stochastic process that satisfies the Markov property. It presents 10 structured chapters that provide a comprehensive insight into the complexity of this subject by presenting many examples and case studies that will help readers to deepen their acquired knowledge and relate learned theory to practice. This book is divided into four parts. The first part thoroughly examines the definitions of probability, independent events, mutually (and not mutually) exclusive events, conditional probability, and Bayes' theorem, which are essential elements in Markov's theory. The second part examines the elements of probability vectors, stochastic matrices, regular stochastic matrices, and fixed points. The third part presents multiple cases in various disciplines: Predictive computational science, Urban complex systems, Computational finance, Computational biology, Complex systems theory, and Computational Science in Engineering. The last part introduces learners to Fortran 90 programs and Linux scripts. To make the comprehension of Markov Chain concepts easier, all the examples, exercises, and case studies presented in this book are completely solved and given in a separate section. This book serves as a textbook (either primary or auxiliary) for students required to understand Markov Chains in their courses, and as a reference book for researchers who want to learn about methods that involve Markov Processes.