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AuthorSimmonds, James G. author
TitleA Brief on Tensor Analysis [electronic resource] / by James G. Simmonds
ImprintNew York, NY : Springer New York : Imprint: Springer, 1994
Edition Second Edition
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Descript XIV, 114 p. online resource


There are three changes in the second edition. First, with the help of readers and colleagues-thanks to all-I have corrected typographical errors and made minor changes in substance and style. Second, I have added a fewmore Exercises,especially at the end ofChapter4.Third, I have appended a section on Differential Geometry, the essential mathematical tool in the study of two-dimensional structural shells and four-dimensional general relativity. JAMES G. SIMMONDS vii Preface to the First Edition When I was an undergraduate, working as a co-op student at North Ameriยญ can Aviation, I tried to learn something about tensors. In the Aeronautical Engineering Department at MIT, I had just finished an introductory course in classical mechanics that so impressed me that to this day I cannot watch a plane in flight-especially in a turn-without imaging it bristling with vecยญ tors. Near the end of the course the professor showed that, if an airplane is treated as a rigid body, there arises a mysterious collection of rather simpleยญ looking integrals called the components of the moment of inertia tensor


I Introduction: Vectors and Tensors -- Three-Dimensional Euclidean Space -- Directed Line Segments -- Addition of Two Vectors -- Multiplication of a Vector v by a Scalar ? -- Things That Vectors May Represent -- Cartesian Coordinates -- The Dot Product -- Cartesian Base Vectors -- The Interpretation of Vector Addition -- The Cross Product -- Alternative Interpretation of the Dot and Cross Product. Tensors -- Definitions -- The Cartesian Components of a Second Order Tensor -- The Cartesian Basis for Second Order Tensors -- Exercises -- II General Bases and Tensor Notation -- General Bases -- The Jacobian of a Basis Is Nonzero -- The Summation Convention -- Computing the Dot Product in a General Basis -- Reciprocal Base Vectors -- The Roof (Contravariant) and Cellar (Covariant) Components of a Vector -- Simplification of the Component Form of the Dot Product in a General Basis -- Computing the Cross Product in a General Basis -- A Second Order Tensor Has Four Sets of Components in General -- Change of Basis -- Exercises -- III Newtonโ{128}{153}s Law and Tensor Calculus -- Rigid Bodies -- New Conservation Laws -- Nomenclature -- Newtonโ{128}{153}s Law in Cartesian Components -- Newtonโ{128}{153}s Law in Plane Polar Coordinates -- The Physical Components of a Vector -- The Christoffel Symbols -- General Three-Dimensional Coordinates -- Newtonโ{128}{153}s Law in General Coordinates -- Computation of the Christoffel Symbols -- An Alternative Formula for Computing the Christoffel Symbols -- A Change of Coordinates -- Transformation of the Christoffel Symbols -- Exercises -- IV The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem -- The Gradient -- Linear and Nonlinear Eigenvalue Problems -- The Del Operator -- The Divergence, Curl, and Gradient of a Vector Field -- The Invariance of ? ยท v, ? ร{151} v, and ?v -- The Covariant Derivative -- The Component Forms of ? ยท v, ? ร{151} v, and ?v -- The Kinematics of Continuum Mechanics -- The Divergence Theorem -- Differential Geometry -- Exercises

Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis


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