Introduction to Vectors and Tensors
This convenient single-volume compilation of two texts offers both an introduction and an in-depth survey. Geared toward engineering and science students rather than mathematicians, its less rigorous treatment focuses on physics and engineering applications, building upon the systematic development of concepts rather than emphasizing mathematical problem-solving techniques. A practical reference for professionals, it is suitable for advanced undergraduate and graduate students.
Volume I begins with a brief discussion of algebraic structures followed by detailed explorations of the algebra of vectors and tensors, in addition to aspects of linear and multi-linear algebra. Topics include vector spaces, linear transformations, determinants and matrices, spectral decompositions, and tensor and exterior algebra. Volume II opens with a discussion of Euclidean manifolds and proceeds to the development of analytical and geometrical aspects of vector and tensor fields. Subsequent chapters survey the integration of fields on Euclidean manifolds, hypersurfaces, and continuous groups.
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ˆ ˆ ˆ A A A A V U algebra arbitrary binary operation called Cartan parallelism Cartesian coordinate system Christoffel symbols component form condition contravariant covariant derivative decomposition defined definition denote differential domain dt dt dual eigenvalues endomorphism equation Euclidean manifold Exercises exists exterior derivative following theorem formula function geodesic given grad Hermitian hypersurface identity element ij ij inner product space inverse isomorphism left-invariant field linear transformation linearly independent mapping matrix multiplication natural basis notation one-to-one orthogonal orthonormal basis product basis Proof prove real numbers reciprocal basis relative representation result scalar field Section skew-symmetric smooth strict components subset subspace surface coordinate system tangent vector tensor field transformation rule unique vector field vector space zero