Tensor and Vector Analysis: With Applications to Differential Geometry

Name: Tensor and Vector Analysis: With Applications to Differential Geometry
Rating: 5.00 (1 reviews)

by C. E. Springer

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ISBN13: 9780486498010
Binding: Paperback
Subject: N/A
Publisher: Dover Publications
Publisher Imprint: Dover Publications
Publication Date: November 21, 2012
Pages: 242
Original Price: USD 19.95
Language: English
Edition: N/A
Item Weight: 318 grams
BISAC Subject(s): Vector Analysis and Geometry / Differential

About The Book

Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics.
Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and differentiation, tensor algebra and analysis, and vector analysis and algebra. Additional topics include differentiation of vectors and tensors, scalar and vector fields, and integration of vectors. The concluding chapter employs tensor theory to develop the differential equations of geodesics on a surface in several different ways to illustrate further differential geometry.

About The Author

C. E. Springer was Professor of Mathematics at the University of Oklahoma.

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A Classical Gateway to Tensors and Differential Geometry

C. E. Springer’s Tensor and Vector Analysis: With Applications to Differential Geometry (Dover) offers a clear, concise introduction to tensor calculus and its geometric applications. Written with an eye toward accessibility, the book balances rigor with pedagogy, making it suitable for advanced undergraduates and beginning graduate students in mathematics, physics, and engineering.

The text begins with fundamental notions of vectors and vector spaces, gradually extending these to tensors, tensor algebra, and transformation laws. Springer’s methodical progression emphasizes intuition: tensors are presented as natural generalizations of vector operations, rather than as abstract constructs introduced abruptly. Subsequent chapters explore contraction, symmetrization, and other operations before moving to applications in differential geometry, including covariant differentiation, Christoffel symbols, curvature, and geodesics.

The pedagogical strengths lie in the clarity of exposition and the steady development from simple concepts to advanced applications. Definitions are accompanied by worked examples that demonstrate both computation and geometric interpretation. Exercises at the end of chapters reinforce understanding, offering a balance of routine practice and conceptual challenge. Springer’s emphasis on concrete applications—particularly the geometry of curves, surfaces, and manifolds—renders the subject more approachable for readers oriented toward physics, especially general relativity.

Nevertheless, the text has limitations. It presumes prior familiarity with linear algebra and multivariable calculus, and its classical coordinate-based approach does not encompass the modern, manifold-theoretic framework. Compared with contemporary works (e.g., Bishop & Goldberg, Schutz, Carroll), it lacks breadth, though it remains an excellent stepping stone toward such advanced treatments.

In summary, Springer’s book endures as an effective introductory text. Its structured exposition, accessible style, and applied orientation commend it as a foundational resource, particularly for those seeking to bridge elementary vector analysis with the differential geometry underlying modern physics.

Tensor and Vector Analysis: With Applications to Differential Geometry

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