"To the reader who wishes to obtain a bird's-eye view of the theory of differential forms with applications to other branches of pure mathematics, applied mathematic and physics, I can recommend no better book." — T. J. Willmore, London Mathematical Society Journal. This excellent text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primaril... Read More
"To the reader who wishes to obtain a bird's-eye view of the theory of differential forms with applications to other branches of pure mathematics, applied mathematic and physics, I can recommend no better book." — T. J. Willmore, London Mathematical Society Journal. This excellent text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primaril... Read More
Description
"To the reader who wishes to obtain a bird's-eye view of the theory of differential forms with applications to other branches of pure mathematics, applied mathematic and physics, I can recommend no better book." — T. J. Willmore, London Mathematical Society Journal. This excellent text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primarily to engineers and physical scientists, but it has also been used successfully to introduce modern differential geometry to students in mathematics. Chapter I introduces exterior differential forms and their comparisons with tensors. The next three chapters take up exterior algebra, the exterior derivative and their applications. Chapter V discusses manifolds and integration, and Chapter VI covers applications in Euclidean space. The last three chapters explore applications to differential equations, differential geometry, and group theory. "The book is very readable, indeed, enjoyable — and, although addressed to engineers and scientists, should be not at all inaccessible to or inappropriate for ... first year graduate students and bright undergraduates." — F. E. J. Linton, Wesleyan University, American Mathematical Monthly.
Reprint of the Academic Press, Inc., New York, 1963 edition.
Details
Price: $12.95
Pages: 240
Publisher: Dover Publications
Imprint: Dover Publications
Series: Dover Books on Mathematics
Publication Date: 29th March 2012
Trim Size: 5.37 x 8.5 in
ISBN: 9780486139616
Format: eBook
BISACs: MATHEMATICS / Applied
Table of Contents
Foreword; Preface to the Dover Edition; Preface to the First Edition I. Introduction 1.1 Exterior Differential Forms 1.2 Comparison with Tensors II. Exterior algebra 2.1 The Space of p-vectors 2.2 Determinants 2.3 Exterior Products 2.4 Linear Transformations 2.5 Inner Product Spaces 2.6 Inner Products of p-vectors 2.7 The Star Operator 2.8 Problems III. The Exterior Derivative 3.1 Differential Forms 3.2 Exterior Derivative 3.3 Mappings 3.4 Change of coordinates 3.5 An Example from Mechanics 3.6 Converse of the Poincaré Lemma 3.7 An Example 3.8 Further Remarks 3.9 Problems IV. Applications 4.1 Moving Frames in E superscript 3 4.2 Relation between Orthogonal and Skew-symmetric Matrices 4.3 The 6-dimensional Frame Space 4.4 The Laplacian, Orthogonal Coordinates 4.5 Surfaces 4.6 Maxwell's Field Equations 4.7 Problems V. Manifolds and Integration 5.1 Introduction 5.2 Manifolds 5.3 Tangent Vectors 5.4 Differential Forms 5.5 Euclidean Simplices 5.6 Chains and Boundaries 5.7 Integration of Forms 5.8 Stokes' Theorem 5.9 Periods and De Rham's Theorems 5.10 Surfaces; Some Examples 5.11 Mappings of Chains 5.12 Problems VI. Applications in Euclidean Space 6.1 Volumes in E superscript n 6.2 Winding Numbers, Degree of a Mapping 6.3 The Hopf Invariant 6.4 Linking Numbers, the Gauss Integral, Ampère's Law VII. Applications to Different Equations 7.1 Potential Theory 7.2 The Heat Equation 7.3 The Frobenius Integration Theorem 7.4 Applications of the Frobenius Theorem 7.5 Systems of Ordinary Equations 7.6 The Third Lie Theorem VIII. Applications to Differential Geometry 8.1 Surfaces (Continued) 8.2 Hypersurfaces 8.3 Riemannian Geometry, Local Theory 8.4 Riemannian Geometry, Harmonic Integrals 8.5 Affine Connection 8.6 Problems IX. Applications to Group Theory 9.1 Lie Groups 9.2 Examples of Lie Groups 9.3 Matrix Groups 9.4 Examples of Matrix Groups 9.5 Bi-invariant Forms 9.6 Problems X. Applications to Physics 10.1 Phase and State Space 10.2 Hamiltonian Systems 10.3 Integral-invariants 10.4 Brackets 10.5 Contact Transformations 10.6 Fluid Mechanics 10.7 Problems Bibliography; Glossary of Notation; Index
"To the reader who wishes to obtain a bird's-eye view of the theory of differential forms with applications to other branches of pure mathematics, applied mathematic and physics, I can recommend no better book." — T. J. Willmore, London Mathematical Society Journal. This excellent text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primarily to engineers and physical scientists, but it has also been used successfully to introduce modern differential geometry to students in mathematics. Chapter I introduces exterior differential forms and their comparisons with tensors. The next three chapters take up exterior algebra, the exterior derivative and their applications. Chapter V discusses manifolds and integration, and Chapter VI covers applications in Euclidean space. The last three chapters explore applications to differential equations, differential geometry, and group theory. "The book is very readable, indeed, enjoyable — and, although addressed to engineers and scientists, should be not at all inaccessible to or inappropriate for ... first year graduate students and bright undergraduates." — F. E. J. Linton, Wesleyan University, American Mathematical Monthly.
Reprint of the Academic Press, Inc., New York, 1963 edition.
Price: $12.95
Pages: 240
Publisher: Dover Publications
Imprint: Dover Publications
Series: Dover Books on Mathematics
Publication Date: 29th March 2012
Trim Size: 5.37 x 8.5 in
ISBN: 9780486139616
Format: eBook
BISACs: MATHEMATICS / Applied
Foreword; Preface to the Dover Edition; Preface to the First Edition I. Introduction 1.1 Exterior Differential Forms 1.2 Comparison with Tensors II. Exterior algebra 2.1 The Space of p-vectors 2.2 Determinants 2.3 Exterior Products 2.4 Linear Transformations 2.5 Inner Product Spaces 2.6 Inner Products of p-vectors 2.7 The Star Operator 2.8 Problems III. The Exterior Derivative 3.1 Differential Forms 3.2 Exterior Derivative 3.3 Mappings 3.4 Change of coordinates 3.5 An Example from Mechanics 3.6 Converse of the Poincaré Lemma 3.7 An Example 3.8 Further Remarks 3.9 Problems IV. Applications 4.1 Moving Frames in E superscript 3 4.2 Relation between Orthogonal and Skew-symmetric Matrices 4.3 The 6-dimensional Frame Space 4.4 The Laplacian, Orthogonal Coordinates 4.5 Surfaces 4.6 Maxwell's Field Equations 4.7 Problems V. Manifolds and Integration 5.1 Introduction 5.2 Manifolds 5.3 Tangent Vectors 5.4 Differential Forms 5.5 Euclidean Simplices 5.6 Chains and Boundaries 5.7 Integration of Forms 5.8 Stokes' Theorem 5.9 Periods and De Rham's Theorems 5.10 Surfaces; Some Examples 5.11 Mappings of Chains 5.12 Problems VI. Applications in Euclidean Space 6.1 Volumes in E superscript n 6.2 Winding Numbers, Degree of a Mapping 6.3 The Hopf Invariant 6.4 Linking Numbers, the Gauss Integral, Ampère's Law VII. Applications to Different Equations 7.1 Potential Theory 7.2 The Heat Equation 7.3 The Frobenius Integration Theorem 7.4 Applications of the Frobenius Theorem 7.5 Systems of Ordinary Equations 7.6 The Third Lie Theorem VIII. Applications to Differential Geometry 8.1 Surfaces (Continued) 8.2 Hypersurfaces 8.3 Riemannian Geometry, Local Theory 8.4 Riemannian Geometry, Harmonic Integrals 8.5 Affine Connection 8.6 Problems IX. Applications to Group Theory 9.1 Lie Groups 9.2 Examples of Lie Groups 9.3 Matrix Groups 9.4 Examples of Matrix Groups 9.5 Bi-invariant Forms 9.6 Problems X. Applications to Physics 10.1 Phase and State Space 10.2 Hamiltonian Systems 10.3 Integral-invariants 10.4 Brackets 10.5 Contact Transformations 10.6 Fluid Mechanics 10.7 Problems Bibliography; Glossary of Notation; Index
