Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA’s Deborah and Franklin Tepper Haimo Award and Brown’s Teaching with Technology Award. He is the author of several books, including Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension.Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton’s Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

I endorse this third edition with enthusiasm. Moving away from Java based applets for the previous edition to Maplesoft based applets is significant. There are many new examples, and these are more clearly identified. There are many significant changes to produce a more useful and attractive teaching too. Figures are well-placed to be easily matched to the text. Adding historical vignettes help students appreciate how the topic developed.--Ken Rosen

• Preface1 Plane Curves: Local Properties1.1 Parametrizations1.2 Position, Velocity, and Acceleration1.3 Curvature1.4 Osculating Circles, Evolutes, Involutes1.5 Natural Equations2 Plane Curves: Global Properties2.1 Basic Properties2.2 Rotation Index2.3 Isoperimetric Inequality2.4 Curvature, Convexity, and the Four-Vertex Theorem3 Curves in Space: Local Properties3.1 Definitions, Examples, and Differentiation3.2 Curvature, Torsion, and the Frenet Frame3.3 Osculating Plane and Osculating Sphere3.4 Natural Equations4 Curves in Space: Global Properties4.1 Basic Properties4.2 Indicatrices and Total Curvature4.3 Knots and Links5 Regular Surfaces5.1 Parametrized Surfaces5.2 Tangent Planes; The Differential5.3 Regular Surfaces5.4 Change of Coordinates; Orientability6 First and Second Fundamental Forms6.1 The First Fundamental Form6.2 Map Projections (Optional)6.3 The Gauss Map6.4 The Second Fundamental Form6.5 Normal and Principal Curvatures6.6 Gaussian and Mean Curvatures6.7 Developable Surfaces; Minimal Surfaces7 Fundamental Equations of Surfaces7.1 Gauss’s Equations; Christoffel Symbols7.2 Codazzi Equations; Theorema Egregium7.3 Fundamental Theorem of Surface Theory8 Gauss-Bonnet Theorem; Geodesics8.1 Curvatures and Torsion8.2 Gauss-Bonnet Theorem, Local Form8.3 Gauss-Bonnet Theorem, Global Form8.4 Geodesics8.5 Geodesic Coordinates8.6 Applications to Plane, Spherical, and Elliptic Geometry8.7 Hyperbolic Geometry9 Curves and Surfaces in n-dimensional Space9.1 Curves in n-dimensional Euclidean Space9.2 Surfaces in Euclidean n-SpaceAppendix A: Tensor NotationIndex

Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA’s Deborah and Franklin Tepper Haimo Award and Brown’s Teaching with Technology Award. He is the author of several books, including Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension.Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton’s Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

I endorse this third edition with enthusiasm. Moving away from Java based applets for the previous edition to Maplesoft based applets is significant. There are many new examples, and these are more clearly identified. There are many significant changes to produce a more useful and attractive teaching too. Figures are well-placed to be easily matched to the text. Adding historical vignettes help students appreciate how the topic developed.--Ken Rosen

• Preface1 Plane Curves: Local Properties1.1 Parametrizations1.2 Position, Velocity, and Acceleration1.3 Curvature1.4 Osculating Circles, Evolutes, Involutes1.5 Natural Equations2 Plane Curves: Global Properties2.1 Basic Properties2.2 Rotation Index2.3 Isoperimetric Inequality2.4 Curvature, Convexity, and the Four-Vertex Theorem3 Curves in Space: Local Properties3.1 Definitions, Examples, and Differentiation3.2 Curvature, Torsion, and the Frenet Frame3.3 Osculating Plane and Osculating Sphere3.4 Natural Equations4 Curves in Space: Global Properties4.1 Basic Properties4.2 Indicatrices and Total Curvature4.3 Knots and Links5 Regular Surfaces5.1 Parametrized Surfaces5.2 Tangent Planes; The Differential5.3 Regular Surfaces5.4 Change of Coordinates; Orientability6 First and Second Fundamental Forms6.1 The First Fundamental Form6.2 Map Projections (Optional)6.3 The Gauss Map6.4 The Second Fundamental Form6.5 Normal and Principal Curvatures6.6 Gaussian and Mean Curvatures6.7 Developable Surfaces; Minimal Surfaces7 Fundamental Equations of Surfaces7.1 Gauss’s Equations; Christoffel Symbols7.2 Codazzi Equations; Theorema Egregium7.3 Fundamental Theorem of Surface Theory8 Gauss-Bonnet Theorem; Geodesics8.1 Curvatures and Torsion8.2 Gauss-Bonnet Theorem, Local Form8.3 Gauss-Bonnet Theorem, Global Form8.4 Geodesics8.5 Geodesic Coordinates8.6 Applications to Plane, Spherical, and Elliptic Geometry8.7 Hyperbolic Geometry9 Curves and Surfaces in n-dimensional Space9.1 Curves in n-dimensional Euclidean Space9.2 Surfaces in Euclidean n-SpaceAppendix A: Tensor NotationIndex

Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA’s Deborah and Franklin Tepper Haimo Award and Brown’s Teaching with Technology Award. He is the author of several books, including Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension.Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton’s Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

I endorse this third edition with enthusiasm. Moving away from Java based applets for the previous edition to Maplesoft based applets is significant. There are many new examples, and these are more clearly identified. There are many significant changes to produce a more useful and attractive teaching too. Figures are well-placed to be easily matched to the text. Adding historical vignettes help students appreciate how the topic developed.--Ken Rosen

• Preface1 Plane Curves: Local Properties1.1 Parametrizations1.2 Position, Velocity, and Acceleration1.3 Curvature1.4 Osculating Circles, Evolutes, Involutes1.5 Natural Equations2 Plane Curves: Global Properties2.1 Basic Properties2.2 Rotation Index2.3 Isoperimetric Inequality2.4 Curvature, Convexity, and the Four-Vertex Theorem3 Curves in Space: Local Properties3.1 Definitions, Examples, and Differentiation3.2 Curvature, Torsion, and the Frenet Frame3.3 Osculating Plane and Osculating Sphere3.4 Natural Equations4 Curves in Space: Global Properties4.1 Basic Properties4.2 Indicatrices and Total Curvature4.3 Knots and Links5 Regular Surfaces5.1 Parametrized Surfaces5.2 Tangent Planes; The Differential5.3 Regular Surfaces5.4 Change of Coordinates; Orientability6 First and Second Fundamental Forms6.1 The First Fundamental Form6.2 Map Projections (Optional)6.3 The Gauss Map6.4 The Second Fundamental Form6.5 Normal and Principal Curvatures6.6 Gaussian and Mean Curvatures6.7 Developable Surfaces; Minimal Surfaces7 Fundamental Equations of Surfaces7.1 Gauss’s Equations; Christoffel Symbols7.2 Codazzi Equations; Theorema Egregium7.3 Fundamental Theorem of Surface Theory8 Gauss-Bonnet Theorem; Geodesics8.1 Curvatures and Torsion8.2 Gauss-Bonnet Theorem, Local Form8.3 Gauss-Bonnet Theorem, Global Form8.4 Geodesics8.5 Geodesic Coordinates8.6 Applications to Plane, Spherical, and Elliptic Geometry8.7 Hyperbolic Geometry9 Curves and Surfaces in n-dimensional Space9.1 Curves in n-dimensional Euclidean Space9.2 Surfaces in Euclidean n-SpaceAppendix A: Tensor NotationIndex