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Differential Geometry of Curves and Surfaces
This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it.
- GenresMathematicsGeometry
374 pages, Hardcover
Published September 27, 2016
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Displaying 1 - 2 of 2 reviews
December 27, 2017
This textbook is as close as one can get to Visual Complex Analysis by Tristan Needham in that it contains much more intuitive geometrical figures than any other Differential Geometry textbook out there.
All Diff Geo textbooks contain the typical figures, but this one helps build the intuition and understand some not-so-easily-to-visualize concepts by including colored figures that explicitly show the various quantities that are of concern.
A nice example is the one where the author shows that the covariant derivative is just the projection of the "normal" derivative(simple del operator) onto the tangent plane of a manifold.
Another example is the figure which shows the a Jacobi field on a sphere; this neatly illustrates what the Jacobi fields are actually about(telling us, in the language of geodesics, how much the manifold-here surface-diverges from being flat).
Apart from this textbook's many helpful figures, the author tries to explain the various results or theorems/lemmas in simple ways.
A beautiful example is when the author calculates the change in the angle(angle displacement) of a vector as it gets parallely transported along a closed curve on a sphere(to be precise, it's curves of constant latitude) and shows that near the north pole, we get a change in the angle by approximately 2π which is the result that we would expect if we did the same parallel transport along a closed curve in 2d flat space(R^2), thus showcasing that a sphere is locally flat without even referring to homeomorphisms/diffeomorphisms(which are arguably more sophisticated concepts).
Due to studying physics rather than mathematics, I chose to skip a course on Differential Geometry and jump straight to Riemannian geometry. I must say that I was using this textbook alongside the much more advanced textbook by Do Carmo(Riemannian geometry) and this textbook helped me understand the geometric meaning behind nearly every concept found in Riemannian geometry.
So, it will surely serve well anybody using this as a main or supplementary textbook for a Differential Geometry course(even graduate students who want to build intuition and understand everything geometrically).
All in all, I was very lucky to have found this book and I am very satisfied with it.
Amazing!
All Diff Geo textbooks contain the typical figures, but this one helps build the intuition and understand some not-so-easily-to-visualize concepts by including colored figures that explicitly show the various quantities that are of concern.
A nice example is the one where the author shows that the covariant derivative is just the projection of the "normal" derivative(simple del operator) onto the tangent plane of a manifold.
Another example is the figure which shows the a Jacobi field on a sphere; this neatly illustrates what the Jacobi fields are actually about(telling us, in the language of geodesics, how much the manifold-here surface-diverges from being flat).
Apart from this textbook's many helpful figures, the author tries to explain the various results or theorems/lemmas in simple ways.
A beautiful example is when the author calculates the change in the angle(angle displacement) of a vector as it gets parallely transported along a closed curve on a sphere(to be precise, it's curves of constant latitude) and shows that near the north pole, we get a change in the angle by approximately 2π which is the result that we would expect if we did the same parallel transport along a closed curve in 2d flat space(R^2), thus showcasing that a sphere is locally flat without even referring to homeomorphisms/diffeomorphisms(which are arguably more sophisticated concepts).
Due to studying physics rather than mathematics, I chose to skip a course on Differential Geometry and jump straight to Riemannian geometry. I must say that I was using this textbook alongside the much more advanced textbook by Do Carmo(Riemannian geometry) and this textbook helped me understand the geometric meaning behind nearly every concept found in Riemannian geometry.
So, it will surely serve well anybody using this as a main or supplementary textbook for a Differential Geometry course(even graduate students who want to build intuition and understand everything geometrically).
All in all, I was very lucky to have found this book and I am very satisfied with it.
Amazing!
February 15, 2022
What a phenomenal introduction to the differential geometry. Even though this book covers only 2-surfaces and it doesn’t introduce (but comes close) the Riemann tensor, it provides one of the best, to my knowledge, explanation of major conceptual points of the differential geometry. It has many nice diagrams and examples that makes this book such a joy to study.
Displaying 1 - 2 of 2 reviews