Description
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Explains the subject well and does not require too much in prerequisites — the first two Acts can be read by someone with only some calculus and linear algebra. The best part are the exercises where the reader is encouraged to construct shapes out of paper or peel strips off a fruit. These remind me of the science books I read as a child. Those exercises are fun and instructive
No formula derivations, but only simple reasonings, as if modern geometry were only intuitive. a lot of design, but rather confuse and little explainments.
I think this book should be recommended to math students as an excellent auxiliary tutorial and to physics student (who learn General Relativity) as a basic textbook in Differential Geometry.
A majority of the book are derivations in 3D and analogies using simple geometries in R3. I would recommend this to an undergraduate student looking to get into GR, but it’s not the best reference book.
Wunderbares Buch, mit leicht nachvollziehbaren Gedankengängen und Entwicklungen, selbst von komplexen Sachverhalten. Hätte ich wirklich gerne während des Studiums gehabt.