Riemann

Riemann

1. What is a torus? o In geometry, a torus is a surface of a revolution generated by revolving a [|circle] in three-dimensional space about an axis coplanar with the circle. In most contexts it is assumed that the axis does not touch the circle and in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit. 2. Why was Riemann an important figure in math? o Riemann was a master analyst. He observed that once one ha a way of measuring speed along any path in a manifold, calculus automatically gives way of measuring lengths of curves in the manifold, and algebra (and trigonometry) automatically gives a way of measuring angles. He explained the concept of curvature on a 3-dimensional plane. He invented the gadget for keeping track of the different curvatures in different directions and that is called Riemann curvature tensor named after Riemann of course. He was a father figure in the mathematics world. He is well known for his concept of the Riemann Sums, which is basically a method in which you would estimate the area under a curve by dividing the curve into rectangular sections and taking the area of the rectangles in order to estimate the area of under the curve. 3. What does the author state about the importance of Riemann’s work with regards to changing our view of the world? o The author states that Riemann’s work opened new possibilities for modern science and mathematics, and fundamentally altered the way geometry and topology would develop. 4. Describe the changing political structure of Europe at the time, and explain the increasing role of the university as an institution of significance with regards to creating knowledge. How do universities work? o The political scene in Europe was changing. The era of Napoleon had ended and France seems to have been leading the way in the knowledge field with great universities. The new century seems to go to France with Great Britain right behind. No one ever expected to see Germany take a power seat in mathematics. Universities in Germany were disunited and were competitive which created many incentives for them to advance in their teachings. Universities worked by joining youth and experience, research and teaching, and that made the universities the exciting arenas for scientific discoveries. Challenges are what make universities great. Students who attended university would go for their doctorate degree and after that they would pursue in later work after their thesis and submitting the results as a paper and giving a presentation to the public, inaugural lecture. 5. Write down any questions you have about the reading. o Why did Gauss choose the third topic for Riemann to discuss when he new that Riemann was uncomfortable, was it jealousy or was it trust that led Gauss to know that his student was capable? o Riemann made some groundbreaking discoveries how did he discover such discoveries, how did he think?