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**About this book :- **

**Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2 ** written by
** James W. Cannon **.

This textbook cover the following topics

(1) The solution of cubic and quartic equations required serious consideration of complex numbers, thought at first to be mysterious. But the mystery disappeared when it was seen that complex numbers simply model the Euclidean plane. Abel and Galois proved that equations of degrees 5 and higher could not be solved in the relatively simple manner by formula as had sufficed in equations of degrees 1 through 4. But Gauss, without giving explicit solutions, managed to prove the Fundamental Theorem of Algebra that ensured that complex numbers sufficed for their solution. Gauss gave proofs involving the geometry and topology of the plane.

(2) Newton showed that the study of motion could be greatly simplified if, instead of examining standard equations, one examined differential equations. Proving the existence of solutions to rather general differential equations led to problems in topology. One of the standard proof techniques involves Brouwer’s Fixed Point Theorem. This volume proves that theorem in dimension 2 and outlines the proof in general dimensions.

(3) Descartes demonstrated that mechanical devices other than straight edge and compass can construct curves of very high degree. Once curves of very general form are accepted as interesting, further delicate questions of length and area arise: finite curves of infinite length, finite curves of positive area, space filling curves, disks whose interiors have smaller areas than their closures, 0-dimensional sets through which no light rays can penetrate, continuous functions that are nowhere differentiable, sets of fractional dimension.

(4) The study of solutions to equations became more unified when all variables were considered to be complex variables. Riemann modelled complex curves by surfaces, which are 2-dimensional manifolds and are called Riemann surfaces. The analysis of 2-dimensional manifolds led naturally to notions, such as triangulation, genus, and Euler characteristic.

James W. Cannon

**Book Detail :- **
** Title: ** Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2
** Edition: **
** Author(s): ** James W. Cannon
** Publisher: ** American Mathematical Society
** Series: **
** Year: ** 2017
** Pages: ** 181
** Type: ** PDF
** Language: ** English
** ISBN: ** 1470437155,9781470437152
** Country: ** US

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**About Author :- **

The author **James W. Cannon ** (1943-2012) was born, in Bellefonte, Pennsylvania. James Cannon received his Ph.D. in Mathematics from the University of Utah in 1969, under the direction of C. Edmund Burgess.

He was a Professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University. He held this position until his retirement in September 2012.

James Cannon gave an AMS Invited address at the meeting of the American Mathematical Society in Seattle in August 1977, an invited address at the International Congress of Mathematicians in Helsinki 1978, and delivered the 1982 Mathematical Association of America Hedrick Lectures in Toronto, Canada.

James Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007. In 2012 he became a fellow of the American Mathematical Society.

**All Famous Books of this Author :- **

Here is list all books, text books, editions, versions, solution manuals or solved notes avaliable of this author, We recomended you to download all.

** • Download PDF Two-Dimensional Spaces: Volume 1, Geometry of Lengths, Areas, and Volumes by James W. Cannon **

** • Download PDF Two-Dimensional Spaces: Volume 2, Topology as Fluid Geometry by James W. Cannon **

** • Download PDF Two-dimensional Spaces: Volume 3, Non-Euclidean Geometry & Curvature by James W. Cannon **

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**Book Contents :- **
**Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2 ** written by
** James W. Cannon **
cover the following topics.

1. The Fundamental Theorem of Algebra

2. The Brouwer Fixed Point Theorem

3. Tools

4. Lebesgue Covering Dimension

5. Fat Curves and Peano Curves

6. The Arc, the Simple Closed Curve, and the Cantor Set

7. Algebraic Topology

8. Characterization of the 2-Sphere

9. 2-Manifolds

10. Arcs in S2 Are Tame

11. R. L. Moore’s Decomposition Theorem

12. The Open Mapping Theorem

13. Triangulation of 2-Manifolds

14. Structure and Classification of 2-Manifolds

15. The Torus

16. Orientation and Euler Characteristic

17. The Riemann-Hurwitz Theorem

Bibliography

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- Abstract Algebra
- Calculus
- Differential Equations
- Engineering Mathematics
- Linear Algebra
- Math Magic
- Real Analysis