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442C Test Three
Definitions, examples, theorems and other things to know on the third 442C exam.
18
Mathematics
Undergraduate 4
04/25/2010

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Term
Write the sum and product formulas for the Riemann zeta function.
Definition
[image]
Term
Information about what is coded in the Riemann zeta function?
Definition
information regarding the distribution of primes
Term
What was Gauss's attitude towards non-Euclidean geometry?
Definition
He verified that it was consistent, but kept it under wraps so as not to have to deal with the philosophers.
Term
What surfaces are Elliptic functions typically associated with?
Definition
Torus
Term
State the Gauss-Bonnet Theorem.
Definition
Let k be Gaussian curvature and g be the genus, then integral kdS=2pi(2-2g)
Term
What is integral kdS for the Torus?
Definition
Term
State Euler's formula for the genus of a surface via a map on the surface
Definition
x = V-E+F
convex, x = 2
Term
Who brought complex numbers out of the shadows and established them as the "real" deal? How?
Definition
It was Gauss with the placing of points on the plane in the year 1790!! (CLUE reference)
Term
Describe the Riemann surface of sqrt(2).
Definition
1850s
This is a concrete non-singular Riemann surface. The two horizontal axes represent the real and complex parts of z and the vertical axis represents the real parts of sqrt(z)
Term
What is a tractrix?
Definition
The curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed.
1670
Term
What is a pseudosphere
Definition
Constant-negative Gaussian curvature surface of revolution generated by a tractrix about its asymptote.
1868
Term
What do the pseudosphere and sphere have in common?
Definition
They have the same surface area and volume. They both have constant curvature. Sphere is positive and pseudo is negative...
Term
Angle sums of triangles in the different geometry.
Definition
Euclid = 180
Hyperbolic < 180
Spherical >180
Term
Parallel lines in geometries.
Definition
Know them (I can't draw them...)
Term
What were the effects of the invention of Lobachevski geometry in mathematics and philosophy?
Definition
1871
Killed philosopher's Absolute Truth (synthetic a priori)
Math now based on axioms.
Term
Compare the effects of Hamilton's Quaternions on algebra and Lobachevski's geometry on geometry.
Definition
Both broke the mould and substantially furthered their fields.
Term
What is the most important property of elliptic functions?
Definition
Elliptic functions are doubly periodic
Term
How did elliptic integrals come about?
Definition
They were originated by Euler from Fagano's study of the lemniscate.
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