Term
Write the sum and product formulas for the Riemann zeta function. |
|
Definition
|
|
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
|
|
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
|
|
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
|
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
|
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. |
|
|