METU Department of Mathematics Math 304 History of Math Concepts Spring 2011

Midterm 2

May 6 2010, 17:30 – 18:40

Name and the student number:

Q 1. Who formulated and who proved the Quadratic Reciprocity Law ? When it happened ?

Q 2. What was the contribution of Galois, Abel, Ruffini, and Gauss related to solutions of equations of degree five? When these works were made ?

Q 3. Who discovered non-Euclidean Geometry and when ? Why was it a revolution in Geometry ?

Q 4. What is the difference between analytic and synthetic geometry ? Who represented these directions of geometry in XIX-th century ?

Q 5. What was the content of

the Habilitation Lecture of Riemann ?

Klein’s Erlangen program ?

Q 6. Give the name of the authors of the following theories, concepts, and terms. Theory of determinants. Theory of eigenvalues and diagonalization. The concept of principal curvatures of surfaces. The concept of double integral. Matrix multiplication. The term “derivative”, and the notation f′, f′′, etc. Theory of convergence of power series. Variational calculus. The term “group”. Theory of periodic continued fractions. Mathematical Physics, namely, theory of electricity. The term “Topology”. The term “matrix”. The concept of matrix multiplication, and of the inverse matrix. ”Cramer’s rule” for 2 and 3 unknowns ?

Q 7. Explain briefly what is it (give a formula or a definition)

the Bezout theorem

the number of points which determines a conic curve the heat equation the Gaussian integers a cyclotomic polynomial ? an elliptic integral ?

Q 8. What were the mathematical interests of the following mathematicians ? Describe their famous results. (Give the names of their works and years of publication, if you know any).

Lagrange Cauchy Gauss Abel Jacobi Riemann Klein Hamilton Q 9. For which values of n a regular n-gon can be constructed using compass and straightedge ? Who proved it and when ?

Q 10. Formulate the Quadratic Reciprocity Law. Determine if 5 is a quadratic residue modulo 29, using this Law.

Q 11. Formulate the Fifth Postulate of Euclid. Give three examples of other claims equivalent to it.

Q 12. Give Gaussian definition of the curvature of a surface.

Q 13. Give a formula for Riemann’s zeta-function. Formulate the Riemann Conjecture.