Sample Second Midterm Exam Problems printable pdf download

Sample Second Midterm Exam Problems

SAMPLE SECOND MIDTERM EXAM PROBLEMS - MATH 5378,

SPRING 2002

THIS WILL BE A CLOSED-BOOK, CLOSED-NOTES EXAM. YOU CAN USE

IN YOUR SOLUTIONS ANY RESULT THAT WAS COVERED IN CLASS OR

BY THE TEXT. YOU CAN ALSO USE ANY OF THE RESULTS FROM THE

HOMEWORK.

Formulas for the ﬁrst and second fundamental form and the Gaussian curvature

from pp. 91 and 100, and the geodesic equations in the form

u +

(u )

+ 2

u v +

(v )

= 0,

v +

(u )

+ 2

u v +

(v )

= 0

will be provided on the exam.

(1) Compute the ﬁrst fundamental form of the following surfaces of revolution:

(a) x(u, v) = ((a + b cos v) cos u, (a + b cos v) sin u, b sin v) (the torus);

(b) x(u, v) = (av cos u, av sin u, bv) (the cone).

(2) Suppose we have surface parameterized by coordinates u and v whose ﬁrst

fundamental form is E = 1, F = 0, G = u

+ a

, where a is a constant.

Find the area of the triangle bounded by the curves

u =

av,

v = 1.

(3) Show that the surface (the conoid)

x(u, v) = (u cos v, u sin v, u + v)

is locally isometric to the surface (the hyperboloid of revolution)

y(s, t) = (s cos t, s sin t,

1),

by a mapping given by

t = v + arctan u,

= u

+ 1.

(4) Compute the second fundamental form and the Gaussian curvature of the

pseudosphere x(u, v) = (a sin u cos v, a sin u sin v, a(ln tan

+ cos u)).

(5) Prove that a regular curve on a surface is a geodesic, if and only if its

curvature is equal to the absolute value of its normal curvature.

(6) Two surfaces are tangent to each other along a common curve α. Prove

that if α is a geodesic on one surface, then it will also be one on the other.

(7) Find the geodesics of a conic surface x(u, v) = uα(v) with α = 1 and

α = 1.

(8) Find the geodesic curvature of a helix u = b on the helicoid x = (u cos v, u sin v, av).

(9) Suppose on a sphere of radius R, a triangle formed by arcs of great circles

is given. Let A be the area of the triangle. Find the sum of the interior

angles of the triangle.

Date: April 24, 2002.

Sample Second Midterm Exam Problems

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