Math 433 HW5 Due April 5, 2013
1. Let v and w be tangent vector elds along a curve : I S . Prove that
d
< v (t), w(t) >=< v, w > + < v, w > .
dt
2. Let (s) be a curve parametrized by arclength s, with
Math 433 HW4 Due March 22, 2013
1. Let : R2 R2 be given by (x, y ) = (u(x, y ), v (x, y ), where u and v and dierentiable
functions satisfy the Cauchy-Riemann equations
ux = vy , uy = vx .
Show that i
Math 433 HW4 Due March 22, 2013
1. Let : R2 R2 be given by (x, y ) = (u(x, y ), v (x, y ), where u and v and dierentiable
functions satisfy the Cauchy-Riemann equations
ux = vy , uy = vx .
Show that i
Math 433 HW3 Due February 22, 2013
1. Compute the rst fundamental forms of the following surfaces
a. (u, v ) = (au cos v, bu sin v, u2 ); elliptic paraboloid.
b. (u, v ) = (au cosh v, bu sinh v, u2 );
Math 433 HW3 Due February 22, 2013
1. Compute the rst fundamental forms of the following surfaces
a. (u, v ) = (au cos v, bu sin v, u2 ); elliptic paraboloid.
b. (u, v ) = (au cosh v, bu sinh v, u2 );
Math 433 HW2 Due February 8, 2013
1. Let (s), s [0, l] be a closed convex plane curve positively oriented (i.e. s > 0). The curve
(s) = (s) rn(s), where r is a positive constant, is called a parallel
Math 433 HW2 Due February 8, 2013
1. Let (s), s [0, l] be a closed convex plane curve positively oriented (i.e. s > 0). The curve
(s) = (s) rn(s), where r is a positive constant, is called a parallel
Math 433 HW1 Due January 25, 2013
Without otherwise mentioned, all curves are smooth and regular.
1. A tractrix : (0, ) R2 is given by
t
(t) = (sin t, cos t + ln(tan ),
2
where t is the angle that th
Math 433 HW1 Due January 25, 2013
Without otherwise mentioned, all curves are smooth and regular.
1. A tractrix : (0, ) R2 is given by
t
(t) = (sin t, cos t + ln(tan ),
2
where t is the angle that th
Math 433
Introduction to Dierential Geometry
March 1, 2013
Midterm Exam
Instructions.
1. Two sides of a 3 5 card of notes allowed.
2. Show your work. Explain clearly.
3. There are 5 problems for a tot