M365G Third Midterm Exam, April 12, 2012
1. Graphs.
Consider the surface z = f (x, y ), where we take coordinates u = x and
v = y . Orient the surface so that the normal has positive z -coordinate.
a) Compute the rst and second fundamental forms in terms
M365G First Midterm Exam Solutions, February 16, 2012
1. Prove the following theorem (which we proved in class): Let p R2 , let
U be an open neighborhood of p, and let : U R3 be a smooth surface
patch of some surface S , and let q = (p). If u (p) v (p) =
M365G Second Midterm Exam, March 22, 2012
1. Surfaces of revolution.
Consider the surface of revolution
(u, ) = (r(u) cos(), r(u) sin(), g (u),
where r(u) and g (u) are smooth functions, r(u) > 0 and r (u)2 + g (u)2 = 0.
(The book calls r f , but in prob
M365G Final Exam, May 14, 2012
1. Cylindrical curves.
a) Consider a curve of the form (t) = (a cos(t), a sin(t), f (t), where a is
a constant and f is a smooth function. Compute the speed, curvature and
torsion at time t in terms of the (unknown) function
HW #2 SOLUTIONS FOR OTHER PROBLEMS
BENNI
1. Problem 1
Converting polar coordinates to Cartesian coordinates is fairly simple. It may not be the most
elegant solution, but we can reduce this problem to a case we know well by this method. Regardless
of the
M365G Third Midterm Exam, April 12, 2012
1. Graphs.
Consider the surface z = f (x, y ), where we take coordinates u = x and
v = y . Orient the surface so that the normal has positive z -coordinate.
a) Compute the rst and second fundamental forms in terms
M365G Second Midterm Exam, March 22, 2012
1. Surfaces of revolution.
Consider the surface of revolution
(u, ) = (r(u) cos(), r(u) sin(), g (u),
where r(u) and g (u) are smooth functions, r(u) > 0 and r (u)2 + g (u)2 = 0.
(The book calls r f , but in prob
M365G Second Midterm Exam, March 22, 2012
1. Surfaces of revolution.
Consider the surface of revolution
(u, ) = (r(u) cos(), r(u) sin(), g (u),
where r(u) and g (u) are smooth functions, r(u) > 0 and r (u)2 + g (u)2 = 0.
(The book calls r f , but in prob
M365G First Midterm Exam, February 16, 2012
1. Prove the following theorem (which we proved in class): Let p R2 , let
U be an open neighborhood of p, and let : U R3 be a smooth surface
patch of some surface S , and let q = (p). If u (p) v (p) = 0, then th
M365G First Midterm Exam Solutions, February 16, 2012
1. Prove the following theorem (which we proved in class): Let p R2 , let
U be an open neighborhood of p, and let : U R3 be a smooth surface
patch of some surface S , and let q = (p). If u (p) v (p) =
M365G Final Exam, May 14, 2012
1. Cylindrical curves.
a) Consider a curve of the form (t) = (a cos(t), a sin(t), f (t), where a is
a constant and f is a smooth function. Compute the speed, curvature and
torsion at time t in terms of the (unknown) function
M365G Third Midterm Exam, April 12, 2012
1. Graphs.
Consider the surface z = f (x, y ), where we take coordinates u = x and
v = y . Orient the surface so that the normal has positive z -coordinate.
a) Compute the rst and second fundamental forms in terms