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)
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
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
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 a
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
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)
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
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
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 s
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
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 a
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)