2.094
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS
SPRING 2008
Homework 1
Instructor:
Assigned:
Due:
Prof. K. J. Bathe
02/07/2008
02/14/2008
Problem 1 (10 points):
Consider the sheet of material shown.
Here
=
=
+
+
Also, the stresses are
=
=
=
Id
18.02 Problem Set 11, Part II Solutions
1. We put the center of the sphere at the origin O as usual, and take the
North Pole N = (0, 0, a) as the xed point. Let P be an arbitrary point on
the surface of the the sphere S, and D the straight-line distance f
6.253: Convex Analysis and Optimization
Homework 2
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
(a) Let C be a nonempty convex cone. Show that cl(C) and ri(C) is also a convex cone.
(b) Let C = cone(cfw_x1 , . . . , xm ). Show that
m
X
ri(C) =
18.02 Problem Set 10, Part II Solutions
1. Base: R: x2 + (y
1)2 1. Top: z = f (x, y) = (x2 + y 2 )1/2 .
In cylindrical coords, base is
0 r 2 sin
0
The top is z = r.
(a)
V
=
Z
Z
2 sin
Z
Z
r
1dz rdr d =
0
0
0
0
Z
Z
8 3
=
r3 /3|2 sin d =
sin d
0
3 0
0
Z
18.075 Practice Test 1 for Exam 3
November 23, 2004
Justify your answers. Cross out what is not meant to be part of your solution.
Total number of points: 75.
I. 1. (5 pts) Find the region of convergence of the series
(x 1)n
.
n
n=0 (n + 1)
2. (5 pts) Fin
18.781 Problem Set 4 part 2
Thursday March 15, with the rest of Problem Set 4.
Collaboration is allowed and encouraged. However, your writeups should be your own, and you
must note on the front the names of the students you worked with.
Extensions will on
Subject 24-242. Logic 11. Homework due Thursday, April 29
A set A of natural numbers is said to be m-reducible (for "many-one reducible") to a set B just in
case there is a total Z functionf such that, for any n, n is in A if and only iffin) is in B.
A is
6.253: Convex Analysis and Optimization
Homework 1
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
(a) Let C be a nonempty subset of Rn , and let 1 and 2 be positive scalars. Show that if C is
convex, then ( 1 + 2 )C = 1 C + 2 C. Show by example
Homework 1
18.086
Spring 2006
for MONDAY 2/13/06
This homework is optionalif you hand it in, that will be recorded (and
a missed homework later will be treated generously) Thank you for patience
with the videotaping process !
MATLAB 1 Draw decent gures of
MASSACHUSETTS INSTITUTE O F TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control I1
Fall 2007
Quiz 2 Solution
Posted on Monday, December 3, 2007
1. The root locus of a feedback system with open-loop poles at 1 , 3 and openloop zeros