Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
Lecture Notes on The Mechanics of Elastic Solids
Volume 1: A Brief Review of Some Mathematical Preliminaries
Version 1.0
Rohan Abeyaratne Quentin Berg Professor of Mechanics Department of Mechanical Engineering MIT Copyright c Rohan Abeyaratne, 1987 All r
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Selected Topics in Mathematics with Applications
Final Exam Spring 2007
Every problem in the calculus of variations has a solution,
provided the word solution is suitably understood.
David Hilbert (18621943)
Work any 5 problems.
Pickup exam: 12:3
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Selected Topics in Mathematics with Applications
SOLUTIONS to the FINAL EXAM Spring 2007
Problem 1: Using rst principles (i.e. dont use some formula like d/dx(F/ )F/ =
0 but rather go through the steps of calculating F and simplifying it etc.) dete
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 1 (Inclass)
Spring 2007
1.5 hours
You may use the notes you took in class and any other handwritten notes in your own
handwriting. No other sources should be used.
Problem 1: Explain each of the following concepts precisely (ye
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 1
Spring 2007
SOLUTION
PROBLEM 1:
a) A vector space is a set V of elements called vectors together with operations of addition and
multiplication by a scalar, where these operations must have the following properties:
(A) Corres
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 2 (Take home)
Spring 2007
My education was dismal. I went to a school for mentally
disturbed teachers.
Woody Allen
INSTRUCTIONS:
Do not spend more than 4 hours.
Please give reasons justifying each (nontrivial) step in your c
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 2
Spring 2007
SOLUTION
Problem 1: Consider the set V of all 2 2 skewsymmetric matrices, i.e. matrices of the form
x=
0x
x 0
< x < .
,
(1.1)
a) Dene the addition of two vectors by
x+y =
0
x+y
x y
0
where
x=
0x
x 0
,
y=
0y
y 0
,
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 1 (In class)
Spring 2004
1.5 hours
You may use the notes you took in class but no other sources.
Please give reasons justifying each (nontrivial) step in your calculations.
Problem 1: Let R be a 3dimensional Euclidean vector sp
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
MECHANICAL 2.035

Spring 2007
2.035: Midterm Exam  Part 2 (Take home)
Spring 2004
Examinations are formidable even to the best prepared, for
the greatest fool may ask more than the wisest person may
answer.
Charles Caleb Colton (17801832)
INSTRUCTIONS:
Do not spend more than 3 ho