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 E
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 Hilb
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 t
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
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 op
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 g
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 vec
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.
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