15-122 Written Homework 3 Solutions
Page 1 of 4
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 3 Partial Solutions
In this homework assignment, we will work with specifying and implementing search in an
array. In lecture, we wor
15-122 Written Homework 1
Page 1 of 9
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 1 Solutions
Name:
Andrew ID:
Recitation:
The theory portion of this weeks homework will introduce you to the way we reason about
C0 code in 15-
15-122 Homework 2
Page 1 of 6
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 2
Due: Thursday, September 12, 2013 at 10pm
Name:
Andrew ID:
Recitation:
The written portion of this weeks homework will give you some practice working
15-122 Homework 2 Solutions
Page 1 of 3
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 2 Partial Solutions
The written portion of this weeks homework will give you some practice working with the
binary representation of integers
Math 225B: Dierential Geometry, Homework 3
Ian Coley
January 22, 2014
Problem 1.
Prove that every vector bundle over Euclidean space is trivial.
Solution.
We use the concept of a pullback bundle. Suppose that F0 , F1 : N M are smoothly
homotopic maps and
Math 225B: Dierential Geometry, Homework 6
Ian Coley
February 13, 2014
Problem 8.7.
Let be a 1-form on a manifold M . Suppose that
Show that is exact.
c
= 0 for every closed curve c in M .
Solution.
We claim that this condition is equivalent to the follo
Math 225B: Dierential Geometry, Homework 8
Ian Coley
February 26, 2014
Problem 11.1.
Find H k (S 1 S 1 ) by induction on the number n of factors.
Solution.
We claim that H k (T n ) = n . For the base case, we know that H 0 (S 1 ) = H 1 (S 1 ) = R, which
k
Math 225B: Dierential Geometry, Homework 2
Ian Coley
January 17, 2014
Problem 5.10
(a) Prove that
LX (f ) = Xf + f LX
LX [(Y )] = (LX )(Y ) + (LX Y ).
(b) Reformulate Proposition 8 with the denition
1
(LX Y )(p) = lim [(h Y )p Yp ].
h0 h
Solution.
(a) We
Math 225B: Dierential Geometry, Homework 4
Ian Coley
January 30, 2014
Problem 6.3.
(a) In the proof of Proposition 2, show that
f
xi
=
p
y i
.
f (p)
(b) Complete the proof of Proposition 2 by showing that if
n
i
,
y i
i
Y =
,
xi
i=1
so that
n
X=
i=1
i
i
i
15-122 Homework 3
Page 1 of 9
15-122: Principles of Imperative Computation, Spring 2013
Homework 3 Programming: DosLingos
Due: Monday, September 23, 2013 by 22:00
This week we will do three short exercises on searching and sorting arrays of integers and
s
Math 225B: Dierential Geometry, Final
Ian Coley
March 15, 2014
Problem Spring 2011, 1.
Show that if X is a smooth vector eld on a (smooth) manifold of dimension n and if Xp is
nonzero for some point of p, then there is a coordinate system dened in a neigh
15-122 Homework 5
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15-122: Principles of Imperative Computation, Spring 2013
Homework 5 Programming: Text Editor
Checkpoint: Monday, October 14, 2013 by 22:00
Due: Monday, October 21, 2013 by 22:00
For the programming portion of this weeks hom
15-122 Homework 4
Page 1 of 6
15-122: Principles of Imperative Computation, Spring 2013
Homework 4 Programming: Clac
Due: Monday, September 30, 2013 by 22:00
For the programming portion of this weeks homework, you will implement a small for a
TM
postx cla
15-122 Homework 1
Page 1 of 3
15-122: Principles of Imperative Computation, Spring 2013
Programming 1: Pixels
Due: Monday, September 9, 2013 by 22:00
This second programming assignment is designed to get you used to writing some preconditions and postcond
15-122 Homework 0
Page 1 of 4
15-122: Principles of Imperative Computation, Spring 2013
Programming 0: Scavenger hunt
Due: Monday, September 2, 2013 by 22:00
Welcome to 15-122! This zeroth programming homework is designed as a low-stress opportunity for y
Math 225B: Dierential Geometry, Homework 7
Ian Coley
February 20, 2014
Problem 8.17.
(a) Let M n and N m be oriented manifolds, and let and be an n-form and an m-form
with compact support, on M and N , respectively. We will orient M N by agreeing
that v1
/ Grad.java
/*
Grad is a subclass of Student - a simple example of subclassing.
-adds the state of yearsOnThesis
-overrides getStress() to provide a Grad specific version
*/
public class Grad extends Student cfw_
private int yearsOnThesis;
/*
Ctor takes a
Math 225B: Dierential Geometry, Homework 1
Ian Coley
January 10, 2014
Problem 3.12.
(a) Let Fp be the set of all C functions f : M R with f (p) = 0, and let : Fp R be
a linear operator with (f g) = 0 for all f, g Fp . Show that has a unique extension
to a
/ Student.java
/*
Demonstrates the most basic features of a class.
A student is defined by their current number of units.
There are standard get/set accessors for units.
The student responds to getStress() to report
their current stress level which is a f
Math 225B: Dierential Geometry, Homework 5
Ian Coley
February 6, 2014
Problem 7.8.
(a) Let 2 (V ). Show that there is a basis 1 , . . . , n of V such that
= (1 2 ) + + (2r1 2r ).
(b) Show that the r-fold wedge product is non-zero and decomposable, and th
15-122 Homework 4
Page 1 of 6
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 4
Due: Thursday, September 26, 2013 at 10pm
Name:
Andrew ID:
Recitation:
In this homework assignment, we will examine asymptotic complexity, searching
15-122 Homework 3
Page 1 of 8
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 3
Due: Thursday, September 19, 2013 at 10pm
Name:
Andrew ID:
Recitation:
In this homework assignment, we will work with specifying and implementing sea
15-122 Written Homework 11
Page 1 of 7
15-122 : Principles of Imperative Computation, Fall 2013
Written Homework 11 [Update 2]
Due: Thursday, November 21, 2013 by 10pm
Name:
Andrew ID:
Recitation:
The written portion of this weeks homework will give you s