MATH 227 : Calculus 3 for Honours Physics
Fall 2016 - Assignment #1
Instructions:
The problems on this assignment cover lines and planes in <3 , and an introduction
to vector curves and motion in <3 , as discussed in Sections 10.1, 10.5, 10.7, and 10.9
o
MATH 227: CALCULUS 3 for HONOURS PHYSICS
FALL 2016
Lectures: 10:30 - 11:20 a.m. MWF in MC 4021
Tutorial: 5:30 a.m. - 6:20 p.m. in MC 4020
Instructor: B. J. Marshman
Email: bjmarshm@uwaterloo.ca
Phone: 519-888-4567, ext. 33762
Office: MC 6444D
Office hours
Math 228: Assignment 1
due Friday, September 25, 2015 at 1:30pm
1. Sketch the direction eld of the dierential equation
y0 =
for t
y(y
2)
0 and y 2 [ 2, 5]. What are the equilibrium solutions?
2. Sketch the direction eld of the dierential equation
y0 = x
x
Math 228: Assignment 2
due Friday, October 2, 2015 at 1:30pm
1. Solve the following rst-order linear dierential equations (or IVP, as it may be).
(a) y 0 = t + 5y
(b) 2xy 0 + y = 6x, with the conditions x > 0 and y(4) = 20
(c) y 0 + y = 5 sin(2t)
(d) y 0
Math 228: Assignment 3
due Friday, October 9, 2015 at 1:30pm
Note: For exact equations, solve means solve implicitly.
1. Determine whether the equation is exact. If so, solve it.
(a) (2x + 3) + (2y 2)y = 0 (b) ( y + 6t) + (ln(t) 2)y = 0, t > 0
t
2. Prove
Math 228: Assignment 4
due Friday, October 16, 2015 at 1:30pm
1. For each of the functions f (t, y) identify regions in the (t, y)-plane where the initial value
problem y 0 = f (t, y), y(t0 ) = y0 has a unique solution:
(a) f (t, y) =
2 + 3y 2
t2 5t + 6
(
Math 228: Assignment 1 Solutions
due Friday, September 25, 2015 at 1:30pm
1. Sketch the direction eld of the dierential equation
y0 =
for t
y(y
2)
0 and y 2 [ 2, 5]. What are the equilibrium solutions?
Solution:
Equilibrium solutions are y 0 and y 2.
2. S
Math 228: Assignment 2
due Friday, October 2, 2015 at 1:30pm
1. Solve the following rst-order linear dierential equations (or IVP, as it may be).
(a) y = t + 5y
(b) 2xy + y = 6x, with the conditions x > 0 and y(4) = 20
(c) y + y = 5 sin(2t)
(d) y y = 2te2
MATH 228: Dierential Equations for Physics and Chemistry
Fall 2015 - LEC 001
MWF 11:30-12:20 in MC 2017
Instructor: Andrew Beltaos
Email: abeltaos@uwaterloo.ca
Phone: 519-888-4567, ext. 33508
Oce:
MC 6503
Oce hrs: Wed 10:0011:00 and Thurs 10:0012:00
or by
MATH 227: CALCULUS 3 for HONOURS PHYSICS
FALL 2014
Lectures: 10:30 - 11:20 a.m. MWF in MC 4021
Tutorial: 5:30 a.m. - 6:20 p.m. M in RCH 301
Instructor: B. J. Marshman
Email: bjmarshm@uwaterloo.ca
Phone: 519-888-4567, ext. 33762
Oce: MC 6444D
Oce hours: Mo
1
MATH 227 : Calculus 3 for Honours Physics
Fall 2015 - Assignment #1
Instructions:
The problems on this assignment cover lines and planes in 3 , and an introduction
to vector curves in 3 , as discussed in Sections 10.1, 10.5, and 10.7 of your text. If
y
1
MATH 227 : Calculus 3 for Honours Physics
Fall 2015 - Assignment #2
Instructions:
The problems on this assignment cover further work on vector curves in 3 , motion,
and line integrals of scalar and vector elds, as discussed in Sections 10.7, 10.8, 10.9
1
MATH 227 : Calculus 3 for Honours Physics
Fall 2015 - Assignment #4
Instructions:
The problems on this assignment cover double integrals in Cartesian and polar
coordinates, as discussed in Sections 12.1-12.4 of your text.
Appropriately labelled diagram
1
MATH 227 : Calculus 3 for Honours Physics
Fall 2015 - Assignment #5
Instructions:
The problems on this assignment cover triple integrals in Cartesian, cylindrical
polar, and spherical polar coordinates, as discussed in Sections 12.5-12.7 of your
text.
1
MATH 227 : Calculus 3 for Honours Physics
Fall 2014 - Assignment #3
Instructions:
The problems on this assignment cover surfaces in <3 , expressed in Cartesian,
cylindrical polar, and spherical polar coordinates, as discussed in Sections 10.6,
12.6, an
Lecture 4
Level Sets/Contours
(Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 14.1)
You are no doubt familiar with the idea of contour plots from geography topographic maps
of a region in which points of identical elevation
Lecture 18
Double Integrals (contd)
Electrostatic eld near an innite at charged plate
Consider a thin, at plate of innite size that is charged, with constant charge density (in appropriate
units of charge/unit area). The problem is to nd the electrostatic
Lecture 7
Gradient and directional derivative (contd)
In the previous lecture, we showed that the rate of change of a function f (x, y) in the direction of a
vector u, called the directional derivative of f at a in the direction u, is simply the dot produ
THE MOBIUS STRIP AND STOKES' THEOREM
1. Stokes' Theorem Let us recall Stokes' Theorem: Theorem 1.1. Let M be an oriented surface in R3 with boundary given by the closed curve , with orientation induced from that of M (by the right hand rule.) Let F(x, y,
LINES AND PLANES IN R3
In this handout we will summarize the properties of the dot product and cross
product and use them to present various descriptions of lines and planes in three
dimensional space.
1. The Geometry of the Dot Product
Let v =< v1 , v2 ,
KEPLERS LAWS OF PLANETARY MOTION
1. Introduction
We are now in a position to apply what we have learned about the cross product
and vector valued functions to derive Keplers Laws of planetary motion. These
laws were empirically determined in the early 160
PHYSICAL INTERPRETATIONS OF CURL AND DIVERGENCE
1. Physical Interpretation of the Curl
Let F(x, y, z) = (P (x, y, z), Q(x, y, z), R(x, y, z) be a vector eld. We can think
of F as representing the velocity eld of some uid in space. We want to give
a physic
THE VOLUME OF AN n-DIMENSIONAL SPHERE IN Rn+1
1. Introduction
A circle of radius R can be thought of as the set of points (x, y) in R2 that are
a distance R from the origin. The equation is written x2 + y 2 = R2 . The region it
encloses has a size (area)
THE DIFFERENT KINDS OF INTEGRALS
We have seen many dierent kinds of integrals in this course. Let us review them
and the relations between them. All of them involve integrating functions or vector
elds over geometric objects of dierent dimensions:
The bas
INTEGRATION AND DENSITIES
1. Introduction
In this course we study many dierent kinds of integrals. However, all of them
share something in common: All integrals are computed to measure some quantity
which is additive. Let us explain what this means. If we
CURVES AND SURFACES IN R3
Let us intuitively dene a curve as a geometric object which, on a very small
scale, looks like a straight line. We need to make this more precise. The simplest
example of a curve is the graph of a function of one variable, y = f
Lecture 15
Multiple Integration
(Relevant section from Stewart, Section 15.1)
We now turn to the integration of scalar-valued functions f : Rn R, i.e., f (x1 , x2 , , cn ), over
regions in Rn . The need to perform such integrations is common in Physics. F
Lecture 27
Line integrals: Integration along curves in Rn
(Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 16.2, pp. 1034-1041)
In this section, we shall be integrating scalar-valued functions f (x, y, z) and vector-valued f