Midterm 1
Math 227
Name
February 10th, 2010
Student Number
No books, notes or calculators are allowed.
Problem 1 (5 points): Let a and b be non-negative real numbers and let Ca,b be the helix given by the
parameterization
x (t) = a cos t i + a sin t j + b
The Pendulum
Model a pendulum by a mass m that is connected to a hinge by an idealized rod that
is massless and of xed length . Denote by the angle between the rod and vertical. The
d
dt
mg
forces acting on the mass are gravity, which has magnitude mg and
Stokes Theorem
The statement
F n dS
F dr =
S
S
provided
The curve S is the boundary of the surface S
The orientations of S and n obey the right hand rule
The vector eld F has continuous rst partial derivatives at every point of S
The proof Both integr
The Astroid
Imagine a ball of radius a/4 rolling around the inside of a circle of radius a. We
shall now nd the equation of the curve traced by a point P painted on the inner circle. You
can nd a java applet demonstrating this curve on the web at
http:/ww
Planetary Motion
with Corrections from General Relativity
Let r(t) be the position at time t of a planet (approximated by a point mass, m) in
orbit around a sun (also approximated by a point mass, M ) whose position is xed at the
origin. According to Newt
QUADRIC SURFACES
name
equation in
standard form
x = const
cross-section
y = const
cross-section
z = const
cross-section
plane
ax + by + cz = d
line
line
line
two lines
two lines
ellipse
elliptic
cylinder
x2
a2
+
y2
b2
=1
parabolic
cylinder
y = ax2
one lin
Poissons Equation
In these notes we shall nd a formula for the solution of Poissons equation
2
= 4
Here is a given (smooth) function and is the unknown function. In electrostatics, is the
charge density and is the electric potential. The main step in ndi
Flux Integral Example
Problem: Evaluate
F n dS where F = x4 + 2y 2 + z k, S is the half of the surface
S
1 2
1 2
2
4 x + 9 y + z = 1 with z 0 and n is the upward unit normal.
Solution 1. Parametrize the half-ellipsoid
x(, ) = 2 cos sin
y(, ) = 3 sin sin
Faradays law
Faradays Law is the following. Let S be an oriented surface with boundary C. Let
E and H be the (time dependent) electric and magnetic elds and dene
H n dS = magnetic ux through S
E dr = voltage around C
C
S
Then the voltage around C is the n
Flux Formulae
Parametrized Surfaces. If the surface is
x = x(, )
y = y(, )
z = z(, )
then, setting,
x y z
, ,
T =
we have
T =
x y z
, ,
dS = T T d d
T T
n=
T T
ndS = T T d d
Level Surfaces. If the surface is g(x, y, z) = 0, then
n=
dS =
ndS =
g
g
g
g
Mathematics 227
12 marks
Midterm February 5, 2014
Page 1 of 2
1. Let C be the curve at the intersection of the cylinder x2 + y 2 = 1 in R3 with the plane
z = ax for some a > 0. Parametrize the curve, and nd its curvature as a function of
the parameter. Is
MATH 227 SAMPLE MIDTERM 1 SOLUTIONS
Winter/Spring 2014
1. Calculate the line integrals:
x2 + y 2 ds, where x(t) = (t cos t, t sin t), 0 t 2.
We have x(t)2 + y(t)2 = t2 cos2 t + t2 sin2 t = t and
(a)
x
ds = x (t) =
(t sin t + cos t)2 + (t cos t + sin t)2 d
Vector Identities
Gradient
1.
(f + g) =
f+
g
2.
(cf ) = c f , for any constant c
3.
(f g) = f g + g f
4.
(f /g) = g f f g /g 2 at points x where g(x) = 0.
5.
(F G) = F (
G) (
F) G + (G
)F + ( F
Divergence
6.
(F + G) =
F+
7.
(cF) = c
8.
(f F) = f
9.
The Physical Signicance of div and curl
Consider a (possibly compressible) uid with velocity eld v(x, t). Pick any time t0
and a really tiny piece of the uid, which at time t0 is a cube with corners at
x0 + n1 (1) + n2 (2) + n3 (3)
e
e
e
(1)
e
n1 , n2 , n
CARTESIAN COORDINATES
z
dz
(x, y, z )
z
dy
y
x
dx
volume element dV = dx dy dz
y
x
z
z
y
y
x
z
x
z
y
x
x
surface of constant y
surface of constant x
y
surface of constant z
CYLINDRICAL COORDINATES
z
r
dr
dz
z
r
r d
y
volume element dV = r dr d dz
x
z
z
r
Midterm 2
Math 227
March 31st, 2010
Student Number
Name
No books, notes or calculators are allowed.
Problem 1 (5 points):
1. The vector eld
F = (2xy 3 + z 2 ) i + (x2 y 2 yz) j + (xz + y 2 ) k
is conservative. Find the values of the constants , , and .
1
1
Ant problem:
Suppose that four ants sit on the corners of a square with side length L. Let x1 , x2 , x3 , x4
be the position vectors of the four ants and suppose that each one travels with constant
speed in the direction of the next ant. That is, the di
S OLVE THE FOLLOWING PROBLEM :
On the ctional planet of Valinor, an elf and a dwarf start at the equator and travel at a
constant speed v0 leagues per day, always in a Northwesterly direction. The planet Valinor
is perfectly smooth and has a radius of R l
MATHEMATICS 317 Section 921
Final exam, July 27, 2012
Show all your work. Use back of page if necessary. Calculators are not allowed.
Last Name:
First Name:
UBC Student Number:
Score
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Questi
The University of British Columbia
Mathematics 317
Final Examination
13 April 2011
Time: 150 minutes
Full Name:
Student # :
Signature:
This Examination paper consists of 12 pages (including this one). Make sure you have all 12.
Instructions:
No extra mate
The University of British Columbia
Final Examination - April 26, 2008
Mathematics 317 Section 202
Instructors: Jim Bryan and Hendryk Pfeier
Closed book examination.
Name
Time: 3 hours
Signature
Student Number
Special Instructions:
Be sure that this exami
The University of British Columbia
Final Examination - April 21, 2010
Mathematics 227
Section 201
Instructor: Jim Bryan
Closed book examination
Time: 3 hours
Name
Signature
Student Number
Special Instructions:
- Be sure that this examination has 10 pages.
The University of British Columbia
Final Examination - April 19, 2007
Mathematics 317
Instructors: Jim Bryan and Hendryk Pfeier
Closed book examination
Name
Time: 3 hours
Signature
Student Number
Special Instructions:
Be sure that this examination has 14
Buoyancy
In these notes, we use the divergence theorem to show that when you immerse a body
in a uid the net eect of uid pressure acting on the surface of the body is a vertical force
(called the buoyant force) whose magnitude equals the weight of uid dis
Torque
Newtons law of motion says that the position r (t) of a single particle moving under the inuence of
a force F obeys mr (t) = F . Similarly, the positions ri (t), 1 i n, of a set of particles moving under the
inuence of forces Fi obey mri (t) = Fi ,
Circulation around a small circle
Let C be the circle which
is centered on r0
has radius
lies in the plane through r0 perpendicular to n
is oriented in the standard way with respect to n. Imagine standing on the circle
with your feet on the plane thr
The Chain Rule
The Problem
You already routinely use the one dimensional chain rule
d
dt f
x(t) =
df
dx
x(t)
dx
dt (t)
in doing computations like
d
dt
sin(t2 ) = cos(t2 )2t
In this example, f (x) = sin(x) and x(t) = t2 .
We now generalize the chain rule t
The Heat Equation
Let T (x, y, z, t) be the temperature at time t at the point (x, y, z) in some body. The
heat equation is the partial dierential equation that describes the ow of heat energy and
consequently the behaviour of T . We start by deriving the
Department of Mathematics
University of British Columbia
MATH 227 (Section 201) Final Exam
April 26, 2011, 8:30 AM - 11:00
Family Name:
Initials:
I.D. Number:
Signature:
CALCULATORS, NOTES OR BOOKS ARE NOT PERMITTED.
JUSTIFY ALL OF YOUR ANSWERS (except as
Integration on Manifolds
Manifolds
A manifold is a generalization of a surface. We shall give the denition shortly. These
notes are intended to provide a lightning fast introduction to integration on manifolds. For
a more thorough, but still elementary di