AMath 231
ASSIGNMENT # 2: Vector Fields
Spring 2011
Due on or before Wednesday, May 18th at 9:30am in drop box 6, slot 11 or 12 across from
MC4066.
1. Given that the Euclidean norm is dened as |x|2 = x x and that x(t) is a C 1 function,
show that
(a) (|x(
AMATH 231: Assignment 3 (Line Integrals)
Due Wednesday 5 October at the beginning of tutorial. Write your solutions clearly and concisely.
Marks will be deducted for poor presentation and incorrect notation.
1. Evaluate the line integral of the scalar eld
AMath 231
ASSIGNMENT # 3*: Line integrals, cons. elds
Spring 2011
Due on or before Wednesday, June 1st at 9:30am in drop box 6, slot 11 or 12 across from
MC4066.
*Note that this assignment is a 2-week assignment, and is double-weighted.
PS2 # 1 (domain is
AMATH 231 COURSE SUMMARY
Curves & Vector Fields
- Curves in Rn:
[Ch. 1]
g : [ a, b] R
g (t ) = ( g1 (t ), g 2 (t ),., g n (t )
n
(ie path of a particle through space with parameter time t)
- Expressing Equations in Parametric Form (above)
- Limits:
lim g
Assignment 7 - Stokess and Divergence theorem.
Due Noon March 04, 2016.
1. The surface integral of a scalar function can be given in a particularly
simple form when the surface is the graph of a function i.e., when
its equation is of the type z = f (x, y)
Assignment 10 - Fourier series.
Due Noon Monday March 28, 2016.
1. A function f (x) is defined in the interval 1 < x < 1 by,
0 1 < x < ,
1
< x < ,
f (x) =
2
0
< x < 1.
where 0 < < 1.
(a) Sketch the graph of f (x) and show that a Fourier series expansi
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Assignment 6 - Surfaces and surface integrals.
Due February 12, 2016.
~ Find an orthonormal basis set lying on the surface.
1. Sketch the following surfaces S.
(u, v) [0, ] [0, 2]
~ = (cos v sin u)i + (1 + sin v sin u)j + (cos u)k,
(a) S
(u, v) [1, 3] [
Assignment 5 - Circulation, Flux and Divergence theorem.
Due Noon February 05, 2016.
1. Compute the area enclosed by the hypocycloid of four cusps
~ : [0, 2] 7 R2 and ~ (t) = (a cos3 t, a sin3 t),
where a > 0 is a constant.
2. Let D be a piecewise C 1 ,
Assignment 4 - Conservative fields and Greens theorem.
Due Noon January 29, 2016.
1. A force F~ acts on a particle that is moving in the plane along the semi-circle ~ (t) = ( cos t)i+(sin t)j; t [0, ].
R
Find the work W = ~ F~ d~s when
p
(a) F~ = x2 + y 2
S. .. nmmann-mi/y"
University of Waterloo, Waterloo, ON
Faculty of Mathematics
Department of Applied Mathematics
AMATH 231
Calculus 4
Midterm Test Fall Term 2005
Time: 1 -21- hours
NO AIDS
Instructor: J. Wainwright
[16]\/ 1. Explain the term eld li
AMath 231
ASSIGNMENT # 1: Review and Curves
Spring 2011
Due on or before Wednesday, May 11th at 9:30am in drop box 6, slot 11 or 12 across from
MC4066.
Comment: The rst two problems are review. If you took Math 247 (Advanced Calculus 3)
then you might not
Problem Set 5:
Fourier series and Fourier transforms
(a) Fourier Series
1. Consider the full-wave rectication function f dened by
f(:1:) =| sins; |, for 7r < a: < 7r.
a) Find the Fourier series of f.
b) Sketch the graph of the function to which the series
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Question 1', 17 Marks.
Consider the ow of a uid in two dimensions across a curve 0.
Suppose the uid has density p and velocity vector eld v, and let
the curve be given by
g(t) = (x(t),y(t), t 6 [(1,1)], Where g(t) is piecewise CI.
Carefully derive
Question 1, ll marks.
Consider the function/(x) = x. 0 5 x 5 I., for some: rcol number L > 0.
I) Find the Fourier cosine series Uf/(X). and sketch the function to which it converges on R.
4.
r l k
. - _L - 1 - 2 J: : 4 /mq
( 5 ffi) _ L (I- , 3" 6pr Va _
Assignment 1 - Review material.
Not to be handed in
Linear algebra
are the unit basis vectors of the Cartesian coordinate system.
Notation: (i, j, k)
~ of a particle of mass m is defined by H
~ = ~r (m~v ), where ~v =
1. The angular momentum vector H
~
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