MAT2705 22S homework and daily class log
Jump to current date!
[where @ is
located]
Daily lectures and your homework will appear here each day as it is assigned, including some PDF
and/or Maple problem solution files or PDF notes, with occasional links to
some MAPLE worksheets when helpful to illustrate some points where technology can be
useful. [There are 56 class days in the semester, 4
each normal week, numbered consecutively below and labeled
by the (first initial of the) day of the week.]
It is your responsibility to check homework here since nmost but not all homework
exercises are online. Homework hints are also found here. (Put a favorite in your
browser to the class homepage.) You are responsible for any hyperlinked
material here as well as requesting any handouts or returned tests or quizzes from classes
you missed. Homework is understood to be done by the
next class meeting (unless that class is a test, in which case the
homework is due the following class meeting), but the online deadline will be
stated as midnight of the due day so that you have time in class or after to ask
questions that you did not submit via the Ask Your Instructor
online tool. You may correct submitted homework after the deadline without
having to request an extension.
[[Homework
problems surrounded by double square brackets
[[...]] are not in the online homework
system but are important to do.]]
Read:
HOMEWORK ADVICE;
quiz BlackBoard access/submission
-
M: (January 10)
DURING CLASS:
Lecture
Notes 1.1a: Differential Equations: how to state them and "check" a solution;
summary handout: odecheck.pdf
New to Maple 3 minute video [and we will check your BlackBoard
access to the textbook portal]
AFTER CLASS (THIS IS THE HOMEWORK):
1) Log on to My Nova, choose the Class Schedule with Photos,
view fellow students.
2) Go to
BlackBoard and look at the class portal and Grade book
for our course: you
will find all your Quiz, Test and Homework grades here during the
semester once there is something to post. Everything we do except for
quizzes, tests and grades will take place through our course website
although you will access our e-text portal inside BlackBoard to avoid an
extra login.
3)
Download Maple
2021
for Windows or Mac
if you haven't already done so and install it on your laptop when you get a
chance (it takes about 15 minutes). If
you have any trouble, email me with an explanation of the errors.
You are expected to be able to use Maple on your laptop when
needed. We will develop the
experience as we go.
No
previous experience is assumed.
4) Enter the e-text MyLab Math portal
if you have not already done so, as explained in the
welcome email message!
5)
Homework Problems:
1.1: 3, 5,
13, 33 online (only a few problems so you can check out our class website and read about the
course rules, advice, bob FAQ, etc, respond with your email).
It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Optional: get acquainted with
Maple clickable calculus DE entry and "odetest" for problems 3,5,13 [even
33!]
[memorize!: "A is proportional to B" means "A = k
B" where k is some constant,
independent of A and B]
Proportionality statements
must be converted to equalities with a
constant of proportionality introduced:
y is
proportional to x: y ∝ x means y
= k x . [y is a multiple of x]
y is
inversely proportional to x: y ∝ 1/x
means y = k/x .
y is
inversely proportional to the square of x: y ∝ 1/x2
means y = k/x2
6) By the end of the week, reply to my
welcome e-mail from your OFFICIAL
Villanova e-mail account (which identifies you with your full name),
telling about your last math courses, your comfort level with graphing
calculators and computers and math itself, how much
experience you have with Maple if any (and Mathcad if appropriate) so far, why you chose your major, etc,
anything you want to let me know about yourself that will give me more of an
idea about you as a person. [For example, I like to do
humorous sketching.
and cooking.] Tell me
what your previous math course was named (if at VU: Mat1500 = Calc 1, Mat1505 = Calc
2, Mat2500 = Calc3).
[In ALL email to me, try to include the string "mat2705" somewhere in the
subject heading if you want me to read it quickly. I filter my email.]
-
T:
Lecture
Notes 1.1b: Differential Equations: initial conditions
[extra:
initial data: what's the deal?];
1:1: 8, 23 [see
Maple plot (execute worksheet by clicking on the !!! icon on the
toolbar)];
formulating DEs: 27 [approach like 29, make a generic diagram like
this one to calculate the slope of the
tangent line and set it equal to the derivative
.mw],
31, 35, 43; perhaps in class together:
[[29. Hint: recall perp lines have slopes which are negative
reciprocals, the normal line is perpendicular to the tangent line and passes
through the same point on the curve; make a diagram of the given point (0,1), a "generic" function graph curve
whose normal at a point (x,y) on the curve passes through
(0,1), and draw in the connecting normal line segment between these two
points and the perpendicular tangent
line, then compute the slope of the normal line from the two points, and then from the
negative reciprocal of the derivative value, finally equate the two
to get the DE: mw ]].
Is your dorm abbreviation missing from
bob's dorm list?
Fill out dorm and cell phone info on signup sheet today.
-
W: read the handout: algebra/calc background
sheet [online only: more rules of
algebra NOT!];
Lecture
Notes 1.2: First order DEs independent of the unknown [some
redundancy with previous lecture, plus two word problems];
[lunar
landing example: mw] [Swimmer
problem example: mw]
1.2 (antiderivatives as DEs): 1, 5, 15,
21 [Hint: write piecewise linear function from graph: v1(t) for first expression,
v2(t) for second expression, solve first IVP
for x1(0) = 0, second IVP for x2(4) = x1(4);
example: pdf,
mw],
25 (like lunar landing problem,
see example), 35,
41 (boundary value problem
mw; first
solve bomb drop IVP, evaluate time when it arrives
at y = h (target height),
then solve projectile IVP with unknown velocity,
then impose that it reach height h at
the given time)
[[43, convert
final answer to appropriate units!]].
-
F: Quiz 1 paper copy handed out in class (read
Test/Quiz rules;
submission instructions):
due in
BlackBoard Sun 11:59pm,
[optional: hand in paper copy in class Monday for bob's feedback];
if time, more discussion of 1.2
river crossing problem;
Lecture
Notes 1.3: First order DEs; Direction (= slope) fields and complications
[directionfields.mw] [ode1-complications.mw]
1.3 direction
fields:
3, 11, 15;
[[in
class drawing exercise: 1.3:
3:
for fun hand draw in all the curves with a pencil on the full page paper
printout:
projectable image;
bob's attempt;
variation ]]
Office hours; Course "Syllabus"
now online.
WEEK 2:
M: MLK Day no classes.
-
T:
Lecture Notes 1.4a: Separable first order DEs
[
example
1: implicit soln, other
complications visualized].
1.4 (separable DEs): 1,
4,
9 (see mw!),
25, 27, 29.
-
W:
Lecture Notes 1.4b: Separable first order DEs
handout on
exponential behavior/ characteristic time
[how to
plot?] [cooking
roast in oven remarks]
[read this worksheet
explicit plot example];
revisit
1.3.3 for next quiz;
1.4: 33,
35, 41
[hint: first we need to evaluate the fraction f of the total which
is only U],
45 [if your cell phone were waterproof: "Can you hear me now?"
attentuation of signal--characteristic length],
49 [cooling problem].
-
F: Quiz 2 (separable DEs and
directionfields, see Quiz 2 in archive);
1.4:
69 [hanging cable problem; hyperbolic functions are important in DEs! [wiki]];
together in class: "fun" with Newton's law of cooling:
[[65.
CSI problem; same as the in lecture example and problem 49 but at
end we want to find a previous time instead of a subsequent time]].
[see bob's hand solution,
mw]
No office hour today, I have to do a 3 hour implicit bias workshop!
WEEK 3[-1]:
-
M:
Lecture Notes 1.5a: Linear first order DEs
online handout: recipe for first order linear DE
1.5: 2, 5, 8, 21, 24;
*[be sure you can check your solutions by checking at least one of these problems, both the general solution and the initial
value problem solution with the dsolve
template].
For future reference:
> deq := y ' = x y
[space implies multiplication]
> sol:=dsolve(deq, y(x))
> solinit := dsolve({deq, y(0)=1}, y(x))
[or enter DE plus IC separated by a comma, right click on output,
choose Solve DE, for
y(x), or Solve DE interactively;
rightclick "Simplify, Simplify"
or "Simplify, Symbolic"
(with radicals) may be necessary to simplify the result]
> y ' = x y , y(0)=1
Use function notation to change independent variable:
> y '(t) = t y(t) , y(0)
= 1
-
T:
Lecture Notes 1.5b: Linear first order DEs:complications
[accumulation functions]
[Khan]
1.5: 15, 24, 27; 30;
[[read
this worksheet 29:
solutions defined by a definite integral are area accumulation functions . Use the fact that the derivative of
the function defined by the integral is the integrand including all
multiplicative factors]];
[[41
in class activity: epc4-1-5-41.htm
]]
remarks
on solving word problems
completely in terms of arbitrary parameters:
cooling.mw.
-
W:
Lecture Notes 1.5c: Linear first order DEs:
Mixing Problems [Maple
template]
[these mixing tank problems are an example of developing and solving a differential equation
that models a physical situation, and one where we have some intuition];
Solute (salt) plus solvent (water) makes a
solution (salt water!); concentration of
solute in solution = ratio of solute to solution;
1.5: Use
Eq. 18 in the book or the boxed equation in the lecture notes, check with
Maple template:
respectively constant, decreasing, increasing volumes: 33, 36, 37;
another constant volume lake application: 45 [solution
worksheets].
Note: we are skipping 1.6: exact DE
etc. These are less important, and Maple can solve
these when needed anyway. A similar integrating factor technique works for
exact equations but
where the independent and dependent variables are on an equal footing.
-
F: Quiz 3 on linear 1st order DEs;
1.R
(review problems are not online!): in class we classify the odd problems
1-35 but don't solve them;
[[ solve 25 [it can be done in two ways by expanding out the square on the RHS
before integrating with integration constant C or by using a u-substitution
with integration constant K; express K in terms of C as given in the book supplied answers by
combining the two terms in the K solution [identity for expanding out (x+1)3 !] and then comparing with the C
solution];
solve 35 in two ways and compare the results---different
constants enter your expressions, show how they agree]].
WEEK
4[-1]: Chapter 2 is just applications of chapter 1. Test Tuesday week 5 in
class on chapter 1
-
M: Lecture Notes
2.1a: First order DEs: Logistic Equation
handout on solution of
logistic DEQ
[Maple: characteristic time
/shape, directionfield,
integral formula,
24];
2.1 (logistic DE):
5 (resolve the DE this one time following the steps in the Lecture Notes),
6
(reverse S curve, choose a horizontal window to show full S curve based on
plus/minus 5 times characteristic time,
answer: >
dsolve({x'(t) =3
x(t) (x(t)-5),
x(0)=2},x(t))
(copy and paste into the input line, rightclick choose RHS, then
from the CSM
choose Plotbuilder, choose horizontal window based on characteristic
time (denominator inside exponential), make sure you then see the S-curve without wasting too much window
where nothing is happening);
17 (see Maple worksheet for setup
problem 15);
21 (initial value of population and its derivative given
instead of population at two times, need to set k and P(0)
with two conditions),
23 [Note dx/dt = 0.004
x (200-x) so k = 0.004,
M=200 for logistic curve
solution formula],
29.
Quiz 3 answer key now online.
-
T:
Lecture Notes 2.1b: First order DEs: More models, Separable: f(y)
2.1 (other population models):
handout on DE's that don't involve the ind
var explicitly; cubic
example;
11 ["inversely propto sqrt": use β = k1/P^(1/2), δ =
k2/P^(1/2) in eq.(1),
beta and delta are fractional
(logarithmic derivative) birth/death rates;
this leads to P' ~ P^(1/2)
soln],
13 [P' ~ P^2 example],
[[30, 31 [ans: find βo = 0.3 from condition dP/dt(0) = 3x10^5
using the DE at t =0, then use solution
α=0.3915 from second condition (solve this yourself or at least check numerically that it satisfies the 6
month condition) to find the limiting population as t goes to infinity (i.e.,
neglect decaying exponential)]].
39 [oscillating growth].
-
W:
Lecture Notes 2.3: First order DEs: acceleration models
2.3 (acceleration-velocity models: air resistance):
air resistance handout
(example of a piecewise defined DE and solution and the importance of dimensionless
variables);
[optional reading to show what is possible:
comparison
of linear, quadratic cases; numerical solution
for any power];
2;3: 1, 3,
9 [remember weight is mg, so mg = 32000 lb determines
mass
m = 1000 in USA "slug" units, convert final speed to mph
for interpretation!],
10
parachute problem: falling from rest in linear model but with piecewise
resistance (air, then parachute in air),
17 [v > 0 falling slowing down
to stop in quadratic model (tan phase), plug in numbers,
more context],
19 [constant thrust = reverse gravity!];
[[optional reading:
22 this is the tanh case: vterminal = 20.7 ft/s, t
= 8 min 5 s].[soln,
.mw]]]
MLRC 2 minute video for students, Spring 2022:
https://vums-web.villanova.edu/Mediasite/Play/8fbfd5b1a2564af285f480e7f459956b1d
-
F: No quiz;
Lecture Notes 2.4: First order DEs: numerical DE solving
with Euler's method
[Euler
template for HW; explanation of
algorithm]
2.4 (Euler numerical solution):
1, 27 [Maple cannot solve this exactly.]
[[read
29 end of worksheet]].
WEEK 5[-1]: Test 1 in
class Tuesday: one separable DE, one linear DE
Detour into linear
algebra:
-
M:
Lecture Notes 3.1: linear systems of equations: elimination, DEs;
[inconsistent 3x3 system; DEs
lead to linear systems];
why LinAlg with DE?;
[maple dsolve and DEplot for 2x2 systems
of DEQs];
Read 3.1 on linear systems (review of high school topic solving 2 or 3 linear equations in
2 or 3 variables),
do
3.1: 3, 5, 7; 9, 23, 25.
-
T: Test 1: one separable DE, one linear DE.
Read
Test Instructions;
Remember bob is trusting you to observe
our Villanova honor system.
-
W: 3.2:
Lecture Notes 3.2: matrices and row reduction "elimination";
handout on
RREF (Reduced Row Echelon Form, section 3.3)
[see bob's examples 2,
3 using Maple];
3.2: 1, 5, 7, 9 [these are partially reduced, only requiring successive
backsubstitution to solve, you can do this on paper];
11,13,15,
[you may do these by hand on
paper or use step-by-step row ops with the MAPLE Tutor; we ALWAYS want a
complete reduction] ;
you must learn a technology method since this is insane to do
by hand after the first few simple examples;
23 [matrix with a parameter, reduction
depends on value];
in class
we reduce two matrices in this Maple file
using the LinearSolveTutor.
-
F: Quiz 4 (see 20F: quiz 5);
3.3:
Lecture Notes 3.3: Row reduction solution of linear systems;
handout on
solving linear systems example
[.mw];
we always want to do a full [Gauss-Jordan] reduction, not the partial Gauss
elimination reduction also described by the textbook;
the Matrix
palette inserts a matrix of a given number of rows and columns;tab between
entries;
3.3: 1, 3, 9, 11, 14, 16, 17, 19, 23 (= 3.2.13), 29 (=3.2.19)
You can use this template to invoke the Tutors and
the reduction and solving commands.
WEEK 6[-1]: Happy Valentines Day!
hearts with Maple
check online grades to make sure I did not
misenter any test grades; answer key
is now online
-
M:
Lecture Notes 3.3: Balancing Chemical Reaction Application (short
lecture); [online balancing]
In class fun: Application:
chemical reaction problem.
(what can you
find out on the web about interpreting this chemical reaction?
chem site reaction balancer
gives balanced reactions but
no description, this one of our exercise turns out to be interesting).
word of the day (semester really): can you say "homogeneous"?
Get used to this word, it will be used the rest of
the semester. It refers to any linear equation not containing any terms
which do not have either the unknowns or any of their derivatives present,
i.e., if in standard form with all the terms involving only the unknowns on
the left hand side of the linear equation, the right hand side is ZERO. [nonhomogeneous
means the RHS is nonzero]
-
T:
Lecture Notes 3.4: Matrix operations [finally matrix multiplication: handout];
3.4: 1, 5, 7, 9, 13, 15, 17, 19;
[[for the mathematically curious read:
29 etc about powers
of square matrices]];
Matrix algebra is easy in Maple
[see this worksheet for how to do Matrix stuff in Maple].
-
W: Lecture Notes 3.5: Matrix inverses; [Maple
inverse template: mw]; [2x2
inverse! mw,
pdf]
3.5: 1,
5, 7, 9, use Maple to do row reductions:
11, 19,
23 [Maple template; this is
a way of solving 3 linear systems with same coefficient matrix
simultaneously, as in the alternative
derivation of the matrix inverse];
30.
Matrix multiplication and matrix inverse, determinant, transpose [or
use context sensitive menu and use Standard Operations menu]
> A B [note space between
symbols to imply multiplication]
> A-1
> |A| [absolute
value sign gives determinant, or just use context sensitive menu]
Memorize:
.
Switch diagonal entries. Change sign off-diagonal entries. Divide by
determinant.
Always check inverse in Maple if you are not good at remembering this.
-
F: Quiz 5;
3.6:
Lecture Notes 3.6: Matrix determinants;
[forget minors, cofactors,
forget Cramer's rule (except remember what it is when old-fashioned texts
refer to it), use Maple to evaluate determinants; we only need row
reduction evaluation to understand determinants];
3.6:
7, 9, 10, 13, 15, 17, 25, 29.
[[7*use
this example to record your Gaussian elimination
steps in reducing this to triangular form and check det value against your
result]].
Optional.
1)
Explanation of why determinants are important for measuring area and volume
etc in higher dimensions. [Calc 3 stuff]
2)
Why
the transpose?
[to combine vectors we arrange them into rows of matrix,
the transpose maps columns to rows, rows back to columns];
WEEK
7[-1]:
-
M:
Lecture Notes
4.1: Linear independence;
R^n spaces and linear
independence of a set of vectors [please read this motivating
example];
now we look at linear system coefficient matrices
A as collections of columns A = < C1| ... |Cn >,
matrix multiplication on the right by a column of
coefficients as taking linear combinations of those columns,
A x = x1 C1 + ... + xn Cn
and solving homogeneous linear systems of equations A x = 0 as
looking for linear relationships among those columns (nonzero solns=linear
rels) ;
4.1: 5, 7; 13; 15, 17; 19, 23; 29, 31, 33
[use technology in
both 1) evaluating the determinant and 2) context sensitive
menu for the ReducedRowEchelonForm
needed
for these problems];
-
T: Lecture Notes
4.2: Vector spaces and subspaces
(homogenous linear systems define
subspaces!);
study the handout on solving linear systems revisited
[remember the original one: solving linear systems example];
[note only a set of vectors x resulting from the solution of a linear homogeneous condition
A x = 0 can be a subspace of a vector
space; interpretation: lines, planes, ..., hyperplanes through the origin];
4.2: 1, 3, 7, 11; 15, 21.
Optional online note: the interpretation of solving linear
homogeneous systems of equations: A x = 0
[visualize
the vectors from this note].
-
W:
Lecture Notes 4.3: Span of a set of vectors;
so far: linear
independence (A x = 0),
express a vector as a linear combination (A x =
b)
[or does b belong to a linear relationship with
the columns of A?];
now give a name to the subspace of all linear combinations of a set of
vectors;
4.3: 1, 3, 5, 7 [this is exactly the case
for the row reduction basis of the solution space of a homogeneous linear
system];
9, 13; 17, 21.
-
F: Quiz 6
due Monday midnight after break;
Lecture Notes 4.4:
Bases;
online note
on
linear combinations, forwards and
backwards [maple to visualize]
4.4: 1, 3, 5, 7;
9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of
parameters, repeat for next 4 problems reducing first],
13; 15, 17, 25 [use technology for all HW row reductions and determinants
(but 2x2 dets are easy by hand!)].
Spring Break.
enjoy and be safe.
WEEK 8[-1]:
-
M: plane grid workshop [Maple version
of the example][printboth]
0) bob explains the
coordinate grid handout example made for the new basis {<2,1>,<1,3>} of the
plane.
Then in pairs or trios, you do the following each separately,
but discussing and critiquing each other's work:
1) Using this example, graphically find the new coordinates of the
point (4,7) reading off the grid. Then find the old coordinates for the
point whose new coordinates are (2,2). Confirm your graphical read outs by the corresponding matrix multiplications.
Then using the above handout as a guide, fill in each work sheet and confirm
that your graphical read outs agree with the corresponding matrix
calculations.
2)
Worksheet 1: new basis b1=
<1,1>, b2= <-2,1> .
Using a straight edge (piece of paper?) draw these
arrows at the origin, and then replicate then tip to tail along the new
coordinate axes, marking off the new tickmarks and reproduce the grid
corresponding to -2..2 in the new coordinate tickmarks.
Graphically represent the position vector of the point (x1, x2) =(-5,1) and
find its new coordinates ( y1, y2 )
using the grid. Similarly draw in the position vector of the point whose new
coordinates are (y1, y2) = (2,-1) and
read off the old coordinates (x1, x2).
3)
Worksheet 2: new basis b1=
<3,-1>, b2= <2,2> .
Using a straight edge (piece of paper?) draw these
arrows at the origin, and then replicate then tip to tail along the new
coordinate axes, marking off the new tickmarks and reproduce the grid
corresponding to -2..2 in the new coordinate tickmarks.
Graphically represent the position vector of the point (x1, x2) =(-2,6) and
find its new coordinates ( y1, y2 )
using the grid. Similarly draw in the position vector of the point whose new
coordinates are (y1, y2) = (2,-1) and
read off the old coordinates (x1, x2).
In each case note how the numerators of the columns of the inverse basis
changing matrix produce the denominator times the old basis vectors,
confirming that these columns are the new coordinates of the standard basis
vectors i and j .
Hand these in tomorrow in class.
Note sections 4.5, 4.6 are not
in the syllabus, covering topics appropriate for an advanced class in linear
algebra.
-
T: 4.7 Lecture Notes 4.7:
Vector spaces of functions;
bridge to vector spaces
whose elements are functions.
for test 2 review (in Tuesday class in
one week):
handout summarizing
linear vocabulary for sets of vectors;
handout summarizing linear system vocabulary
for linear systems of equations
Part 2 of course begins: transition back to DEs:
read the 4.7 subsection (p.279)
on function spaces and then examples 3, 6, 7, 8;
if you are curious, example 5 illustrates the partial fraction decomposition
needed in engineering integration;
read the worksheet on the
vector space of (at most) quadratic functions [quadratics.mw,
pdf];
4.7: 9. 11, 14 ( c1 f1(x)+ c2 f2(x)
= 0, set x = 0 to get one condition on the two coefficients,
set x = 1 to get a second, then solve 2x2 system),
15 [Hint: solve c1(1 + x) + c2(1 - x) + c3(1 - x2) = 0, for unknowns c1,c2,c3;
are there nonzero solutions? if not, these are linearly independent polynomials, in terms of which any quadratic
expression in x can be written; note that any two (nonzero!)
functions of x that are not proportional are automatically linearly
independent],
18 [same approach].
Optional Reading: Why do we care
about linear coordinate grids in the plane?
-
W:
Lecture Notes
5.1a:
2nd order linear DEs: an intro;
read 5.1 up to the subsection on
linear independence;
Memorize: y ' = k y < -- > y = C e k x
y '' + ω2 y = 0 < -- > y =
C1 cos(ω x) + C2 sin(ω x)
[when x is a time variable, "omega" = ω is the angular frequency "radians per time unit" as opposed to
just "frequency" f =
ω/(2π ) or "revolutions or cycles per time unit" as in physics, but in our class we
will just say "frequency" for ω, assumed to be expressed in radians per time
unit or converted to revolutions per time unit as convenient;
see also
damped
harmonic oscillators and
RLC circuits, and
Hertz
(computers now have GHz clock speeds),
not just a rental
car company but "cycles per second"]
5.1:
1, 3, 9, 11, 17.
-
F: No quiz;
check out answer key for Quiz 6 online;
5.1: W: Lecture Notes
5.1b:
constant coefficient 2nd order linear homogeneous DEs;
repeated root plot
[problem 49 worked here is useful for working with exponentials];
5.1[from linearly independent subsection]: 33, 35, 39.
WEEK [9]:
-
M:
Review for Test 2; [16F
test 2 (answer key)];
What's with this coordinate
grid stuff?
[grid exercise
solutions (scroll to bottom)]
-
T: Test 2 on chapters 3, 4. Maple may be used for
all row reductions and matrix inverses and determinants without showing
details, but do show matrix multiplication explicitly.
-
W: Lecture Notes
5.2:
The Wronskian and higher order constant coefficient 2nd order linear DEs;
online handout:
wronskian and higher order constant coefficient linear homogeneous DEs;
[Maple Wronskian]
5.2: 1, 7,
13, 17, 23.
-
F: Quiz 7
(like 18F:quiz7 real exponentials only) thru 5.1 (no homework from text);
Lecture Notes
5.3.0: Complex arithmetic and complex exponentials
handout on complex
arithmetic, exponentials
[Maple
commands; the complex number i is
uppercase I in Maple].
WEEK 10[-1]:
-
M:
Test 2 back [read summary remarks]
Lecture Notes
5.3a:
higher order constant coefficient linear homogeneous DEs and complex
exponentials: distinct roots;
[lecture
example,
decayingsinusoidals.mw]
5.3: 1, 5, 9, 17, 21, 23, 33; -> [1,
9, 17, 23, 33]
[[optional
examples for 5.4 when we need to find amplitude and phase shift: 23
example expressed in
phase-shifted cosine form,
another example:
25]];
[phase
shifted form summary: mw, pdf]
[ignore instructions
to factor by hand or polynomial long divide: use technology for all factoring
(bad roots!)].
-
T:
Lecture Notes
5.3b:
higher order constant coefficient linear homogeneous DEs: repeated roots;
[complex exponentials and phasors?]
[solving IC's using Maple Wronskian
examples];
5.3 (higher order DEs): 11, 18, 23, 25; -> [5, 13, 25, 35]
33, 35 [ignore instructions to
use one solution to find more; just find roots with Maple!]
-
W:
Lecture Notes
5.4: Linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillators);
[example weakly damped oscillations: 14;
summarizing handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)
5.4: 3 [use
meters for consistent MKS units!],
13
(goal: maximum positive displacement), 15,
17, 19
[[ read 23,
pdf; this is a useful application
problem]].
to plot two expressions in t together as
requested here to compare with a choice of plots:
> plot([x1(t),x2(t)],t=0..10,gridlines=true) USE FOR QUIZ 8
-
F: Quiz 8 (complex root pair DE like
this quiz); Quiz 7 answer key online;
Lecture Notes
5.5: NON-homogeneous 2nd order DEQ with constant coefficients;
[example mw];
summary handout on driven
(nonhomogeneous) constant coeff linear DEs;
[final exercise on this
handout sheet: pdf
solution];
5.5: 1, 3, 8
[express in terms of exponentials],
10, 33,
35;
[not many of the book RHS driving
functions are physically interesting here;
we will not cover "variation of parameters"; the book
presentation of the method of undetermined coefficients is a recipe with
little justification (educated guess), instead the undetermined coefficient
method handout shows exactly how and why one gets
the particular solutions up to these coefficients].
WEEK 11[-1]:
-
M: Lecture Notes
5.6a: Driven damped harmonic oscillators;
summary
handout on damped harmonic oscillator driven by sinusoidal driving function
[1 sheet printable version];
[in class example.mw pdf:
resonance
calculation side by side], maple resonance plots: general,
explore];
2 class day HW assignment:
5.6: 1,
3, 11, 13, 17 (template
for final 3 problems, HW portal wants phase-shifted cosine form for both
steady state and transient in 11, 13, phase shift between 0 and 2 Pi,
decimal solutions);
Wiki:
driven
harmonic motion,
amplitude plots:
they introduce ζ = 1/(2Q)]. [ignorable bonus: specific,]
-
T: Lecture Notes
5.6b: Driven damped harmonic oscillators: special cases; [earthquake!]
summary handout on beating and
resonance [optional
beating plot exercise using this
worksheet];
Online HW
catchup day. Don't let these go unfinished. Get bob's help if needed.
We will watch the 4 minute Google linked video of resonance NOT! (but see the
engineering explanation linked PDF for the detailed explanation):
Tacoma Narrows Bridge collapse (resonance NOT!):
Wikipedia;
Google (You-Tube video)]
(4 minutes);
a real resonance bridge problem occurred more
recently: the Millenium Bridge
resonance (5 minutes).
-
W: Quiz 8 answer key is online
here;
Lecture Notes
6.1a: Eigenvectors and eigenvalues;
Transition back to linear algebra:
In
class watch bob use the Maple worksheet
ExploreEigenDrag.mw to
graphically determine the eigenvectors and eigenvalues of three 2x2
matrices. It turns out once you release the mouse cursor, you have to
re-execute the procedure to continue, not very user friendly. Sorry.
[in the lecture we
recall the old handout
coupled system of DEQs and its
directionfield: direction fields for Maple help visualize eigendirections of a 2x2 matrix; see the 6
examples in this
Maple worksheet DEPlot directionfield
phaseplot template]
6.1: 1, 2, 7 [1, 2, 7 mw examples
of EigenTutor etc];
13; Ignore
polynomial division discussion, use technology for roots of polynomials!
-
F: Quiz 9 due Tuesday (Final Four travel delay); beating, resonance response calculation;
Lecture Notes
6.1b: Eigenvectors and eigenvalues: more (linear independence,
complex eigenvectors);
[3x3 examples
real;
complex];
in class find eigenvectors of
,
only
need quadratic
formula!
6.1: 15, 19, 23 (upper triangular so diagonal values are eigenvalues!),
(complex eigenvalues:) 29, 31
[do everything by hand for 2x2 matrices;
for 3x3 or higher, go thru process: use Maple determinant: |A-λI|
= 0 (right click on matrix, Standard Operations), then solve to find characteristic equation and
its solutions, the eigenvalues,
back sub them into the matrix equations (A-λI)
x = 0 and solve by rref, backsub, read off eigenvector basis of
soln space, NO POLYNOMIAL DIVISION! compare to Maple's Eigenvector result,
from the context sensitive menu].
WEEK 12[-1]:
-
M:
Lecture Notes
6.2: Diagonalization;
summary: diagonalization;
[diagonalize: 2x2 examples, 3x3 examples
real;
complex];
6.2: 3, 10, 13, 19, 25 [use Maple for det and
solve for eigenvalues, then by hand find eigenvectors, use Maple for matrix
multiplication; check with Eigenvalues etc]
-
T:
Lecture Notes
6.2b: Geometry of diagonalization;
class handout:
grid page for three homework problems below.;
[[HW: read
the worksheet explanation: 6.2.34
epc4-6-2-34.mw]];
If you have time, find the eigenvalues/eigenvectors
by hand, if not use Maple:
0)
Lecture example done as an example for those below.
1)
For the matrix: A = <<-3|2>,<-3|4>> entered by rows,
find the eigenvalues and order them by increasing
value, then rescaling if necessary to obtain two independent integer component eigenvectors {b1, b2}, make the basis changing matrix
B = <b1|b2>, use the coordinate transformations
x = B y and y = B-1
x to find the new coordinates of the point
x = <x1, x2> =
<0,5> in the plane, then make a grid diagram with the new (labeled)
coordinate axes associated with this eigenbasis
(labeled also by the eigenvalue λ =
<value> ) together with basis vectors and
the projection parallelogram of this point; finally
evaluate the matrix product AB = B-1A B to see that it is
diagonal and has the corresponding eigenvalues in order along the
diagonal.[multiply b1 and b2 by A to confirm the eigenvalues];
2) For the matrix: A = <<-5|4>,<4|-5>>,
repeat with <x1, x2> =
<4,2>.
3)
For the matrix: A = <<-40|8>,<12|-60>>,
repeat with <x1, x2> =
<-4,5>.
Return next class.
No more sections of chapter 6 will be
covered.
7.1 is better done with "reduction of
order" instead of "increasing the order" to decouple
systems.
7.2 covers the matrix form of first
order linear DE systems which we will do in context with 7.3 over 4 days.
-
W:
Lecture Notes
7.3a: 1st order linear homogeneous DE systems: real
eigenvalues:
summary:
the
geometry of diagonalization and first order linear
homogeneous DE systems;
[2-d
examples: lecture phaseplot,
more phaseplot examples]
7.3: 3, 5
(use as template), 7, 17, 20, 25.
-
F: Quiz 10 on diagonalization with both real and complex eigenvalues;
Lecture Notes
7.3b: 1st order linear homogeneous DE systems:
complex
eigenvalues:
[example 2]
7.3:
15, 25;
[[3x3 with complex eigenvalues, good
practice example 26]];
summary handout
on 1st order linear
homogeneous DE systems (complex eigenvalues) (2-d example: [phaseplot]);
[[ignorable optional worked examples:
Find the general solution for the DE system x ' =
A x
for the matrix A = <<0|4>,<-4|0>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,0> (solution
on-line:
.pdf);
Repeat the process for problem 7.3.11: x ' =
A x
for the matrix A = <<1|-2>,<2|1>>,
initial conditions x(0) = <0,4> (solution
on-line: .mw,
.pdf)]];
WEEK 13[-2]:
-
M:
Lecture Notes
7.3c: nonhomogeneous 1st order linear DE systems, etc;
[lecture 3x3 example: first order
epc4-8-3-ex2.mw, second order
epc4-8-3-ex2-secondorder.mw;
2x2
example: .mw];
summary
handout on
extending eigenvalue
decoupling
to nonhomogeneous and special second order constant
coefficient DE systems
8.2:
1,
11 (this is the 0 eigenvalue case,
so a constant driving function requires a constant times t for the trial
function, OR just integrate the decoupled first order DE using the
standard first order algorithm, see page 2 of today's lecture).
-
T:
Lecture Notes
7.3d: mixing tanks, etc;
[example
7.3.4 closed system, complex eigenvalues;
driven case
example open
system, real eigenvalues];
(compartmental analysis: Pharma,
environmental and epidemiology modeling
applications);
[USE
MAPLE templates to get the matrices from the parameters, and the plots];
7.3:
33 (open, real),
37 (closed,
complex);
8.2:
15 (read
text for example 2, then 15 = driven open 2 tank, like lecture example).
Quiz 9 answer key online
here,
please read carefully in preparation for Test 3.
-
W: Open resource take home Test 3 started in class:
[template for directionfield]
chapter 5, first order linear homogeneous DE systems (through day 46: real
and complex eigenvalues);
please read the
test instructions this time.
Easter Recess:
WEEK 13[+1]:
-
T: last day in JB!
reminder of decaying sinusoidal functions:
decayingsinusoidals.mw [draw a
picture! visualize your angle]
work on test 3 in class.
-
W: 7.5 last topic: undamped coupled harmonic oscillator systems;
Lecture Notes
7.5a: mass spring systems; [figure8curve.mw];
7.5: 3, 5 [Maple quick check template:
epc4-7-5-template.mw]
in class exercise results and visualization:
2mass2spring-short.mw.
-
F: Test 3 due in class, test sheet stapled as page 1, pledge signed.
If
the weekend will help improve your work, email bob for permission to turn it
in Monday.
driven undamped coupled harmonic oscillator systems. There is
no penalty!
Lecture Notes
7.5b: driven mass spring systems;
[again figure8curve.mw];
result for the system in the previous class exercise driven
by a specific frequency oscillating force :
2mass2spring-short-driven.mw; we start problem 9 online in class;
(this is the final exam topic,
stopping short of the resonance calculation with general frequency):
7.5:
9 (the driving force F2 stated
is actually the force per unit mass: f 2= F2/m2)
11, 13 [use
the
epc4-7-5-template.mw
template worksheet to get the matrix].
WEEK 14[-1]:
-
M: No more homework but bob
will explain how we can take damping into account in our couple harmonic
oscillator problem;
Lecture Notes
7.5d: mass spring system: reduction of order and damping:
[again figure8curve.mw;
redorder4x4.mw].
Explore 3 oscillation
modes.
-
T (F): exercise doing the core
calculation from the
20F final exam.
No HW, but
end of semester feedback
form in Word for Wednesday in person submission (see email)
-
@ W (M): CATS review time
(first 10 minutes);
movie day:
earthquakes and tall buildings.
Sat: 004 exam
Mon: 005 exam
- scroll up for current day
Weeks 2 and 3 thru 4: come by and find me in my office hour or Zoom
office hour, tell me how things are
going.
This is a required visit. Only takes 5 minutes or less.
If you are at all confused, try to do this way before Test 1 in week 4.
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: week 5
Test 2: week 9
Test 3: Take home out-in week 12
FINAL EXAM:
2705-04 MWF/T 10:30:
2705-05 MWF/T 11:35: [switch okay with permission]
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
21-apr-2022 [course
homepage]
[log from last time taught]
hearts with Maple
enjoy and be safe.
