MAT2500 24S [Jantzen] homework and daily class log
Jump to current date!
[where @ is
located]
Your homework will appear here each day as it is assigned, 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, numbered consecutively below and labeled
by the (first initial of the) day of the week.]
It is your responsibility to check homework here
(not all problems are in WebAssign). (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). WebAssign "deadlines" are at 11:59pm of the
day they are due, allowing you to complete problems you have trouble with after
class discussion. [If Friday is quiz day, HW due Friday has a WebAssign deadline
of midnight Monday.] The WebAssign extension tool can be used to request more time or more attempts for a given
homework assignment, so persistence can give any student 100 percent credit on
the homework cumulative grade.
*ungraded asterisk marked problems are to be done with MAPLE as explained in the separate
but still tentative
MAPLE homework log,
which will be edited as we go.
Textbook technology:
WebAssign homework management/grading is required,
giving you access to an incredible wealth of multimedia tools together with the
online e-book textbook you can access from any internet connection.
WebAssign deadlines are suggested to keep students on track, but
extensions in time and number of tries are always granted.
Textbook problems
are labeled by chapter and section and problem number, and identify
each WebAssign problem. Those problems which are not available in WebAssign will be square bracketed
[[...]]
and
should be done
outside of WebAssign. Check this page for hints and some linked Maple worksheet
solutions. Use the Ask Your Teacher tool to get help on any
problem for which you cannot get the correct answer. There is no reason for
anyone not to get 100 percent credit for the homework assignments. This
is the most important component of learning in this class, doing problems to
digest the ideas. Use the Extension tool for
requesting more tries or more time. All requests are granted.
If you have any questions, drop by my office St Aug 370 (third floor
facing Mendel by side stairwell) or just come to see where
you can find me in the future when you might need to. I welcome visitors.
-
W (January 17, 2024): Introduction and Overview
We will access our e-textbook/HW
WebAssign
portal through BlackBoard with your laptop or
phone.
[Use appropriately: "Ask your Teacher" and "Request Extension"]
Homework
(light first day assignment):
Make sure you read my
welcoming email
sent before the first class, and register with
WebAssign (immediately, if not already done) and send an email with your
schedule attached as described there (follow up with
a reply about yourself by the weekend). Explore the on-line resources.
Read pages linked to our class home page.
Read what Villanova students say about the most important
things you can do to succeed here.
Make sure you have Maple 2023 on your local
computer, available by
clicking here. if you haven't already done so and install it on your laptop when you get a
chance (it takes about 15 minutes or less total), 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 for calc 2 topics.
[There is also a Maple Calculator app for your phone!]
No problem if you never used it before.
just skim briefly:
Lecture Notes12.1 rectangular coordinate systems, distance formula, spheres
(and course intro)
WebAssign Problems:
WebAssign101 is a very quick intro to WebAssign.
Read 12.1 reviewing 3d
Cartesian coordinate systems, distance formula and equations of spheres;
12.1:
13, 17, 19,
26, 49
(WebAssign has random numbers in your problem);
This short list is so you can check out our class website and
read about the course rules, advice, syllabus, etc. It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Anyone who wishes access to all the
textbook homework problems for extra practice, ask bob for the enrollment
key.
-
Th:
Lecture Notes12.2a
vectors;
look
over the handout on diff/int/algebra;
Problems
which are not available in WebAssign will be red square bracketed
[[...]] and should
be done outside of WebAssign but not turned in.
[[Optional.
Read this 12.1:
23
hint: show the distance from P1 to M is the same as from P2 to M
and equal to half the total distance; this is the hard way with points and
not vectors]],
12.2: 1, [[2]], 7,
9, 15, 17,
21, 25. [Stop reading at Applications section, saved for tomorrow]
Remember WebAssign vector input requires the Math Vector palette
using boldface i, j, k notation.
-
F:
Quiz 1 on 12.1 (see
archive) distance
formula, sphere analysis (completing
the square) due Monday in class
SNOW DAY.
So we are now one day behind... :-(
WEEK 2[-1]:
-
M:
Lecture Notes12.2b
plane
vectors and trig;
12.2: vector diagram problems in
the plane [example
7, showing DOC versus WS modes,
wind speed example];
32 (express the components of each vector, add them
exactly (symbolically), evaluate to decimal numbers, find magnitude and
angle with maximum number digits, then round off as requested; this
worksheet shows how to do vectors in Maple),
35 [like example 7], 39 [like wind
speed problem],
45
[linear combinations in the plane].
-
W:
Lecture Notes12.3a
vector ops: dot product [page numbers rearranged];
check
sign up roster for data; paper copy will be printed and distributed
Thursday;
12.3 online summary handout on
dot product: [example 3];
1, 2, 5, 9,11, 15, 21, 23, 33;
optional (only for math lovers) a fun problem if you like math:
[[55
geometry soln: pdf,
mw]],
-
Th:
Lecture
Notes12.3b vector ops: projection;
motivation:
"orthogonal projection" visualization is
a trio of vectors with the same
initial point;
called "resolving a vector" in physics/engineering
contexts;
handout on resolving a vector;
[orthogonal >= perpendicular];
-
F: Quiz 2
(see
archive)
on projection
[animation,
slider]
Lecture Notes 12.4
vector ops: cross product; [mw];
USE MAPLE FOR ALL CROSS PRODUCT EVALUATIONS except some simple i, j,
k notation examples;
12.4 cross product:
online summary handout on
geometric definition;
[why component and geometric definitions agree:
crossprodetails.pdf];
1, 7, 11, 13,
16, 19,
27 (find 2 edge vectors from a mutual corner first, use 3 vectors
by adding a zero third component and then use the cross
product),
31 (Maple example: trianglearea.mw),
33, 35, 37 [use Maple; zero triple scalar
product => zero volume => coplanar],
39 (first redo diagram with same initial points
for F and r).
[[ignorable:
54, but why are a) and b) obvious?
visualization]]
Note:
>
<2,1,1>
· (<1,-1,2>
x <0,-2,3>)
[boldface "times" sign and boldface
centered "dot" from Common Symbols palette parentheses required]
>
<2,1,1> ·
<2,1,1> then take sqrt (Expressions palette) to get length [example
worksheet: babyvectorops.mw]
WEEK 3[-1]: -
M:
Lecture Notes 10.1
parametrized curves;
detour: calc2 review parametrized
curves in Maple
[summary handout on parametrized curves]
[ignorable: textbook
example curves: s9-10-1.mw (wow!)
if interested open this worksheet and execute it by hitting the !!! icon on the
toolbar, then read];
it is not very useful to try to draw parametrized curves based on what
the graphs of x and y look like: technology is meant for
visualizing math! it is harder in 3d.
BUT REMEMBER, WE ONLY NEED TO WORK WITH SIMPLE
CURVES.
This detour now is because we want to describe equations of
simple straight lines, later spacecurves in Chapter 13.
10.1: 3
[evaluate the endpoints to determine direction], 8, 9,
22 [hyperbolic
functions: cosh2 x- sinh2
x = 1, recognition is enough], 24,
26 [use 1-1 and show black
grid].
-
W: Lecture Notes
12.5a lines and planes; check out
the answer key to quiz 2 please;
summary of
equations for lines and planes [linesplanes.mw];
vector equations rule!;
never use the symmetric equations of a line: they are useless for all
practical purposes!;
12.5 (equations): 1 (draw a quick sketch to understand each statement),
3,
4,
5,
13, 17, 19, 23, 31,
41, 45, 51, 57 [.mw].
Remember,
these problems are exercises in using vector operations: the goal is to
use them and not some other way to do them.
-
Th: Lecture Notes
12.5b points lines and planes: separations;
[mw]
summary online handout on geometry of
points, lines and planes
(distances between);
in these problems do not just plug into a formula: this is practice
in vector projection geometry, we really don't care about the distance!
draw a picture, make the projection diagram, evaluate
the appropriate projection:
12.5: 69 ( find point on line, project their difference vector
perpendicular to the
line ),
71 (find point on plane, project their difference along the normal) ,
73 (find pt on each plane, project their difference vector along the normal);
76 (Don't use distance formula; draw a figure, find a point on the plane and move from it along the
normal in both directions to get a point in the two desired parallel
planes, then write plane equations),
78
(find pt on each line (set parameters to zero!), project the
2 point difference vector along the normal to the parallel planes that
contain them);
[optional challenge problem for the
curious: 81].
Ignorable but fascinating worksheet:
vector projection takes you to nearest point
-
F: Quiz 3 on lines and planes (see
archive);
view figure in color online; previous Quiz answer keys are online;
12:R (Review problems are not in WebAssign,
but in the e-text at Chapter 12 Review, Exercises section, this worksheet
link
has solutions):
13 [resolution in the plane],
14 [torque is position vector crossed
into the force vector, use geometric definition, vectors must have
same initial point to compare for included angle, MKS units mean convert to
meters],
22 [distance between point and plane],
23 [Start by going
back to the parametrizations by setting each ratio to the parameter t or s,
as in t = (x-1)/(2) -> x =
1+2 t (solve for x)]
24 [2 intersecting planes],
25
[find a point on the line of intersection, use the stated normal, write the
eqn of the plane],
26 [4 part problem, use parametric equations for part
b)],
check with bob if you have questions about any of these;
27 [2
parallel planes]
in class we work together on 26
optional:
read section 12.6 only for fun, since we will be using quadric surfaces
during the course so why not skim through this material quickly?
for mathematically curious: 38 [what is the distance of a point from
any coordinate plane?---what is the distance of the point
(x,y,z) from
the plane y = 1? [soln is ellipsoid]]
WEEK 4[-1]: TO EDIT: -
M: Lecture Notes
13.1 vector calculus: space curves; [cubic,
cutcylinder]; [eliminate
parameter]
Use Maple to plot spacecurves always;
trying to figure out what curves look like from their equations is a waste
of time here;
13.1: 1, 3 ,5, 7, 9, 15, 23,
33, 35 [just copy and paste together,
but choose t =-10..10 for curve], 39,
47.
Maple stuff:
> F(t):=
<t, t2 , t3>
: F(t) creates
a Maple function, PlotBuilder will plot it (not the function definition)
> F '(t)
> with(plots):
> spacecurve(F(t), t=0..1,color=red)
or use PlotBuilder
-
W: Lecture Notes
13.2a vector calculus ops; [twisted cubic: mw];
[integrate velocity, angles: mw2];
13.2 : [[ignorable 1:
pdf, 2]],
3, 5 [recall: exp(2t) = (exp(t))2, what kind of curve
is this?],
7, 10, 15, 17, 19, 23, 27;
(see Maple video:
secantlinevideo.mw;
vectorfunctionlinearapprox.mw)
-
Th: Lecture Notes
13.2b vector calculus and vector ops followup;
Maple worksheet vectoravg.mw to
see how the integral of a vector-valued function can be interpreted
visually; rules of differentiation extend from
scalar-functions to vector-valued functions easily (example below).
13.2:
35 [angle between tangent vectors], 37, 43, 49,
50;
[[product rule examples: 56, use combined dot/cross product
differentiation rules 4,5 from this section:
(a ·
(b x c)) ' =
a' ·
(b x c)) +
a ·
(b' x c)) +
a ·
(b x c'))
]].
[product rule holds for all the products involving vector factors as long as you keep the
order of factors the same in each resulting term if cross products are
involved; usual sum rules always apply]
To use
dot product from Common Symbols palette:
>
with(VectorCalculus): BasisFormat(false):
[ignorable
template
for plotting curve and tan line together]
[ignorable
technology to do vector integrals]
-
F:
Quiz 4 (see
archive):
Lecture Notes
13.3a arclength; [numerical
integration]
13.3 (arclength): arclength toy problems require squared length of tangent
vector to be a perfect square [or a factorization that makes a u-sub work!]
to be integrable usually!
[ignorable summary arclength and arclength parametrization];
[integral of speed with respect to time is distance traveled! so the length
of a curve is the integral of the magnitude of the tangent vector! easy.]
3, 5 [perfect square]; 9;
11 [need to
abort Maple's integration, convert to inert form here],
13 [hint: to parametrize the curve, first express
y
and then z in terms of x, then let x = t; another
perfect square],
14 [hint: let x = cos(t), y = 2 sin(t)
for the ellipse, then solve the plane eqn for z to get the
parametrization, for one revolution of this ellipse;
for the
approximate integration, use the circled exclamation mark on the top line
center of the toolbar to stop Maple from going on forever, then make
the integral inert selecting it in the input region and using the Format
menu, Convert to, Inert Form. Then you can numerically approximate it no
problem.]
WEEK 5[-1]: -
M: Lecture Notes
13.3b curvature; [osculating
circle play];
13.3 (curvature): 20, 23, 31 [do not use formula 11: instead use the parametrized curve form
r
= <t,a t4,0> of the curve
y = a x4,
then let t = x to enter in
WebAssign, adjust for the red number which enters the WebAssign version];
51 [twisted cubic:
perfect square!], 53,
55
[Same as for problem 31:
r
= <a cos(t),b sin(t),0> . For osculating circle, normals at axis intercepts are pointing inward
along axes, just go distance ρ along axis to get center, write equation
of circle with radius of curvature there.]
>
with(Student[VectorCalculus]):
SpaceCurveTutor(<t,t2,0>,t=-1..1)
from the Tools Menu, Tutors, Vector Calculus,
Space Curves [choose
animate osculating circles]
-
W: Lecture Notes 13.4
motion in space;
13.4 (splitting
the acceleration vector, ignore Kepler's laws):
[[numerical derivatives:
read 1
, so you don't waste time entering data]],
3, 5,
11 [v = exp(t) + exp(-t) since
v2 is a perfect square],
17 [optional: graph; rotating the curve around,
see the animation],
19 (minimize a function when its derivative is zero
(critical point)! confirm minimum by plotting function);
40 [perfect square!], 45 [read
this for the answer and visualize
it!].
[ignorable summary
on geometry of spacecurves and space curve curvature and acceleration;
Maple
worksheets for these handouts:
rescaled
twisted cubic, helix]
Valentine's Day: 
but 
are better;
heart in a box
-
Th:
suggested review problems: 13.R: [4, 6, 10,
24]; [11: TNB frame]
in class we play with 24 using this
worksheet
(without continuous curvature but just continuous value and slope, such
piecewise curves joining points are called cubic splines, important in
engineering and design);
[ignore this one 14a: use parametrized curve r
= <t, t4 - t2,0> to
evaluate the curvature; by symmetry
N(0)
easily must point down find osc circle: x2 + (y+1/2)2 = 1/4,
14b*: edit the template and do the zoom plot to see the close match
of the circle to the curve, including the separate original plot showing the
large scale behavior];
[highway
banking,
pdf soln]
-
F: Quiz 5 on line!
(see
archive)
good prep for Test 1;
Lecture Notes 14.1 real functions of n>1 independent variables [text
graphs,
plots,
Plotbuilder];
Maple is the appropriate tool to make multivariable plots.
14.1: 3
[WebAssign graphing? report back, I vetoed the rest],
5, 11, 16, 18, 20,
35 [let me know if you get hassled for your guesses!], 36, 45,
81 (common answer
here, showing how the data is fit).
WEEK 6[-1]: Test thru
chapter 13 Thursday. Change: Tuesday 5:30pm review
session in OldFal 105-
M:
Lecture Notes 14.2
multivariable limits; [plot-explore.mw];
14.2:
2, 11, 23, 39, 42, 45, 53
[all the continuous function limits were skipped as trivial]
[toolbar
plot option: contour, or "style=surfacecontour" or PlotBuilder
style "surface with contour"].
-
W:
Lecture Notes 14.3a
1st order partial derivatives; [visualize
mw],
14.3 (partial
derivatives, finally): 5, 7, 9, 13, 19, 20, 25,29, 32, 37, 38.
-
Th: Test 1 thru chapter 13
-
F: no quiz (test week);
Lecture Notes 14.3b
higher order partial derivatives, etc; [example:
decimal formulas]
14.3: second and higher derivatives
(and
implicit differentiation!;
optional:
why partials commute)
50, 51, 52
[handwork simpler!], 53, 59 [use this example for higher
derivatives in Maple],
72 [sign of fxy? is fx
increasing or decreasing in the y direction?],
73 [Read
this problem, but only do parts a and b. Part c has 0 credit to help you
avoid doing too many calculations. just average the adjacent secant line
slopes on either side of the point where the partial derivative is to be
evaluated, as in the opening example: mw, pdf,
this is not a testing problem! tedious so I show you how to work through
it, in fact no red numbers, this is the solution, just read through it and
enter the final numbers],
83, 85 [nonideal gas; implicit
differentiation, each partial derivative assumes the third variable is
held constant; it is easier to do this
by hand than by Maple!].
[[read:
81 PDE
example]]
WEEK 7[-1]: -
M:
Lecture Notes 14.4a
linear approximation;
[linear approximation and tangent planes:
differentiability illustrated;
moving tangent plane,
differentiability-zoom]
summary:
linear approximation;
14.4:
3, 5, 20, 25, 27,
28 [tabular data linear
approximation].
-
W: 14.4:
Lecture Notes 14.4b
differential approximation;
summary:
differential
approximation and error estimates;
31, 37, 39, 41,
42 [hint: differential of volume but change in height is twice thickness],
48 [first solve for P],
49 [compute differential d (1/R)
=-R^(-2) dR etc, this worksheet explains how a
general derivation without specific values answers the question once and for
all; hand derivation: pdf].
Optional examples.
4x6prints?;
[diff
approx example]
-
h: Test 1 back; check BlackBoard grades against those on your paper
quizzes and test;
come discuss with bob if grade not high enough for
your tastes;
14.5:
Lecture Notes 14.5a chain
rule; [Laplacian];
(I never use tree diagrams);
chain rule:
1 [just repeat answer for part 2, don't waste time], 9, 19, 27, 39, 40.
-
F: Quiz 6 (see archive) due Wednesday after break; more
chain rule practice:
Lecture Notes 14.5b chain
rule etc; [2nd
derivatives] [[read
wave equation:
51 ]];
14.5:
(related rates, etc):
[[41:
solution explained, read, insert answer]],
42, 44, 45,
related rates: 47
[cute animation],
48 [Doppler effect],
51 [wave superposition].
Spring Break.
WEEK 8[-1]: -
M:
Lecture Notes 14.6a
directional derivatives; [visualize
it]
14.6 (directional derivatives; stop at tangent planes to level
surfaces):
1 [graphical estimation], 7, 11, 17, 25, 27, 35.
optional: [44 graphical
estimation of gradient]
-
W:
Lecture Notes 14.6b gradient and contours;
[visualize it 3d (temperature
example)];
level surface tangent planes; note z
= f(x,y) corresponds to F(x,y,z) = z
- f(x,y)
= 0):
14.6: 33, 37,
49, 51 [visualize it 3d],
55, 60 [set gradient equal to k n, where
n is a normal to the given plane and solve for
x,y,z , then backsub results into original equation for surface to
determine k],
65 [plug parametrized coordinates into
equation of surface to solve for the parameter],
71.
ignorable: visualize it 2d
[optional lecture notes
example: 67].
-
Th: Lecture Notes 14.7a
max-min stuff;
summary
of
2D
max-min 2nd derivative test [ignorable
derivation];
14.7 (textbook example 4): 1, 3, 5,
13, 14,
25 (because of the symmetry, in first
quadrant, y = x, then solve by hand to get critical
points, including second derivative test and evaluation of f at critical points,
plot x=0..2,y=0..2,z=0..10
with context menu),
27 (symmetry also here
y = x).
you can use the plot3d command form context menu to see
if it looks right!
optional: if you are interested in the more realistic case of example
4 where numerical root finding is required, read
this worksheet.
No office hour today. The dr is sick.
-
F: Quiz 7
(see
archive);
Lecture
Notes 14.7b more max-min stuff; [visualize
it];
14.7:
boundaries:
35 [ignorable gene fraction example: mw, pdf]
word problems
(box):
43
[minimize square of distance],
51, 55 [soln: pdf,
mw].
WEEK 9[-1]: Test 2 on chapter 14 in class
Thursday. -
M:
work together on 14.7.54 first
[solution handouts/s9-14-7-54.mw];
[answer: height
is 2.5 times the square base; obviously cost of materials is not the design
factor for normal aquariums, no?],
14.R: (review problems; note some of the highest numbered problems
not chosen here refer to
14.8, which we did not do): some in class if time:
18, 21,
25 [tan
plane, normal line but F(x,y,z)=z-f(x,y)=0], *** [solution
25, 29]
29 [tan
plane, normal line but F(x,y,z)=LHS-RHS=0],
33 [linear approx], 34 a or b
[error analysis with differential], ***
39 [chain rule], 42 [implicit
differentiation],
53 [critical pts
analysis]. ***
useful
online
summary derivatives of 2d and
3d
functions;
with important summary of
multivariable
derivative and differential notation
Tuesday 5:30pm voluntary problem session room Old Falvey 104 (next to
our classroom).
-
W: Quiz 7 back
(see
archive);
remarks; more review.
error estimation problem [soution:
pdf,
mw];
14.7:
8, 16 [Maple gives the critical
points, classify them, check contourplot and plot3d]
-
Th: Test 2 on chapter 14.
-
F:
Lecture Notes 15.1a interated and Riemann integrals; [cross-sections]
See Maple Tools Menu,
select Tutors, Calculus Multivariate,
Approximate Integration Tutor (midpoint evaluation usually best);
Only do midpoint part b) in 1, 3:
15.1:
1 [do this
problem using the Maple Approximate Integration Tutor in this template
worksheet];
3, [don't do upper
right],
6 [midpoint sampling:
(m,n)=(2,3), x along 20 ft side, y along 30 ft side: answer = 3600,
edit worksheet for your red numbers or set up yourself],
7 [for
average value divide integral by area of rectangle, we talk about this
next time].
WEEK 10: -
M:
Lecture Notes 15.1b integration over rectangles, average value;
[visualize it];
15.1:
15, 21, 29 [this factors into two 1d integrals!], 33 [which order avoids integration by
parts?],
35 [corners are connected by straight lines], 49, 53, 56.
step by step checking of multiple integration::
>
with(Student[MultivariateCalculus]): (or Tools, Packages, Student
MultvariateCalculus)
> MultiInt(
x + y, y = c..d , x
= a..b,output=steps)
[extra
explanation on iterated integrals in Maple (how to enter,
how to check step by step)]
-
W: Quiz 7b due Wednesday after Easter Break: open resource but no
discussion;
Answer key to test 2
(see
archive);
tangent plane to graph of f(x,y) is not a line, normal line to graph of
f(x,y) is a line in space;
when using linear approximation or
differentials, all partial derivatives are evaluated at the "simple"
point you start with.
Easter Recess: 
-
W:
Lecture Notes 15.2 on double
integrals, nonrectangular regions
[lecture
ex.4, optional final page lecture
example]
online handout on double
integrals [visual];
15.2:
5, 13 [it is clear you have to
integrate in the x direction first],
17, 25 [make a
diagram!], 35 [what is the base of this solid?],
43, 55, 61.
No office hour
today: blood drive donation.
-
Th: Lecture Notes
10.3 on polar coordinate grid and curves;
[detour on polar
coordinates for integration in the plane (review from MAT1505)];
online handout on
polar coordinates and polar coordinate integration
(page 1, the integration is next time);
(stop reading 10.3 after polar curves;
but we only need very simple
polar curves r = r (θ) this semester, no limacons or cardiods!);
10.3: 3, 5 [in both cases make your own diagram first, reference angles
are multiples of Pi/4 or Pi/3];
9, 11, 22,
25 [substitute directly for
r 2 then divide through by r ].
-
F: Quiz 8
(see
template quiz 8 from F22 in
archive);
15.3 (Lecture Notes
15.3 on polar
coordinate integration; [lecture examples,
pool example,
Explore];
online handout:
more polar coordinate integration
examples: page 2;
view this worksheet to
understand how to draw an iteration diagram for polar coordinates;
use Maple to evaluate trig integrals that you cannot do; it is
the set up of the limits which is important;
15.3:
4, 9, 13, 22 [express in polar
coordinates, find intersection to get angular limits],
32 [u-sub!], 35
[find out where the graphs intersect to find radius of circular region
of integration!],
39,
46 [the red number is irrelevant,
everyone has the same problem].
No office hour today: 2 meetings!
WEEK 11:
4 weeks to go! -
M: Check Blackboard grades; Test 2 and Quiz 7b back together(answer keys
in
archive);
Lecture Notes 15.4a on centers of mass/centroids [alternative
story]
["lamina" just means a 2d region, like a
flat aluminum piece with boundary where we ignore the thickness];
15.4 (center of mass, "centroid" =
geometric center when constant
density): 1, 5, 7, 9, 13,
17 [same
for everyone, but let Maple do the evaluation of the double integral,
setting up the limits is the important thing here, not tedious algebra!].
optional read handout on
distributions of stuff (center of
mass, centroid, averaging with a weighting function! this is similar to
probability distributions)
we skip moments of
inertia---
of course who cares about centers of
mass or geometric centers of regions (CENTROIDS!)--- but this is typical of many
"distribution" problems, including probability, and we have some intution
about where these points should lie so they are good practice in setting up
integrals and seeing results which agree with our intuition.
Don't
look at the sun! [without proper glasses] no office hour today, viewing
party!
-
W: late quizzes?;
Lecture Notes
15.4b on probability (too ambitious review etc, just read the short
textbook section);
[textbook examples]
[if you wish to review one variable probability,
view the 3 minute video opening section 8.5];
15.4: 29,
exponential distributions 31, 32
(just use Maple to evaluate double integrals directly to decimal
values).
if you are curious:
[[normal distribution:
33
(no WebAssign problems!)]].
We
are skipping 15.5 surface area.
-
Th:
Lecture Notes 15.6a triple integrals;
read first two sections, stop before applications;
short version handout: example of iterating triple
integral 6 different ways
[tetrahedrons:
tripleintexample.mw,
tripleintegralexample2.mw,
explore];
changing iteration to adapt to rotational symmetry
15.6.example3
[pdf];
15.6: 4
[resist temptation to just plug into Maple, go thru the hand steps at least
this once: step by step Maple check],
9
[diagram suggests y first, then z, then x],
12 [once x int done,
what is projection into y-z plane? draw 2d diagram of outer double
integral],
17 [diagram for out
double integral enough],
21 [use polar coords in the unit circle in the
y-z plane for outermost double integral, do the innermost integral in
the x direction],
23, 25.
-
F: Quiz 8 answer key posted, please consult first;
Quiz 9 on double integrals in cartesian and polar coords (variation
of Q8 20S in
archive);
Lecture Notes 15.6b on deconstructing
triple integrals (changing order of integration)[15.6.38];
in class exercise:
exercise in setting up triple integrals in Cartesian
coordinates; [result
discussed; pdf];
then we
will try setting up the figure for 35 together;
15.6:
31, 35,
37 [setup],
[soln];
39 [plot of
solid to help you visualize after you try to draw this yourself].
WEEK 12: -
M:
Lecture Notes 15.6c on center of mass;
[centroid of hemisphere,
wedge of cylinder];
Friday in
class we tried together to sketch the
following solid
(3d centroid example):
25: Use a triple integral to find the volume of
the wedge in the first octant that is cut from the parabolic cylinder
y = x2 by the planes
z = 0, y+z = 1 then use Maple to evaluate the triple integral you set up for its volume;
then find the
centroid: .mw,
pdf;
15.6:
44,
45 ,
46 [These worksheets help you set up
the problem.]
-
W:
Lecture
Notes 15.7 on cylindrical coordinates;
[weddingring.mw,
snowcone.mw],
15.7 (cylindrical coords): 1, 3, 6, 7,
15,
[22], 25,
31, 32.
-
Th:
Lecture
Notes 15.8 on spherical coordinates;
[snow cone,
sharp donut];
15.8 (spherical coords): 1, 4, 5, 10,
15
[make an r-z diagram!],
17, 20 [cyl coords NOT!],
24,
27,
31
[by symmetry: centroid on z axis],
43 [set up so easy also in cyl
coords].
<<<< make sure you can do these supporting diagrams for spherical
integration.
-
F: 15.R:
Take home test 3 on chapter 15 out, due
back a week from Monday on Day 53;
we try
selected problems in class:
43a [rotational symmetry centroid; draw r-z diagram for
the solid of revolution, slope intercept formula for z as functioni of r!],
48 [cartesian
to spherical],
53 [change cartesian order] ; [see solutions
48, 53];
18, 20 [switch order!],
polar coords: 27, 38;
47 [2d cartesian to
polar],
email bob if you have trouble with any of
these.
online
handout on cylindrical and
spherical triple integrals: examples;
be sure you understand these
supporting diagrams in the r-z half plane,
summarized here for all
multiple integrals: integration over 2d and 3d
regions (useful for Test 3).
WEEK 13: -
M: Bring your Test 3 materials
to class. You will spend the period working on that.
-
W: Lecture
Notes 16.1 on vector fields!; [fieldplots]
16.1: 2
[use signs of components and slope to understand direction], 7, 15,
23 [factor components], 29 [read this
worksheet],
26, 31 [orthogonal to level curves, points towards
larger values of function],
37 [change in position is velocity at
given point times increment in time].
-
Th:
Lecture Notes
16.2a on scalar line integrals;
[example mw];
online handout on scalar line integrals
(ignore text discussion of "scalar" integrals with respect to dx, dy,
dz separately: these are really vector line integrals);
16.2 ( ∫ f ds scalar line integrals only):
2
[u-sub will work], 3 [use polar coords, then u-sub],
11 [write vector eq of
connecting line segment, t = 0..1];
33 [perfect square simplification using
cos^2+sin^2=1, ds/dt independent of sin/cos],
35 [35, scale up from lecture
notes on unit circle],
optional 0 credit example of
application: 50 (read?).
-
F:
Lecture Notes
16.2b on vector line integrals;
[visualize];
( ∫ F · dr =
∫ F
· (dr/dt) dt =
∫ F
·
T ds = ∫ F1 dx +F2 dy
+ ... ; always use vector notation!):
16.2: 7 [ ∫C <x+2y, x2>
· <dx, dy>;
piecewise parametrized curve: x = t or just substitute y=y(x)
for y and dy],
17 [r = r1 +
t (r2 - r1), t=0..1],
19,
23 [u sub works], 42 [let y = t; work = line
integral of force vector field],
44
[straight line parametrization];
optional no
credit if you are interested otherwise ignore:
47
[see worksheet for set up],
53
[notice parallel projections along this line are constant on each line segment, so can multiply
it by the length, add two separate results].
WEEK 14: -
M: test 3 due in class; check current grades in BlackBoard;
Lecture Notes
16.3 on conservative vector fields; [inverse square force
line integral 16.3 example 1;
counterexample];
16.3: 3, 11 [find potential
function to evaluate],
22, 30 [find potential to evaluate],
31 [any
circle around the origin has nonzero line integral, no?]
[Read only if curious 41].
Optional
note: the final section of 16.3 on conservation of energy is
really
important for physical applications and so it is worth reading even
if it is not required for this course.
-
Tu (F):
Lecture Notes
16.4 on Green's theorem; "oint"!
[example: 22]
16.4 (Green's theorem): 3, 7, 9, 13 [this
can be done by hand!],
21
[optional:
the line integral technique for integrating
areas of regions of the plane is cute but we just don't have time for it
so you can ignore it.]
-
W (M):16.5: divergence and curl [not on final exam]
Lecture Notes 16.5 on
grad, div, curl
Lecture Notes 16.5b on
visualizing div, curl properties: visualize;
[Maxwell's eqns,
magnetic
field lines,
electric field lines]
no office hour: dentist apptmnt
-
@
Th: CATS reviews at beginning of class;
Lecture Notes
16.6-9 on surface integrals and Gauss and Stokes (pages 4 and 5 only);
parabola and
hemisphere scalar and
vector line integral examples;
[flux motivation: sunshine on
Earth surface and
seasons].
Friday 2pm Mendel G92 last minute problem session. Test
3 answer key posted online.
Final Exam
(not cumulative 16.1-4: scalar and vector line integrals, Green's
Thm, potential functions, polar coords):
Come by my office for a 5 minute visit
to see where I am when you might need
help, especially if you are having any difficulty in the class so far. Discuss
Test 1 once back if appropriate.
Tests:
Test 1: Week 6:
Test 2:
Week 9-10:
Test 3: Week 12-13:
FINAL EXAM:
[MWF 11:45 AM] Sat, May 4 01:30 pm - 04:00 pm
[MWF 12:50 PM] Thurs, May 9
02:30 pm - 05:00 pm
[exchanging
of exam slots is possible; ask bob]
Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS
3-jan-2024 [course
homepage]
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