MAT2500 20S [Jantzen] homework and daily class log
Jump to current date!
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.] Red numbered problems are a relic
of the old online HW hints that are now superceded by the e-book addons.
*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.
Problems
which are not available in WebAssign will be square bracketed and must be done
outside of WebAssign. Check this page for hints and some linked Maple worksheet
solutions.
- M (January 13, 2019):
GETTING STARTED STUFF. By Friday, January
18,
reply to my welcome
e-mail [robert.jantzen@villanova.edu]
sent to your OFFICIAL Villanova e-mail account (which identifies you with your
full name),
telling me about your last math courses, your comfort level with graphing
calculators and computers and math itself, [for sophomores only] how much
experience you have with Maple (and Mathcad if appropriate) so far, why you chose your major, etc,
anything you want to let me know about yourself. Tell me
what your previous math course(s) was(were) named (Mat1500 = Calc 1, Mat1505 = Calc
2, Mat2705 = DEwLA).
HINT: Just reply to the welcome
email I sent you before classes started.
[In ALL email to me, include the string "mat2500" somewhere in the
subject heading if you want me to read it. I filter my email.]
Intro example motivating
multivariable calculus: smooth riding on square wheels.
On your laptop/tablet if you brought it:
1) Open
your favorite browser.
(You can open Maple files linked to web pages
automatically if Maple is installed on your computer.)
2)
Log in to MyNova on the Villanova home page
in a browser and check out our class photo roster and visit the link to
my course homepage from it by clicking on my home page URL under my photo and
then on our class homepage, or you can Google me with the string "dr bob
jantzen" to find my home page and then this class home page:
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2500/ ]. Check out the class
photo roster for future reference.
3) bob will
quickly show you the computer environment supporting our class
[like
clicking here to register
with WebAssign; class keys 10:30-001:
villanova 9185 9639,
11:30-002:
villanova 1366 6813
]
If you already have an existing WebAssign account, login
at this page, if not, register to create one free of charge.
You have a
grace period of 2 weeks to pay for the e-textbook if you have not already done
so in a previous semester.
4)
Here is an example of a problem [Stewart 12.1.46] embedded in 3d space but
quickly reduced to a Calc 2 volume of revolution problem:
hand solution [ s12-1-46.pdf],
[notice how different the hand written and Maple
typeset formatting of the solution]
corresponding Maple worksheet solution [
s12-1-46.mw] and illustration of the
problem. All you
have to do for such solution worksheets is read them if you are interested. Don't worry, we will take it slow with Maple.]
During class in the first part of the semester, a signup sheet will be
passed around for your signature. Make sure you sign at the end if it bypasses
you. Today please put your nickname or your first name to be used in class,
and include your cell phone number and your
3 letter
dorm abbreviation listed on the short list side of the signup sheet.
My cell number will be written on the whiteboard.
and click on my home page URL
under my photo. Click on our class URL there.
Check out the on-line links describing aspects of the course (no need yet to look at the
MAPLE stuff).
[You can
drop by my office St Aug 370 (third floor, Mendel side, by
side stairwell) to talk with me about the course if you
wish and to see where you can find me in the future when you need to.]
Homework
(light first day assignment):
Make sure you read my
welcoming email
sent on the weekend (and reply to it within a few days, at least by Friday), and register with
WebAssign (immediately, if not already done).
Explore the on-line resources. Read the pages linked to our class home
page. [Read computer classroom
/laptop etiquette.]
Fill out the paper schedule form bob handed out in class [see
handouts]; use the
3 letter
dorm abbreviations to return in class the next class day.
WebAssign Problems:
WebAssign101 is a very quick intro to WebAssign due Wednesday midnight. Some
of you have already done this.
Read 12.1 reviewing 3d Cartesian coordinate systems, distance formula and
equations of spheres;
and also due Wednesday midnight so you can ask
questions in class if necessary:
12.1:
23, 45
(WebAssign has random numbers in your problem, solved above for the textbook
numbers);
This short list is so you can check out our class website and read about the course
rules, advice, bob FAQ, etc, respond with your email; those who do not yet
have the book should be using the e-book through WebAssign. It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Download Maple
2019
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), I will help you in my office if you
wish.). 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 problem if you never used it before.
- W:
return your schedule forms at the beginning of
class;
check cell phone number, dorm info on daily signup sheet from first day entry;
look
over the class paper handout on diff/int/algebra;
Problems
which are not available in WebAssign will be square bracketed and must
be done outside of WebAssign.
12.1: 11, 15, 17, [21a
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], 3,
5, 7, 13, 15, 19,
21,
25,
29. [Stop reading at Applications section, for tomorrow]
- Th:12.2: vector diagram problems [numbers for example 7];
34 (draw a picture, express the components of each vector, add them
exactly (symbolically), evaluate to decimal numbers, think significant
digits),
37 [same as example 7, different numbers],
45 [grid].
-
F:
Quiz 1 on 12.1 distance
formula, sphere analysis (completing
the square), remember: midpoint coordinates are average of endpoint
coordinates!
12.3 handout on
dot product: [example 3];
1, 2, 5, 9, 11, 15, 21, 23, 33;
optional (only for math lovers) fun problems if you like math: [55]
(geometry [pdf,.mw]),
[57] (chemical geometry
[ soln.mw, plot:
methane.mw]);
WEEK 2[-1]:
M: MLK DAY.
- W:
check online answer key for
Quiz 1;
Office hours and course info document handout;
handout on resolving a vector;
[orthogonal >= perpendicular];
and read this Maple worksheet: [using
Maple (for dot and cross products and projection)];
12.3 projection: 39,
41, [45], 46, 49.
MLK Freedom School Thursday 10:30 Radnor Room
Villanovans Against Sweatshops (bob is advisor), other interesting talks
5:30 Keynote Address today, very powerful woman speaker, don't miss it if you can
help it. [wow!]
- Th: OPEN MAPLE at beginning of class,
wait for bob to explain;
12.4 cross product: 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 and cross
product),
31 (Maple example: trianglearea.mw),
33, 35, 37 (zero triple scalar
product => zero volume => coplanar),
39 (first redo diagram with same initial points
for F and r).
[54, 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]
- F:
Quiz
2 on projection (easy! identical to 19S Quiz 1 with 3 new
points);
detour: handout on parametrized curves
[also]:
[textbook
example curves: s10-1.mw (wow!)][parametrized
curve tutorial];
open these worksheets and execute them by hitting the !!! icon on the
toolbar (then read them!);
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!;
BUT REMEMBER, WE ONLY NEED TO WORK WITH SIMPLE
CURVES.
This detour now is because we want to describe equations of
simple straight lines.
10.1: 4, 9,
13, 17 [hyperbolic
functions, Stewart 8e section 3.11: cosh2 x- sinh2
x = 1, recognition is enough],
21,
just for fun:
[28
: jpg; it does not hurt to use technology if you cannot guess them all];
33, [37].
WEEK 3[-1]:
- M: Everybody is politely requested to stop by my
office for a 5 minute chat during the next few weeks just to see where it is
and report on how things are going; [Test 1 week 5 thursday?];
handout on lines and planes [.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,
5, 7 (parametric only,
but WebAssign forces you to enter the symmetric equations just this once and
never again),
13, 16,
19; 24, 31,
41, 45, 51 ,
57[.mw].
reminder:
"orthogonal projection" visualization is
a trio of vectors with the same
initial point
- W: handout of class lists for communication;
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!:
12.5: 69 (DO NOT PLUG into FORMULA, find point on line, project their difference vector
perpedicular to the
line as in handout),
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 [.mw finding the closest
points!] (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; ans: D = 2);
[optional challenge problem:
81].
Ignorable worksheets:
vector projection takes you to nearest point,
bob's approach to 76.
- Th: 12:R (Review problems are not in WebAssign): 13,
14, 22, 24,
25, 26 (parametric equations for part b),
for fun: 38 [what is the distance of a point from any coordinate
plane---we did this problem in 12.1.23, so what is the distance of the point
(x,y,z) from
the plane y = 1? [soln is ellipsoid]];
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?
in class
we work together on 26.
[Hint on 25: you get the normal to the desired
plane from the cross product of the orientation vector of the line of
intersection, and the normal to the third plane, since the third plane is
orthogonal to the desired plane hence its normal is parallel to the plane].
- F: Quiz 3 on cross product, projection, eqns of lines, planes;
[see archived quiz 2];
Maple assignments start (read these
instructions): note asterisks *;
13.1:
vector calc! [cubic,
cutcylinder] [1], 3,[5], 7,
13, 21, 25
[21-26, for fun only, do quickly by thinking, note technology is not necessary here to distinguish the
different formulas: .mw],
27,
33, [39*:
refer back to similar problem 27: note that z2 = (x2+y2)! plot the spacecurve
and the surface together as in the template, adjust the ranges for the
surface so it is just contains the curve and it not a lot bigger; clue: look
at boxed axes ranges for the curve to chose your window for the surface;
HINT: the template example is a paraboloid, this exercise surface is a cone];
43
[eliminate z first by setting: z2 (for cone) = z2
(for plane) and solve for y in terms of x, and then express z in terms of x and finally let x be t;
Maple can solve the pair of equations for {y,z} in terms of x],
[12.5.57*:
using the answer in the back of the book and this template, plot the two planes and the line of intersection and confirm that
visually it looks right. Adjust your plot to be pleasing, i.e., so the line
segment is roughly a bit bigger than the intersecting planes (choosing the
range of values for t)].
WEEK 4[-1]:
- M: Find at least one (at most 3) Maple
partners for the first assignment; ask bob in class for help if you have
trouble;
13.2 : [1:
pdf, 2],
5 [recall: exp(2t) = (exp(t))2, what kind of curve
is this?],
7, 10,
15, 19, 21,
31, 29a (by hand),
29b* [graph your results using this
tangent line template; make a comment about how it
looks using text mode].
on-line handout:
key idea of vector-valued
functions and the tangent vector
(see Maple video:
secantlinevideo.mw;
vectorfunctionlinearapprox.mw);
Maple stuff:
> F(t):=
<t, t2 , t3>
: F(t) creates a Maple function,
PlotBuilder will plot it.
> F '(t)
> with(Student[VectorCalculus]):
> <1,2,-3> × <1,1,1>
5 ex - 4 ey - ez
[this is just new notation for the unit vectors i, j, k;
> BasisFormat(false): returns to column
notation]
> ∫ F(t) dt
> with(plots):
> spacecurve(F(t), t=0..1,
axes=boxed)
- W: 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: 22 [check
your answers against Maple worksheet],
29 [another
template
for plotting curve and tan line together),
33 [angle between
tangent vectors],
39 [use
technology to do the integrals],
[43], 48, 49,[54, 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]
- Th: 13.3 (arclength): arclength toy problems require squared length of tangent
vector to be a perfect square to be integrable usually! [or a factorization
that makes a u-sub work!]
online
handout on
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 [note the input of the sqrt in the integrand is a perfect square in
this problem];
5, [note the
factorization to make an obvious u-sub];
10 [use numerical integration either with your graphing calculator
or if you use Maple, right-click on output of integral, choose
"Approximate";
for the curious:
oops! what a mess!];
11 [hint: to parametrize the curve, first express
y
and then z in terms of x, then let x = t; another
perfect square],
12 [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].
- F: Quiz 4; HW due Wednesday, but get a start on it
on the weekend!; [short story]
13.3 (curvature): handouts on geometry of spacecurves
(page 1 for 13.3)
and space curve curvature and acceleration
(pages 2-3 for 13.4 later, 4 for both, print together);
[example Maple worksheets on these handouts: rescaled
twisted cubic (page 1), helix
(page4)]
13.3: 17, 25,
27 [do not use formula 11: instead use the parametrized curve form r
= <t,t4,0> of the curve y = x4,
then let t = x to compare with back of book or to enter in
WebAssign];
47 [twisted cubic:
perfect square!], 49;
51
[standard eqn: x2/9+y2/4 = 1, so r = <3 cos(t),2 sin(t),0>,
Hint: 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 or 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]
2d parabola osculating circle zoom.
[quick show and tell 2d curves:
osculate-parabola.mw,
osculate-ellipse.mw]
WEEK 5[-1]: Test 1
this Thursday.
- M: 13.3: [in class derivation,
plot]; Taylor approximation of a
spacecurve:
spacecurvetaylorapprox.mw
optional on-line only:
osculating circle, how to
describe mathematically using vector algebra [.mw].
Finish HW, work on Maple problems, look over archived tests;
Test
prep: online only: projections revisited
just for those who like vector geometry;
on-line reminder of dot and cross products and
length, area, volume.
- W: 13.4 (splitting
the acceleration vector, ignore Kepler's laws):
1,
[2 avg velocity = vector displacement / time interval],
5,
11 [recall v = exp(t) + exp(-t) since
v2 is a perfect square],
17a, 17b*[graph your spacecurve using the
template; pick the time interval t
= -n π..n
π, where n is a small
integer, and by trial and error, reproduce the figure in the back of the
book with 6 peaks, rotating the curve around
and comparing with the back of the book sketch (note the horizontal axis tickmarks); if you wish, then animate the curve with
the template provided],
19 (minimize a function when its derivative is zero
(critical point)! confirm minimum by plotting function);
[37 note that v2 = 32(1 + t2)2
is a perfect square],
39,
[41? also perfect square, see 11] , 45 [visualize
it!]
Maple13.mw HW due
anytime next week, through the weekend. Please read the instructions
carefully.
Wednesday 5:30pm MLRC voluntary
problem session
- Th: Test 1 through 13.3, evaluation of formulas for
various projections along curve.
heart in a box
Happy
V-Day! [Maple]
- F: 13.R: (p.882-883): [4, 6, 10, 16, 24];
in
class we play with 24 and use the worksheet
(without continuous curvature but just continuous value and slope, such
piecewise curves joining points are called cubic splines, important in
engineering and design);
[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 with your hand
results including comments and also 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];
Problems
Plus 13.2. [ READ
soln]
WEEK 6[-1]:
Maple13.mw assignment due anytime
this week thru the weekend
- M: 14.1 [text
examples]: 1, 3,
11,
15, 25, 31, [33],
35, 41,
49;
|maple14.mw problems begin:
55*, using this template just do a single appropriate
plot3d and
contourplot after loading plots and defining the maple
function f (x,y) or us the PlotBuilder];
81a (read only b,c; if you are interested to see how the data is fit
see example 3 among the interesting examples
from 14.1 shown in Maple);
after finishing the
preceding, for fun only if you have extra time look at 61-66 (maple plots
reveal relationships, try to see correlations between formulas and 3d plots,
then the contourplots).
- W: limits [14.2-plot.mw;
extra explanation for those who miss bob's commentary in class 14.2-explore.mw];
14.2: [1], 2,
[4, 5], 9, 13, [15],
17,
23* [toolbar
plot option: contour, or "style=surfacecontour" or right-click
style "surface with contour", explain in comment],
25, 31, 37* [does a 3d plot of the expression support your
conclusion? that is, your conclusion drawn before looking at the back of the book obviously,
plot and explain], 39.
- Th: 14.3 (partial
derivatives, finally):
[1], 3,
5,
11, 15, 17,
21, 31, 33, 41,
51
[in class if time: 22, 24, 28, 30].
- F: Quiz 5 (tabular data);
14.3: second and higher derivatives
(and
implicit differentiation!;
partials commute)
47, 49; 53,
56, 59, 63,
65 [use this example for higher
derivatives];
73 [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: pdf,
this is not a testing problem! tedious so I show you how to work through
it],
[81], 83, [84, 88
ideal gas law],
90.
WEEK 7[-1]:
- M: online
handout on linear approximation;
14.4 (linear approximation and tangent planes:
differentiability illustrated;
moving tangent plane):
1, 3,
[7,]
11, [15], 17,
21, 22 [.mw];
7*[calculate by hand, then do two plots: first > plot3d( [f(x,y),L(x,y)],x = a..b, y = c..d);
choose appropriate ranges centered about the point of tangency to show a good part of the surface behavior
together with
the tangent plane, then zoom in by choosing a smaller window about the point of
tangency as instructed by the textbook, check that they agree, make a
comment that it looks right confirming differentiability].
- W: MWF classroom change! TOL 316;
14.4: handout on
differential
approximation and error estimates; [diff
approx example]
25, 27, 31, 33, 35,
38, 39 [remember partials of this function from 14.3.83]; extra:
In the USA (inch units), the 4x6 photo prints have dimensions 4 in by 6 in. In Europe (cm units) the 10x15 prints have dimensions 10cm by 15 cm.
Unit conversion: 1 inch = 2.54 cm.
Use the differential approximation to estimate the absolute change and percentage change in the
(computed) area of the USA format (new) compared to the European format
(old):
A = x y
. Then compare your linear estimates for both to the corresponding exact changes.
[HINT: apply the differential
approximation using the x and y values of the European format,
with the differentials dx and dy given by the differences USA
format dimensions minus the European dimensions.] [Solution:
4x6prints.mw]
- Th: 14.5: chain rule: 1, 11, 15,
17 (I never use tree diagrams), 21,
31 31 (no need to remember these equations, just implicit
differentiate and solve in practice),
35,
39, [wave equation:
49].
- F: Quiz 6 on linear approximation, differential
approximation [see 18s quiz 5];
Spring Break. :-) Enjoy. Be safe in your travels.
WEEK 8[-1]:
- M: 14.5: 41;
14.6 (directional derivatives; stop at tangent planes):
1,
3,
5, 7, 9,
11, 15,
19 [first find a unit vector in the given direction! sketch the two
points],
23[just split the gradient into its length and unit vector direction
information], 29.
- W:
CHECK BLACKBOARD
GRADES; handout
on derivatives of 2d and 2d
functions [maple 2d-gradient and
directional derivative example][3d: Stewart
Example 14.6.7];
14.6 (level surface tangent planes; note z
= f(x,y) corresponds to F(x,y,z) = z
- f(x,y)
= 0):
14.6: 27, 31, [36, 38,]
41, 45, [47 (derive
equations of plane and line by hand)], 49, 52,
[this one is fun: _
61 [soln]],
65 [soln];
47*
[plot your results in an appropriate window (using the 1-1 toggle or the
option "scaling=constrained" to see the right angle correctly), i.e., adjust windows of function,
plane, line to be compatible, after doing problem by hand];
head start on problem 52 in class with any partner?
- Th: handout on 2D
max-min 2nd derivative test
[derivation; with ignorable bonus handout on multivariable
derivative and differential notation];
14.7: [1], 3,
5, 11, 12, 23 (do by hand,
including second derivative test and evaluation of f at critical points);
23* [next time
do 25, template
shows how to narrow down your search to find extrema by trial and error,
record your tweaked image or images confirming your hand results, include
commentary, see additional comments on Maple HW page summary
(inconclusive saddle point?),
.mw];
optional: if you are interested in the more realistic case of example
4 where numerical root finding is required, read
this worksheet.
- F: Quiz 7 due Monday via email;
4.7.58 example exercise;
least squares fit application;
max volume box example;
14.7: [21] (a warning that extrema are not always isolated
points);
boundaries: 31;
37 [plot3d: >
plot3d(y^4+2*x^3, x = -1 .. 1, y = -sqrt(-x^2+1) .. sqrt(-x^2+1)),
express circular boundary in terms of polar angle to extremize there >
plot(2*cos(t)^3+sin(t)^2, t = 0 .. 2*Pi)],
word problems
(box):
41
[minimize square of distance],
49 (similar to 45 only with different
coefficients in the constraint equation),
53, [soln: pdf,
mw];
[58: word problem with triangular boundary, use constraint to eliminate r,
maximize resulting function of 2 variables on triangular region, consider
triangular boundary extrema; plot3d: (try
first, then sol:
pdf,
mw)];
read
59 [this explains least squares fitting of lines to data, and perhaps
the most important application of this technique to practical problems].
Friday CANCELLED Almost Pi Day bake sale 11-3pm at
C-Center to raise money for the
Starfish Foundation which benefits poor kids in Equador (started by a VU
math major!).
Sat:
Pi Day:
π!
[and Einstein's birthday]
WEEK 9 [-1]: TEST 2 Th?
- M: 14.R:. (review problems; note some of the highest numbered problems refer to
14.8, which we did not do):
some in class if time: 1, 7, 15,
18, 21, 25,
29, 31, 33, 34a, 39,
53, 64;
14.7: 52 [ans: the height is 2.5 times the square
base; obviously cost of materials is not the design factor for normal
aquariums,
no?].
In our online "class" i went over 14.7.53
53
[pdf, mw]
and then had students work on 52 using the chat to communicate. Only
section 01 was recorded but in the future both sections will be recorded.
- W: Multivariable Integration!
Lecture Notes in two parts: the
geometrical interpretation of nested (iterated) calc I integrals comes from
the Riemann integration approach;
See Maple Tools Menu, Select Calculus Multi-Variable, Approximate
Integration Tutor (midpoint evaluation usually best), example:
Riemann on rectangles (second
example):
however, we can
also get the meaning from the idea of integrating cross-sectional length to
get area, integrating cross-sectional area to get volume (tomorrow!).
15.1:
1 [do this
problem using the Maple Approximate Integration Tutor in this template
worksheet, the function takes is upper value in the upper
right corner of each gridbox];
[3, do
this problem using the Maple Approximate Integration Tutor in this template
worksheet, input x*exp(-x*y) etc];
[5 in which corner of each ij box do the max and
min values occur consistently?],
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,
11
[trapezoid y-z cross-section, constant thickness in x
direction, volume = area x thickness;
[optional 15* ; note: (m,n) = (1,1), (2,2),
(4,4), (8,8), (16,16), (32,32) = (2p,2p)
for p = 0..5 is what the problem is asking for (see 3 line Maple
template); what can you conclude
about the probable approximate value of the exact integral to 4
decimal places?] .
- Th: 15.1
(Lecture Notes on iterated integrals on rectangles,
average values): 15,
21, 29, [33, which order avoids integration by
parts?], 35,
43,
[45* (use boxed axes!)],
47, 49;
[iterated integrals in Maple (how to enter,
how to check step by step)]
step by step checking of multiple integration (Shift-Enter allows
multiple line inputs):
> x + y
>
∫ % dx
> eval(%,x=b) - eval(%,x=a)
> ∫ %
dy
> eval(%,y=d) - eval(%,y=c)
> etc... if triple integral
(and simplify may help along the way)
- F: TEST 2 Review questions in class,
read take home test rules;
Test 2 over
weekend, due Monday, as email attached scanned to PDF file named:
lastname-test2.pdf [chapter 14 and previous vector operations from 12 and 13
as needed: partial derivatives, gradient, directional derivatives, linear
approx, differential approx, error propagation, max-min, second derivative
test]
Test 2 released in class, published at end of both classes in
usual spot.
WEEK 10[-1]:
-
M: 15.2 (Lecture
Notes on double
integrals, nonrectangular regions)
[ex.4]:
handout on double
integrals [visual1,
visual2];
explanation of supporting diagram for
double integrals]
5, 13, 15,
17, 23, 25, 27;
35,
39, 49,
51, [57, 59], [read worksheet
70].
Keep in mind multivariable integration is really about parametrizing the
bounding curves of regions in the plane or the bounding surfaces of regions
in space (still to do), to set up iterated integrals, whose evaluation is just a
succession of calc2 integrations, easily done by Maple. Setting up the
integrals, Maple cannot do. This is your job. And to re-express their
parametrization when needed to change the order of integration.
No further Maple assignments. Any asterisk marked problems are optional.
- W: 10.3 detour on polar coordinates for integration
in the plane (review from MAT1505)
online only: review
polar
coordinate trig;
inverse trig;
[angles
in all 4 quadrants!];
Lecture Notes on polar coordinate grid; handout on
polar coordinates and polar coordinate integration
(page 1, the integration is next time);
10.3 (stop before: tangents in polar coords, unnecessary
for us),
read about graphing in polar coords [more
polar fun]);
10.3:
3, 5, 8, 11, 12,
17, 19, 21,
25, 30,
33,
37 (all short review problems);
part of Maple15.mw:
67* [Nephroid
of Freeth: starting at θ = 0 how far does theta have to go for the sine to undergo one full cycle?
i.e., stop at θ/2 = 2 π ;
this
is the plotting interval];
keep in mind that our most important curves for later use are circles
centered at the origin or passing through the origin with a center on one of
the coordinate axes, and vertical and horizontal lines, and lines passing
through the origin, as in the handout examples.
- Th:
15.3 (Lecture Notes on polar
coordinate integration; handout: polar coordinate integration: page 2)
[examples];
view this worksheet to
understand how to draw an iteration diagram for polar coordinates;
15.3: [browse: 1, 4, 5], 7, 11, 14, 15,
17, 21,
23 [twice the volume under the hemisphere
z = sqrt(a2 - x2 - y2) above
the circle x2 + y2
≤ a2 or
the volume between the upper and lower hemispheres],
25 [integrand is difference of z values from cone (below) to sphere
(above) expressed as graphs of functions in polar coordinates].
35, 36.
Supporting diagrams for iterating double integrals
yet again.
-
F:
Lecture Notes on centers of mass/centroids;
read handout on
distributions of stuff (center of
mass, centroid, averaging with a weighting function!)
15.4 (center of mass, "centroids" when constant
density;
skip moment of
inertia---of course who cares about centers of
mass or geometric centers of regions?
--- but this is typical of many
"distribution" problems, including probability, and we have some intution
about where these points should lie):
[2], 5, 7, 11 [see example 3],
25* [integrals, visualize etc use one of these sections as a template:
15-4-5.mw].
Weekend take
home
Quiz 8 on double integrals in Cartesian and polar coordinates
(19s:q8,17s:q8, 16s:q6, 13s:q8).
Published
in usual location
after 12:30pm.
WEEK 11[-1]: View
this video
to save somebody's life.
[See
this worksheet to help with
polar coord iteration diagrams, this
one for cartesian examples too.]
-
M: 15.4 (probability) Lecture Notes on probability;
[take the time to view these: 1-d case 3 minute video: section 8.5, 2-d
case 4 minute video 15.4.3]
read this worksheet on probability
(Stewart section 8.5); [standard
deviation?,
Poisson
distribution?];
15.4: (probability):
27, 29,
30 (30a:
P(x<=1000,y<=1000) =??, 30b: P(x+y<=1000)
= ??] [Maple is
the right tool for evaluating probability integrals!],
[31*, use this template for the
normal distribution; do a 3d plot as in the example with boxed axes to
estimate the volume to compare with the numerical value of the integral to
see if it makes sense, roughly].
- W: Pause to understand supporting diagrams for
double integral iteration;
"In class" exercise:
int2dexercise.pdf. Take out some
paper, and methodically respond to the steps in this problem. At most you
need the quadratic formula in this problem (Maple does it without human
error). Don't expect integer answers. Use
the private Zoom chat to get help from bob at any point. This is not for
credit, but you can email me the scan if you want my feedback. Finish this
for homework tonight.
Triple
integrals in space will be a step up in difficulty, so now is the time to
get the 2d case down.
- Th: [we are skipping surface area 15.5]
Lecture Notes on triple integrals;
15.6 (triple integrals: setup):
handout: example of iterating triple
integral 6 different ways [tripleintexample.mw];
changing iteration to adapt to rotational symmetry 15.6.ex3
[pdf];
15.6: 4, 5
[resist temptation to just plug into Maple, go thru the hand steps at least
this once],
13 [optional
visualization],
17 [polar coords in y-z plane!],
21,
("deconstructing a triple integral"): 27.
-
F: 15.6 (triple integrals: "deconstruction"):
Lecture Notes on deconstructing
triple integrals [15.6.34];
handout exercise:
exercise in setting up triple integrals in Cartesian
coordinates (please take this seriously,ask bob if
you need help, send scan of work;
[soln]);
try to sketch 31
using the same technique: solid enclosed by y = x2, y =
x, y+2z = 4;
[then see 3d Maple
Maple plot; note two projections
of the solid onto coordinate planes are actually faces
of the solid, the third face has a border obtained by eliminating y from
the two equations given in the figure to describe the condition on x
and z for that edge curve];
use the standard maple expression palette icon for the definite triple
integral of the constant function 1 to check the agreement of two different
iterations with two different variables for the innermost integration step.
we go over
previous day handout exercise.
15.6: 31 [careful explanation
from scratch after you give it your best shot], 33, 35
Week 12[:
solution worksheets and pdfs for all the previous HW problems:
here Monday.
- M: 15.6 (center of mass):
Lecture Notes on center of mass;
[centroid of hemisphere,
wedge of cylinder];
In
class?? we try to sketch this solid
(3d centroid example): 23 [find the volume of the wedge in the first octant that is cut from the
cylinder y2 + z2 = 1 by the planes y = x, x
= 1 then use Maple to evaluate triple integral you set up for its volume;
how many ways can this be done as a single triple integral? Then find the
centroid.]
15.6: 40, 41, 42.
- W: 15.7: Lecture
Notes on cylindrical/spherical coordinates;
read
handouts on
cylindrical and spherical coordinates and cylindrical and spherical
regions of space and their bounding surfaces: examples (cylindrical part
only until tomorrow, bob example);
15.7 (cylindrical
coords): 1, 3, [5, 7], 9
(in addition give ranges of cylindrical coordinates describing interior of
this sphere), 12, [15], 17,
20,
21, 29, 30.
Easter "Recess":
- W: Lecture
Notes on cylindrical/spherical coordinates;
snow cone exercise;
handout on cylindrical and
spherical triple integrals: examples; [Wiki:
orthog coords,
oblate
spherical coords]
handout on
radial integration diagrams for simple circles
and lines (cylinders, spheres, planes);
[bob example
again]; [if you like math maybe this might be fun:
spaghetti curves and the cosmic microwave background]
15.8 (spherical coords): 1, 3, 5, 7, 9
(use double angle formula!), 11, 13 , 15,
17,
25, 29, 35, 41.
-
Th: 15.R (using arctrig functions in
a
polar coordinate integration
example pdf,
mw);
review
online: integration over 2d and 3d
regions; snow cone exercise
solution;
15.R: 12, 18, 19 [2d
reorder],
25, 27, 38,
43a [rotational symmetry centroid:
cycloids;
HINT: put
cone on z-axis, base at z = 0, draw r-z diagram for solid of revolution],
47 [2d cartesian to polar],
48 [3d cartesian to spherical],
**
53 [3d re-order]. **
"in
class" we try ** problems
See email about test 3 content.
Similar format to previous test 3.
For last minute triple integral
explanation see this parametrization
worksheet.
- F: 16.1 (Lecture
Notes on Vector fields!): 1, 5, 11; 21, 23, 29, 31, 33;
optional
comparison shopping (think of this as matching game, to see how to
distinguish some feature of the formula by its graphical representation):
11-14: < x,-y>,
<y, x - y>, <y, y + 2>, <cos(x+y),
x> ;
15-18, <1, 2, 3>, <1, 2, z>, <x, y, 3>, <x, y, z>
;
29-32: x2+y2, x(x+y),
(x+y)2, sin(x2+y2)1/2;
19*; just try the template for this
problem, no need to submit, or just read the worksheet and then
the result, with bonus problem
25 done as well.
2:30-3:30pm test 3 Zoom office hour, email
bob problem numbers you wish discussed
Okay, no one showed up. Will send
mail for anytime office hour.
WEEK 13:
- M: Take home Test 3 out just before class, work on it
during class, any questions use private chat. Due back Friday;
read take home test rules;
Submit test by email attached scanned to PDF file named:
lastname-test3.pdf.
-
W:
handout on scalar line integrals
(ignore text discussion of "scalar" integrals with respect to dx, dy,
dz separately);
Lecture Notes on scalar line integrals;
example mw;
16.2 ( ∫ f ds scalar line integrals):
2,
3, 11 [write vector eq of
line, t = 0..1];
31,
33,
36 [if radius 1, ans: <4.60,0.14,-0.44>, worksheet
compares with centroid: obvious midpoint, also bonus: 33 solution].
- Th: 16.2: handout with
exercise on
vector
line integrals (solution
of exercise);
Lecture Notes on vector line integrals;
[.mw]
( ∫ F · dr =
∫ F
· (dr/dt) dt =
∫ F
·
T ds = ∫ F1 dx +F2 dy
+ ... ; always use vector notation!):
7 [ ∫C <x+2y, x2>
· <dx, dy>],
15,
17,
[read: 27 Maple template for vector line
integrals], 32, 39;
45,
51
(notice projection along line constant on each line segment, so can multiply
it by the length, add two separate results);
scalar line int:
48;
inverse square force
line integral example from 16.3 example
1.
- F: Test 3 due, unless extenuating circumstances (ask
for extension by email);
Lecture Notes on conservative vector fields;
handout on
"antidifferentiation" in multivariable calculus: potential function for conservative vector field;
16.3: 3, 7, 11 (b: find potential
function and take difference, or do straight line segment line integral), 15
(potential function); 23,
25, 33, 35.
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. Enough said.
Optional glimpse at surface
integrals for the curious.
WEEK 14:
-
@
M: Lecture Notes on Green's theorem; "oint"!
16.4 (Green's theorem):
3, 7, 9, 17,
"WA:501"
[polar coord problem not in WebAssign: 18,
convert double integral to polar coordinates];
[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.]
- T(F): Lecture Notes on
grad, div, curl;
16.5: handout on
divergence and curl, Gauss
and Stokes versions of Green's Theorem;
handout on interpretation of circulation
and flux densities for curl and div [.mw: visualize];
[magnetic
field lines;
electric field lines];
16.5 (curl, div): 1,
3,
9-11,
12
[easier to interpret vectorially if convert to "grad = del, div = del dot,
curl = del cross"
form],
13, 19 [a magnetic field has div H = 0, so
H =
curl A is a way of representing it in terms of a vector potential so that it
automatically has zero divergence, see problem 38; a static electric field
has curl E = 0 so a scalar potential E = - grad φ is
relevant---the minus sign is another story!],
31 [but read 37, 38 and look at identities
23-29], 32 [pdf].
- W(M): Lecture Notes on surface integrals and Gauss and Stokes
[optional material; not for final exam!]:
[flux motivation: sunshine on
Earth surface and
seasons;
Maxwell's eqns]
Optional 16.6-7: surface integrals
for fun;
Optional 16.8-9: read lightly Stokes' Theorem and Gauss's law if you are
interested, when you have time;
[example1: surface area and centroid of
hemisphere,
Gauss law example, wedge of cylinder 16.7.example3;
example2: parabola of revolution (Stewart16.7.23 expanded
into Gauss/Stokes examples)].
- Th: CATS first 10 minutes; final exam discussion.
Let's turn on our videos so we can see each other on this last class
meeting.
Saturday. Final Exam = take home exam released to all at exam 1 time slot,
due back by Friday, May 8.
Line integrals and Greens Theorem, grad, curl,
div, potential functions.
∞ scroll up for current day
MAPLE HW files:
maple13.mw due: Week 6
maple14.mw due: Week 10
maple15.mw due: Week 12?
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: Week 5: ; MLRC
5:30 problem session .
Test 2:
Week 9-10: ; MLRC
5:30 problem
session .
Test 3: WEEK 13: Take home out
; in ; MLRC problem session
FINAL EXAM:
take home exam released to all at exam 1 time slot, due back by Friday May 8;
001: MWF 10:30 Sat, May 2 10:45 – 1:15;
002: MWF 11:30 Thurs, May 7 2:30
– 5:00; (switching days allowed but notify bob)
Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS
23-apr-2020 [course
homepage]
[log from last time 19s taught with Stewart
Calculus 8e]
does anyone ever scroll down to the end?