MAT2500 22F [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.

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 (August 24, 2022): Introduction and Overview
Lecture Notes12.1 rectangular coordinate systems, distance formula, spheres (and course intro)
(review by reading section). Maple example: [ExploreTanLine.mw, stripped, plot?]

We will access our e-textbook/HW WebAssign portal through BlackBoard with your laptop or phone.
[ "Ask your Teacher" "Request Extension']

FUN (ignorable): Intro example motivating multivariable calculus: smooth riding on square wheels.

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. Explore the on-line resources. Read the pages linked to our class home page.
Make sure you have Maple 2022 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.
No problem if you never used it before.

Read what Villanova students say about the most important things you can do to succeed here.

WebAssign Problems: WebAssign101 is a very quick intro to WebAssign.
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: 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, bob FAQ, etc. It is important that you read the section in the book from which homework problems have been selected before attempting them.
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.
[Here are the worksheets bob used in class to illustrate document versus worksheet mode in Maple:
document mode, worksheet mode; short intro to Maple]
F: Quiz 1 on 12.1 (see archive) distance formula, sphere analysis (completing the square), remember: midpoint coordinates are average of endpoint coordinates!
Lecture Notes12.2b plane vectors and trig;
12.2: vector diagram problems in the plane [numbers for 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),
34 [same as example 7, different numbers; are the lengths of the ropes relevant?],
45.

WEEK 2[-1]:
M: Lecture Notes12.3a vector ops: dot product;
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]],
W: 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];
and read this Maple worksheet: [using Maple (for dot and cross products and projection)];
12.3 projection: 39, 41, [[45]], 46, 49,
57 [chemical geometry]
Th: 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]
F: Quiz 2 (see archive) on projection [animation, slider]
Lecture Notes 10.1 parametrized curves;
detour: handout on parametrized curves;
[ignorable: textbook example curves: s10-1.mw (wow!); parametrized curve tutorial];
if interested 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: 3 [intercepts: where is x=0, y=0?], 8, 9,
22 [hyperbolic functions: cosh² x- sinh² x = 1, recognition is enough], 24, 26.

Week 3[-1]: Monday is Labor Day!

Quiz 1 back, check answer key in archive.
W: Lecture Notes 12.5a lines and planes;
summary online handout on equations for 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, 4, 5, 13, 17, 19, 23, 31, 41, 45, 51, 57 [.mw].

Homework intended due for Friday night now has a Sunday midnight deadline rather than Saturday. That intended for Monday has a Tuesday midnight deadline.
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!:
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 worksheet: vector projection takes you to nearest point
F: Quiz 3 (see archive); 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 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[-2]: class roster list handed out
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: vector calc! 1, 3 ,5, 7, 15, 17 [eliminate the third coordinate to get projection onto a coordinate plane], 23, 25, 33,
47 [plot this first in Maple! to see that it lies on a cone: eliminate z first by setting: z² (for cone) = z² (for plane) ];
W: Lecture Notes 13.2a vector calculus ops; [twisted cubic: mw]; [mw2];
13.2 : [[ 1: pdf, 2]], 3, 5 [recall: exp(2t) = (exp(t))², what kind of curve is this?], 7, 10, 15, 17, 18, 23, 27;
(see Maple video: secantlinevideo.mw; vectorfunctionlinearapprox.mw);
Maple stuff:
> F:=t-> <t, t² , t³> : F(t) creates a Maple function, PlotBuilder will plot it.
        [arrow definition needed here due to a Maple glitch]
> F '(t)
> with(Student[VectorCalculus]):
> <1,2,-3> × <1,1,1>
                 5 e_x - 4 e_y - e_z
[this is just new notation for the unit vectors i, j, k; > BasisFormat(false): returns to column notation]
or just:
> with(VectorCalculus): BasisFormat(false):
> ∫ F(t) dt
> with(plots):
> spacecurve(F(t), t=0..1, axes=boxed)
Th: Lecture Notes 13.2b vector calculus and vector ops followup;
[another template for plotting curve and tan line together]
[use technology to do vector integrals],
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;
[[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]
F: Quiz 4 on 13.2 [12.5] (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 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, 5 [note the input of the sqrt in the integrand is a perfect square in this problem];
9 [use numerical integration with Maple, choose "Approximate" from the context sensitive menu; see worksheet for change of variable];
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[-2]: Thursday Test 1 thru 13.3 arclength (see archive);
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 compare with back of book or to enter in WebAssign, adjust for the red number which enters the WebAssign version];
51 [twisted cubic: perfect square! (slightly different rescaling)],
55 [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 of curvature there. Decimal values of the radius of curvature help choose the window for implicit plotting.]

> 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; [highway banking]
13.4: motion along spacecurves;
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)]
13.4 (splitting the acceleration vector, ignore Kepler's laws):
Read 1 [so you don't waste time entering data],
3, 5, 11 [recall 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, 45 [visualize it!].

See archive for quiz answer keys.
Th: Test 1 thru 13.3 arclength (see archive). Come a bit early if you can, stay a bit late if you can.
F: no quiz;
Lecture Notes 14.1 real functions of n>1 independent variables [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, 36, 45,
81 (common answer here, showing how the data is fit).

WEEK 6[-2]: Check grades on BlackBoard
M: Lecture Notes 14.2 multivariable limits; [plot-explore.mw];
14.2: 2, 11, 23, 25, 28, 29, 31, 32, 34, 39, 45, 53 [all the continuous function limits were skipped as trivial]
[toolbar plot option: contour, or "style=surfacecontour" or right-click 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: Lecture Notes 14.3b higher order partial derivatives, etc; [ ideal gas law, decimal formulas, PDEs]
14.3: second and higher derivatives (and implicit differentiation!; optional: why partials commute)
50, 51, 52, 53, 59 [use this example for higher derivatives in Maple], 72,
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: 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].
[[read: 81 PDE example]]
F: Catchup day; Quiz 5;

WEEK 7[-2]:
M: Lecture Notes 14.4a linear approximation;
[linear approximation and tangent planes: differentiability illustrated; moving tangent plane, differentiability-zoom]
online handout on linear approximation;
14.4: 3, 5, 20, 25, 27, 28.
W: 14.4: Lecture Notes 14.4b differential approximation;
summary handout on differential approximation and error estimates; [diff approx example]
31, 37, 39, 41, 42 [hint: change in height is twice thickness], 48,
49 [this worksheet explains how a general derivation without specific values answers the question once and for all].

Optional example.
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: 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.
[Due date next Monday night so I can download HW grades for midterm grade calculation. Let me know if you need extensions to catch up on any HW.]
F: Quiz 6 on linear approximation and differentials due Wednesday after break (posted in archive);

Fall Break.

WEEK 8[-2]: check BlackBoard grades before tomorrow
M: 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, 48, 51.
W: Lecture Notes 14.6a directional derivatives; [visualize it]
14.6 (directional derivatives; stop at tangent planes to level surfaces): 1, 7, 11, 17, 25, 27, 35.
Th: Lecture Notes 14.6b gradient and contours; [visualize it 2d, visualize it 3d (temperature problem), visualize it 3d ];
online handout on derivatives of 2d and 2d functions; [44 graphical estimation of gradient]
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, 55, 60, 65, 71 [lecture example 67].
F: Quiz 7 on 14.6;
Lecture Notes 14.7a max-min stuff;
summary handout on 2D max-min 2nd derivative test
[ignorable derivation; with ignorable bonus handout on m ultivariable derivative and differential notation];
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).

optional: if you are interested in the more realistic case of example 4 where numerical root finding is required, read this worksheet.

WEEK 9 [-2]: Test 2 thru chapter 14 Thursday
M: Lecture Notes 14.7b more max-min stuff; [visualize it];
14.7: [21] (a warning that extrema are not always isolated points);
boundaries: 35, 39, [gene fraction example, pdf]
word problems (box): 43 [minimize square of distance],
51 (same problem as 47 only with different coefficients in the constraint equation and different product: soln: pdf, mw].

read 61 [this explains least squares fitting of lines to data, and perhaps the most important application of this technique to practical problems].

Check Answer key for quiz 6. Quiz 7 answer key online Wednesday;
W: answer key to quiz 7; work together on 14.7.54 first;
[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: 1, 7, 15, 18, 21,
25 [tan plane, normal line], ***
29 [implicit diff],
31 [find where grad is proportional to a given vector],
33 [linear approx], 34a [error analysis with differential], 39 [chain rule],
53 [critical pts analysis]. ***

Recall summary in prep for Test 2:
ignorable bonus handout on m ultivariable derivative and differential notation];
handout on derivatives of 2d and 2d functions.
Th: Test 2 on chapter 14.
F: Lecture Notes 15.1a interated and Riemann integrals; [cross-sections]
See Maple Tools Menu, Select Calculus Multivariate, Approximate Integration Tutor (midpoint evaluation usually best);
15.1: 1 [do this problem using the Maple Approximate Integration Tutor in this template worksheet];
3,[the Maple Approximate Integration Tutor only works for midpoint, since "upper" refers to the largest value of the function in each rectangular contribution];
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];
11 [identify as trapezoid plane cross section in y direction, thickness in x direction, volume is product].

No office hour today (dept meeting).

WEEK 10[-2]:
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)]

Check BlackBoard grades to make sure entered correctly. Please study Test 2 answer key (see archive), come discuss your test with bob.
W: 15.2 (Lecture Notes on double integrals, nonrectangular regions) [ ex.4, bob lecture example]
online handout on double integrals [visual];
15.2: 5, 13, 17, 19, 25 [only one direction allows a single double integral], 27; 43, 55, 61
[[optional: read worksheet 82 ]].

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.
Th: Lecture Notes 10.3 on polar coordinate grid and curves;
[detour on polar coordinates for integration in the plane (review from MAT1505)]; [polar graphing];[angles in all 4 quadrants!];
online only: review polar coordinate trig; inverse trig;
online handout on polar coordinates and polar coordinate integration (page 1, the integration is next time);
(stop reading 10.3 before: tangents in polar coords, unnecessary for us), read about graphing in polar coords [ignorable: more polar fun]);
10.3: 3, 5, 9, 11, 17, 19, 22, 25, 30, 35, 43, 45 (all short review problems);
[[ 59, 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 ]];
limacons? special case: cardioids?

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 (page 5 of today's notes), and vertical and horizontal lines, and lines passing through the origin, as in the handout examples.
F: Quiz 8 on 15.2 (see archive Quiz 8);
15.3 (Lecture Notes 15.3 on polar coordinate integration; [examples, pool example, Explore];
online handout: polar coordinate integration examples: page 2;
view this worksheet to understand how to draw an iteration diagram for polar coordinates;
15.3: 4, 9, 13, 22 [express in polar coordinates, find intersection to get angular limits], 28, 32, 35, 39, 46.

optional: comparing Cartesian and polar coordinate iteration diagrams:
explanation of supporting diagram for double integrals, yet again

WEEK 11[-2]:
M: Lecture Notes 15.4a on centers of mass/centroids (long 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.
W: Lecture Notes 15.4b on probability; [textbook examples] [[optional normal distribution: 33]].
[if you need to review one variable probability, view this 3 minute video: section 8.5];
15.4: 29, 31, 32,

We are skipping 15.5 surface area. Better done in Calc 3.
Th: integration diagram practice day: pdf, mw;
This is a perfect example of why the choice of polar coordinate origin can be crucial in an application: 15.4: 35
[same for everyone, answer given in worksheet for WebAssign, READ it for this nice example.].
F: Quiz 9 on polar coords, center of mass;
Lecture Notes 15.6a triple integrals;
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],
9, 12, 17,
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.

WEEK 12[-2]:
M: Lecture Notes 15.6b on deconstructing triple integrals [15.6.38];
in class exercise: exercise in setting up triple integrals in Cartesian coordinates; [result discussed]
after this exercise we try to sketch 35 using the same technique: solid enclosed by y = x², z = 0, y+2z = 4 (textbook parameters);
[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];
15.6: 31, 35,
37 [setup], [soln];
39 [plot of solid to help you visualize after you try to draw this yourself].
W: Lecture Notes 15.6c on center of mass; [centroid of hemisphere, wedge of cylinder];
in class we try 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; how many ways can this be done as a single triple integral? Then find the centroid.@] ;
15.6: 44, 45, 46 [These worksheets help you set up the problem.]
Th: Lecture Notes 15.7 on cylindrical coordinates;
[wedding ring / snow cone setup example, weddingring.mw, snowcone.mw],
15.7 (cylindrical coords): 1, 3, 7, 9, 15, 22, 25, 27, 31, 32.
F: Lecture Notes 15.8 on spherical coordinates;
[snow cone, sharp donut];
[Wiki: orthogonal coords];
15.8 (spherical coords): 1, 4, 5, 10, 15, 17, 20, 24, 27,
31, 43. <<<< make sure you can do these supporting diagrams for spherical integration.

No office hour today:
2pm today in Villanova Room: Mendel Medal Lecture (usually a terrific science talk! "Decision-Making, Brain, and Free Will")

online handout on cylindrical and spherical triple integrals: examples;
sure you understand these supporting diagrams in the r-z half plane,
summarized here for all multiple integrals: integration over 2d and 3d regions.

WEEK 13[-1]:
M: 15.R:;
we try selected problems in class: 19 [switch order!], 48, 53; [see solutions 19; 48, 53];
12, 18, 20 [switch order!],
25, polar coords: 27, 38;
43a [rotational symmetry centroid; draw r-z diagram for the solid of revolution],
47 [2d cartesian to polar].
email bob if you have trouble with any of these.

T-Day break
M: Take home test 3 out in class (see archive). [Read test rules please.]
W: Lecture Notes 16.1 on vector fields!; [fieldplots]
16.1: 2 [use signs of components and slope to understand direction], 7, 15, 23, 29, 26, 31, 37.
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, 3, 11 [write vector eq of line, t = 0..1];
33 [perfect square simplification], 35,
[not assigned: 38 worksheet compares with centroid: obvious midpoint, also bonus: 35 solution],
50.
F: Lecture Notes 16.2b on vector line int egrals; [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: t = x],
17 [r = r1 + t (r2 - r1), t=0..1],
19, 23,
42 [t = y],
44;
47 [see worksheet for set up],
53 [notice projection along line constant on each line segment, so can multiply it by the length, add two separate results].

HW due Wednesday.

WEEK 14 [+1]:
M: Take home test 3 due in class unless an extension is requested via email.
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 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.
W: Lecture Notes 16.4 on Green's theorem; "oint"! [example: 22]
16.4 (Green's theorem): 3, 7, 9, 13, 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.]
Th: Lecture Notes 16.5 on grad, div, curl; [use formula or Maple to calculate curl];
16.5: 1, 3, 9, 11 [read this worksheet for understanding answers given in textbook exercises],
14 [ to interpret vectorially convert to "grad = del, div = del dot, curl = del cross" form as in this worksheet],
21 [is the divergence zero? since div(curl(F))=0].
F: what does all that stuff mean?
a gradient vector field is visualized by the level curves/surfaces of its scalar potential (done!);
Lecture Notes 16.5b on visualizing div, curl properties: visualize;
the next step after we stop (ignorable):
Lecture Notes 16.6-9 on surface integrals and Gauss and Stokes;

No office hour today (department meeting).

Monday, quick?:
[flux motivation: sunshine on Earth surface and seasons; Maxwell's eqns].
[magnetic field lines; electric field lines]

Check archived final exams.
Catch up on HW. (Ask for extensions on any outstanding assignments! You have till the end of the final exam period to complete them.)
@ M: last day of class. CATS reviews first 10 minutes of class.
Final exam discussion. Material since test 3. Weighed as fourth test.
Line integrals in the plane and space, conservative vector fields and scalar potential functions, Green's theorem evaluation.

Tuesday 1-4pm, Thursday 1-4pm OFFICE HOURS by appointment (email me to let me know if you wish to come in).
Saturday 10:45 final exam in Mendel 260.
WebAssign extensions upon request till Dec 20.

Tests:
Test 1: Week 5:
Test 2: Week 9-10:
Test 3: Week 12-13:

FINAL EXAM:
MAT2500 [MWF 11:45 AM] Sat, Dec 17 10:45 am - 1:15 pm
MAT1505 [MWF 12:50 PM] Wed, Dec 14 2:30 pm - :00 pm
[some exchanging of exam slots might be possible; ask bob]

Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS

3-aug-2022 [course homepage]
[log from last time 21s taught with Stewart Calculus 8e]

does anyone ever scroll down to the end?