MAT2500 21S [Jantzen] homework and daily class log

Jump to current date! [where @ is located]

This class is offered in synchronous online Zoom format.

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 Red numbered problems are a relic of the old online HW hints that are now superceded by the e-book addons. The 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.

*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 red double square bracketed [[...]]
and must be done outside of WebAssign. Check this homework log page for hints and some linked Maple worksheet solutions.

  1. M (January 25, 2021): Introduction and Overview
    Lecture Notes12.1 rectangular coordinate systems, distance formula, spheres (and course intro) (review by reading section);

    Here is an (ignorable) 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 [] and illustration of the problem. All you have to do for such solution worksheets in the future is read them if you are interested. Don't worry, we will take it slow with Maple.]

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

    Homework (light first day assignment):
    Click here to register with WebAssign; class keys 10:30-001: villanova 9185 9639, 1:50-003: 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.

    Make sure you read my welcoming email senta week before the first class, and register with WebAssign (immediately, if not already done). Explore the on-line resources. Read the pages linked to our class home page.

    Send bob an email as directed here within 24 hours so I can assemble a class list with your chosen nicknames.

    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: 11, 15, 17, 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. It is important that you read the section in the book from which homework problems have been selected before attempting them.

    Download Maple 2020 
    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.

  2. W: Lecture Notes12.2a vectors;
    look over the  handout on diff/int/algebra;   short intro to Maple;  "Ask your Teacher" "Request Extension';
    Problems which are not available in WebAssign will be red  square bracketed [[...]] and should be done outside of WebAssign but not turned in.
    [[Read this 12.1:  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]

    Remember WebAssign vector input requires the Math Vector palette using boldface i, j, k notation. 

  3. Th: Lecture Notes12.2b plane vectors and trig;
    12.2: vector diagram problems in the plane [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; are the lengths of the ropes relevant?]

  4. F: Lecture Notes12.3a vector ops: dot product;
    BlackBoard release: Quiz 1 on 12.1 distance formula, sphere analysis (completing the square), remember: midpoint coordinates are average of endpoint coordinates! Due Saturday midnight;
    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 soln: pdf, mw]],
     [[57 chemical geometry; soln: .mw, plot: ]].

    WEEK 2:
  5. M: SNOW DAY. sneak attack!

  6. W: Lecture Notes12.3b vector ops:  projection;
    motivation:  "orthogonal projection" visualization is a trio of vectors with the same initial point;
    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.

  7. Th: Lecture Notes 12.4 vector ops: cross product; [mw];
    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 by adding a zero third component  and then use the cross product),
    31 (Maple example:,
    33, 35, 37 (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]]

    > <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:]

  8. F: Quiz 2 on projection ;
    Lecture Notes 10.1 parametrized curves;
    detour: handout on parametrized curves [also]:
    [ignorable: textbook example curves: (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!;
    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]].

    bob forgot the name of the cycloid curve on Friday! Mark your bicycle tire at any point and ride in a straight line, that point moves along a cycloid path (common root "cycle"!)

    WEEK 3:
  9. M: Lecture Notes 12.5a lines and planes;
    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, 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].

  10. W: Lecture Notes 12.5b points lines and planes: separations; [mw]
    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: mw),
    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 worksheet: vector projection takes you to nearest point

  11. Th: 12:R (Review problems are not in WebAssign, but in the e-text at Chapter 12 Review, Exercises section):
    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 conver to meters],
    22 [distance between point and plane],
    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;
    for fun: 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]];
    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?
    in class we work together on 26.

  12. F: Quiz 3 on cross product, projection, eqns of lines, planes;
     Lecture Notes 13.1 vector calculus: space curves;  [cubic, cutcylinder];
    13.1: vector calc! [1], 3 ,[5], 7, 13, 21, 25
    [optional: 21-26, for fun only, do quickly by thinking, note technology is not necessary here to distinguish the different formulas: .mw],
    , 33,
    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];
    Maple assignments start (read these instructions): note asterisks *;
    [39*: refer back to similar problem 27: note that  z2 = (x2+y2) so you need to take the square root! 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];
    [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)].

    Valentine's Day. pink lace heartMaple hearts!

    WEEK 4:
  13. M: Find at least one (at most 3) Maple partners for the first assignment; ask bob in class for help if you have trouble;
     Lecture Notes 13.2a vector calculus ops; [
    13.2 : [[1: pdf, 2]], 5 [recall: exp(2t) = (exp(t))2, what kind of curve is this?], 7, 10, 15, 19, 21, 31 [or], 
     [[do 29a (by hand),then
    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:;;
    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]
    or just:    
      with(VectorCalculus): BasisFormat(false):
    >  ∫  F(t) dt
    > with(plots):
    > spacecurve(F(t), t=0..1, axes=boxed)

    Wednesday-Thursday are "no due date" days so our HW assignments are delayed "due" till Saturday midnight.
    Extensions are always granted if you need more time.

  14. W:  Lecture Notes 13.2b vector calculus and vector ops followup;
    Maple worksheet 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 the sum rule is verified by expressing both sides in components]],
    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]

  15. Th: Another snow day.

  16. F: Quiz 4 due Sunday midnight;
    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.]
    [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 with Maple, choose "Approximate" from the context sensitive menu];
    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;
     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:
  17. M: Lecture Notes 13.3b curvature; [osculating circle play];
    13.3 (curvature): 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, adjust for the red number which enters the WebAssign version];
    47 [twisted cubic: perfect square! (slightly different rescaling)], 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]):
    2,0>,t=-1..1)      from the Tools Menu, Tutors, Vector Calculus, Space Curves [choose animate osculating circles]

  18. 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):
    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],
    [41? also perfect square, see 11] , 45 [visualize it!]

    Quiz 3,4 answer keys posted in archive

  19. Th:  13.R: [4, 6, 10, 16, 24];
    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);
    [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]

  20. F: Test 1 (take home) due by class time Monday.

    WEEK 6:
  21. M: assignment due anytime this week thru the weekend (no pressure!)
    Lecture Notes 14.1 real functions of n>2 independent variables
    [graphs, Plotbuilder]; ignore intro video in e-book for now!
    14.1: 1, 3, 11, 15, 25, 31, 35, 41, 49;
    | problems begin:
     [[55*, do a single appropriate plot3d and contourplot after loading plots and defining the maple function f (x,y) or use 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).

  22. W: Lecture Notes 14.2 multivariable limits; [];
     [1], 2, 9, 13, [15], 17,
    [[23*,[toolbar plot option: contour, or "style=surfacecontour" or  right-click style "surface with contour", explain in comment]],
    , 31, 37 [does a 3d plot of the expression support your conclusion?], 39.

  23. Th: Lecture Notes 14.3a 1st order partial derivatives; [visualize mw]
    14.3 (partial derivatives, finally): [1], 3, 5, 11, 15, 17, 21, 31, 33, 41, 51 [in class if time: 24, 26, 30, 38, 42].

  24. F: Quiz on 14.3;
    Lecture Notes 14.3b higher order partial derivatives, etc;
    14.3: second and higher derivatives (and implicit differentiation!; optional: why partials commute)
    47, 49; 53, 56, 59, 63,
    65 [use this example for higher derivatives];
    [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],
    [[read: 81]], 83, [[ 84, 88 ideal gas law]], 90.

    WEEK 7:                                      International Women's Day! today
  25. 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: 1, 3, [7,] 11, [15], 17, 21, 22];
    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].

  26. W: 14.4:  Lecture Notes 14.4b differential approximation;
    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:
    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:]

  27. Th: 14.5:  Lecture Notes 14.5a chain rule; [Laplacian];
    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.

  28. F: Quiz 6 on linear approximation and differentials;
      Lecture Notes 14.5b chain rule etc; [2nd derivatives];
    14.6: (related rates, etc) 37,  39, 44.
    [[related rates 43, wave equation: 49 ]].

    Sunday was π Day!   Pi Day:  π!  [and Einstein's birthday]

    WEEK 8:                     Test 1 Answer key online, please check
  29. M:  Lecture Notes 14.6a directional derivatives; [visualize it];
    14.5: 41;
    14.6 (directional derivatives; stop at tangent planes): 1, 3, 5, 7, 9, 11, 15,
    [first find a unit vector in the given direction! sketch the two points],
    [just split the gradient into its length and unit vector direction information], 29.

    Tomorrow is a no  submission/deadline day...
  30. W: Lecture Notes 14.6b gradient and contours;  [visualize it];
    handout on derivatives of 2d and 2d functions [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 ]] , 65;
    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.

  31. Th:  Lecture Notes 14.7a max-min stuff;
     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.

  32. F: Quiz 7 on critical points, second derivative test, directional derivatives and gradient (available now);
     Lecture Notes 14.7b more max-min stuff; [visualize it];
    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),
    ,  [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].

    WEEK 9:  Check Answer key for quiz 6. Quiz 7 answer key online Wednesday;
  33. M: 14.R: (review problems; note some of the highest numbered problems not chosen here refer to 14.8, which we did not do): 
    in class if time: 1, 7, 15, 18, 21, 25, 29, 31, 33, 34a, 39, 53, 64;
    We try to do a split screen side-by-side Zoom (bob Maple screen)  and Maple (your screen)  exercise together with classifying critical points:
    14.7.12 [14.7.12-nocomment];
     then students work on the max-min word problem 14.7.52 in the breakout rooms:
      [answer: [pdf, mw],  the height is 2.5 times the square base; obviously cost of materials is not the design factor for normal aquariums, no?].

    Recall summary in prep for weekend Test 2:
    ignorable bonus handout on multivariable derivative and differential notation];
    handout on derivatives of 2d and 2d functions.

  34. W: 15.1a: Lecture Notes 15.1a interated and Riemann integrals; [cross-sections; Riemann on rectangles]
    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, in class we do this problem together using the Maple Approximate Integration Tutor in this template worksheet, input x*exp(-x*y) etc];
    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],
    [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].

  35. Th: Lecture Notes 15.1b integration over rectangles, average value; [visualize it];
    15.1: 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)   OR:
    >  with(Student[MultivariateCalculus]):

    >  MultiInt( x + y, y = c..d , x = a..b,output=steps)

  36. F: Test 2 on chapter 14 started in class and finished on the weekend. Due by Tuesday.

    WEEK 10:
  37. M: 15.2 (Lecture Notes on double integrals, nonrectangular regions)  [bob example, [ex.4]]
    handout on double integrals [necessary to change order: visual1; visual2];
    explanation of supporting diagram for double integrals]  <<< this is important for communicating how you iterate a double integral
    5, 13, 15, 17, 23, 25, 27; 35, 39, 49, 51 [[ optional: 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.

  38. W: 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;
     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, 8, 11, 12, 17, 19, 21, 25, 30, 33, 37 (all short review problems);
    part of
    [[ 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 (page 5 of today's notes), and vertical and horizontal lines, and lines passing through the origin, as in the handout examples.

  39. Th: 15.3 (Lecture Notes 15.3 on polar coordinate integration; [examples, poolFreeth];
    handout: polar coordinate integration examples: page 2;
    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],
    [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 in both Cartesian and polar coordinates yet again to hammer home the point. The quiz will involve all three for the same region.

    Easter Weekend.  

    WEEK 11 [-1]:  recall where we were:  polar coord interation; factorizable integrals;
  40. M: 15.4: Lecture Notes 15.4a on centers of mass/centroids; [visualize it];
    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:].

  41. W: Lecture Notes 15.4b on probability;
    [if you need to review one variable probability, view this 3 minute video: section 8.5, then 2-d case 4 minute video 15.4.3];
     [Wiki: standard deviation?, Poisson distribution?];
    15.4:  27, 29,
    30 (30a: P(x<=1000,y<=1000) =??, 30b: P(x+y<=1000) = ??] [Maple is the right tool for evaluating probability integrals!],
    [[optional normal distribution: 31*, use this template for the normal distribution;]].

    We are skipping 15.5 surface area.

  42. Th: Lecture Notes 15.6a triple integrals;
    handout: example of iterating triple integral 6 different ways [tetrahedrons:,, explore];
    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!],
     ("deconstructing a triple integral"): 27.

  43. F: Quiz 8 on double integration in both Cartesian and polar coordinates;|
    Lecture Notes 15.6b on deconstructing triple integrals [15.6.34];
    in class exercise: exercise in setting up triple integrals in Cartesian coordinates (please take this seriously, ask bob if you need help, send scan of work);
    after this exercise together in breakout rooms,
    try to sketch 31 using the same technique: solid enclosed by y = x2, yx, 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.

    15.6: 31, 33, 35

    WEEK 12[-1]: test 2 answer key and graded tests online;
  44. M in class we try together (bob on his white board, you on paper)  to sketch the following solid (3d centroid example):
    23 [Use a triple integral to 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 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.] ;
    next  y = x2 , y + z = 1, z = 0 ;
    Lecture Notes 15.6c on center of mass; [centroid of hemisphere, wedge of cylinder, wedge of parabolic cylinder];

    15.6: 40, 41, 42.

    Warning: the textbook fails to identify triplets of values of cylindrical or spherical coordinates!

  45. W:15.7-8: Lecture Notes 15,7-8a on cylindrical/spherical coordinates;
    handout on cylindrical and spherical regions of space and their bounding surfaces:
    handout on cylindrical and spherical triple integrals: examples wedding ring / snow cone setup 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.

  46. Th: 15.7-8: Lecture Notes 15,7-8b on cylindrical/spherical coordinates; [snow cone, sharp donut];
     [Wiki: orthogonal coords,  spherical coordsoblate spherical coords (Earth and black holes are oblate spheriods)];
    15.8 (spherical coords): 1, 3, 5, 7, 9 (use double angle formula!), 11, 13 , 15, 17, 25, 29, 35, 41.

    [WARNING: failure to identify coords! in problems 1-2]

    Triple integral exercise solution.

  47. F: Quiz 9 on triple integration in Cartesian coordinates;
    15.R  (using arctrig functions in a polar coordinate integration example pdf, mw);
     review: integration over 2d and 3d regions;
    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],
    Breakout Room problem 2: 48 [3d cartesian to spherical]
    , **
    Breakout Room problem 1: 53 [3d re-order]
    . **
    "in class" we try ** problems []

    WEEK 13[-1]:
  48. M: Lecture Notes 16.1 on vector fields!; [visualize it; linear flow]
     16.1: 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)2sin(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.

  49. W: Test 3 on integration in class, then take home. Due Monday.

  50. Th: Lecture Notes 16.2a  on scalar line integrals; [example mw];
    handout on scalar line integrals (ignore text discussion of "scalar" integrals with respect to dx, dy, dz separately)
    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].

  51. 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!):
    7 [ C <x+2y, x2> · <dx, dy>], 15, 17, [read: 27 Maple template for vector line integrals], 32[use template .mw] , 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.

    WEEK 14[-1]:
  52. M: Test 3 due today;
    Lecture Notes 16.3 on conservative vector fields; [inverse square force line integral 16.3 example 1; counterexample];
     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.

  53. W:  Lecture Notes 16.4 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.]

  54. Th:: Lecture Notes 16.5 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],
    , 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].

  55. F: Optional [not for final exam!]:
    this is the culmination of 3 semesters of calculus bringing together all elements of what preceded this!
    Lecture Notes 16.6-9 on surface integrals and Gauss and Stokes;
    [flux motivation: sunshine on Earth surface and seasons; Maxwell's eqns]
    16.6-7: surface integrals for fun;
    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)].

  56.  @ M: CATS first 10 minutes (link in BlackBoard); final exam discussion, see archived finals: (and test 3 answer key);
    line integrals, Green's theorem, curl, divergence, potential functions for conservative vector fields, polar coordinates.

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MAPLE HW files??: due:  Week 6 due:  Week 10

*MAPLE homework log and instructions [asterisk "*" marked homework problems]

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

FINAL EXAM: take home exam released to all at section 001 exam time slot, due back by Monday May 10;
001: MWF 10:20 AM Wed, May 5 10:45 - 1:15
003: MW 1:50/MWF 1:50 PM Sat, May 8 2:30 - 5:00

Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS

3-dec-2020 [course homepage]
[log from last time 20s taught with Stewart Calculus 8e]

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