MAT2500 20S [Jantzen] homework and daily class log

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 Red numbered problems are a relic of the old online HW hints that are now superceded by the e-book addons.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

  1. M (January 13, 2019):
    . By Friday, January 18, reply to my welcome e-mail  [] 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.)
    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: ]. 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.
    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 [] 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.

  2. 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]
  3. 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]

  4. 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 [, plot:]);

    WEEK 2[-1]:
    M: MLK DAY.

  5. 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.

  6. 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:,
    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]

    > <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:]

  7. F:  Quiz 2 on projection (easy! identical to 19S Quiz 1 with 3 new points);
    detour: handout on parametrized curves [also]:
    [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].

    WEEK 3[-1]:
  8. 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

  9. 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.

  10. 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].

  11. 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],
    , 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]:
  12. 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:;;
    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)

  13. W: 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], 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]

  14. 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.]
    [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].

  15. 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]):
    2,0>,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:,]

    WEEK 5[-1]: Test 1 this Thursday.
  16. M: 13.3: [in class derivation, plot]; Taylor approximation of a spacecurve:
    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.

  17. 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],
    [41? also perfect square, see 11] , 45 [visualize it!] HW due anytime next week, through the weekend. Please read the instructions carefully.

    Wednesday 5:30pm MLRC voluntary problem session

  18. Th:  Test 1 through 13.3, evaluation of formulas for various projections along curve.

    heart in a box Happy V-Day! [Maple]

  19. 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]: assignment due anytime this week thru the weekend
  20. M: 14.1 [text examples]: 1, 3, 11, 15, 25, 31, [33], 35, 41, 49;
    | 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).

  21. W: limits [; extra explanation for those who miss bob's commentary in class];
     [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],
    , 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.

  22. 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].

  23. 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];
    [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]:
  24. 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].

  25. 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:]

  26. 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].

  27. F: Quiz 6 on linear approximation, differential approximation [see 18s quiz 5];

    Spring Break.  :-) Enjoy. Be safe in your travels.

    WEEK 8[-1]:
  28. M: 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.

  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?

  30. 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.

  31. 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);
    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)];
    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?
  32. M: 14.R:. (review problems; note some of the highest numbered problems 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;
    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.

  33. 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],
    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?] .

  34. 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)

  35. 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]:
  36. 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.
  37. 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
    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.

  38. 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],
    [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.

  39. 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:].

    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.]
  40. 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].

  41. 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.

  42. 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 [];
    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: 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, 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.
    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.
  44. 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.

  45. 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": 

  46. 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.

  47. 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.

  48. 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)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.

    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:
  49. 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.

  50. 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].

  51. 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.

  52. 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:
  53. @  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.]

  54. 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],
    , 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. 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)].

  56. 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: due:  Week 6 due:  Week 10 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?