MAT2500 24S [Jantzen] homework and daily class log

Jump to current date! [where @ is located]

Your homework will appear here each day as it is assigned, with occasional links to some MAPLE worksheets when helpful to illustrate some points where technology can be useful. [There are 56 class days in the semester, numbered consecutively below and  labeled by the (first initial of the) day of the week.]

It is your responsibility to check homework here (not all problems are in WebAssign). (Put a favorite in your browser to the class homepage.) You are responsible for any hyperlinked material here as well as requesting any handouts or returned tests or quizzes from classes you missed. Homework is understood to be done by the  next class meeting (unless that class is a test, in which case the homework is due the following class meeting). WebAssign "deadlines" are at 11:59pm of the day they are due, allowing you to complete problems you have trouble with after class discussion. [If Friday is quiz day, HW due Friday has a WebAssign deadline of midnight Monday.]  The WebAssign extension tool can be used to request more time or more attempts for a given homework assignment, so persistence can give any student 100 percent credit on the homework cumulative grade.

*ungraded asterisk marked problems are to be done with MAPLE as explained in the separate but still tentative MAPLE homework log, which will be edited as we go.

Textbook technology: WebAssign homework management/grading is required, giving you access to an incredible wealth of multimedia tools together with the online e-book textbook you can access from any internet connection. WebAssign deadlines are suggested to keep students on track, but extensions in time and number of tries are always granted.
Textbook problems
are labeled by chapter and section and problem number, and identify each WebAssign problem. Those problems which are not available in WebAssign will be square bracketed [[...]] and should be done outside of WebAssign. Check this page for hints and some linked Maple worksheet solutions. Use the Ask Your Teacher tool to get help on any problem for which you cannot get the correct answer. There is no reason for anyone not to get 100 percent credit for the homework assignments. This is the most important component of learning in this class, doing problems to digest the ideas. Use the Extension tool for requesting more tries or more time. All requests are granted.

If you have any questions, drop by my office St Aug 370 (third floor facing Mendel by side stairwell) or just come to see where you can find me in the future when you might need to. I welcome visitors.

  1. W (January 17, 2024): Introduction and Overview
    We will access our e-textbook/HW WebAssign portal through BlackBoard with your laptop or phone.
    [Use appropriately: "Ask your Teacher" and "Request Extension"]

    Homework (light first day assignment):
    Make sure you read my welcoming email sent before the first class, and register with WebAssign (immediately, if not already done) and send an email with your schedule attached as described there (follow up with a reply about yourself by the weekend). Explore the on-line resources. Read pages linked to our class home page.
    Read what Villanova students say about the most important things you can do to succeed here.

    Make sure you have Maple 2023 on your local computer, available by clicking here.  if you haven't already done so and install it on your laptop when you get a chance (it takes about 15 minutes or less total), If you have any trouble, email me with an explanation of the errors. You are expected to be able to use Maple on your laptop when needed. We will develop the experience as we go for calc 2 topics. [There is also a Maple Calculator app for your phone!]
    No problem if you never used it before.


    just skim briefly:
    Lecture Notes12.1 rectangular coordinate systems, distance formula, spheres (and course intro)

    WebAssign Problems: WebAssign101 is a very quick intro to WebAssign.
    Read 12.1 reviewing 3d Cartesian coordinate systems, distance formula and equations of spheres;
    12.1: 13, 17, 19, 26, 49 (WebAssign has random numbers in your problem);
    This short list is so you can check out our class website and read about the course rules, advice, syllabus, etc. It is important that you read the section in the book from which homework problems have been selected before attempting them.

    Anyone who wishes access to all the textbook homework problems for extra practice, ask bob for the enrollment key.

  2. 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. 

  3. F: Quiz 1 on 12.1  (see archive) distance formula, sphere analysis (completing the square) due Monday in class
    SNOW DAY. So we are now one day behind... :-(



    WEEK 2[-1]:
  4. M: Lecture Notes12.2b plane vectors and trig;
    12.2: vector diagram problems in the plane [example 7, showing DOC versus WS modes, wind speed example];
    32 (express the components of each vector, add them exactly (symbolically), evaluate to decimal numbers, find magnitude and angle with maximum number digits, then round off as requested; this worksheet shows how to do vectors in Maple),
    35 [like example 7], 39 [like wind speed problem],
    45
    [linear combinations in the plane].

  5. W: Lecture Notes12.3a vector ops: dot product [page numbers rearranged];
    check sign up roster for data; paper copy will be printed and distributed Thursday;

    12.3 online summary handout on dot product: [example 3];
    1, 2, 5, 9,11, 15, 21, 23, 33;
    optional (only for math lovers) a fun problem if you like math:
     [[
    55 geometry soln: pdf, mw]],

  6. Th: Lecture Notes12.3b vector ops:  projection;
    motivation:  "orthogonal projection" visualization is a trio of vectors with the same initial point;
    called "resolving a vector" in physics/engineering contexts;
    handout on resolving a vector; [orthogonal >= perpendicular];

  7. F: Quiz 2 (see archive) on projection   [animation, slider]
    Lecture Notes 12.4 vector ops: cross product; [mw]; 
    USE MAPLE FOR ALL CROSS PRODUCT EVALUATIONS except some simple i, j, k notation examples;

    12.4 cross product:  online summary handout on geometric definition;
     [why component and geometric definitions agree: crossprodetails.pdf];
    1, 7, 11, 13, 16, 19, 27 (find 2 edge vectors from a mutual corner first, use 3 vectors by adding a zero third component  and then use the cross product),
    31 (Maple example: trianglearea.mw),
    33, 35, 37 [use Maple; zero triple scalar product => zero volume => coplanar],
    39 (first redo diagram with same initial points for F and r).
    [[ignorable: 54, but why are a) and b) obvious? visualization]]

    Note:
    >
    <2,1,1> · (<1,-1,2> x <0,-2,3>)  
    [boldface "times" sign and boldface centered "dot" from Common Symbols palette parentheses required]
    >
    <2,1,1> · <2,1,1>     then take sqrt (Expressions palette) to get length [
    example worksheet: babyvectorops.mw]


    WEEK 3[-1]:
  8. M: Lecture Notes 10.1 parametrized curves;
    detour: calc2 review parametrized curves in Maple
    [summary handout on parametrized curves]
    [ignorable: textbook example curves: s9-10-1.mw (wow!)
    if interested open this worksheet and execute it by hitting the !!! icon on the toolbar, then read];
    it is not very useful to try to draw parametrized curves based on what the graphs of x and y look like: technology is meant for visualizing math! it is harder in 3d.
    BUT REMEMBER, WE ONLY NEED TO WORK WITH SIMPLE CURVES.
    This detour now is because we want to describe equations of simple straight lines, later spacecurves in Chapter 13.

    10.1: 3 [evaluate the endpoints to determine direction], 8, 9,
    22 [hyperbolic functions: cosh2 x- sinh2 x = 1, recognition is enough], 24,
    26 [use 1-1 and show black grid].

  9. W: Lecture Notes 12.5a lines and planes; check out the answer key to quiz 2 please;
    summary of equations for  lines and planes [linesplanes.mw]; vector equations rule!;
    never use the symmetric equations of a line: they are useless for all practical purposes!;
    12.5 (equations): 1 (draw a quick sketch to understand each statement),
    3, 4, 5, 13, 17, 19, 23, 31, 41, 45, 51, 57 [.mw].

    Remember, these problems are exercises in using vector operations: the goal is to use them and not some other way to do them.

  10. Th: Lecture Notes 12.5b points lines and planes: separations; [mw]
    summary online handout on geometry of points, lines and planes (distances between);
    in these problems do not just plug into a formula: this is practice in vector projection geometry, we really don't care about the distance!
    draw a picture, make the projection diagram, evaluate the appropriate projection:
    12.5: 69 ( find point on line, project their difference vector perpendicular to the line ),
    71 (find point on plane, project their difference along the normal) ,
    73 (find pt on each plane, project their difference vector along the normal);
    76 (Don't use distance formula; draw a figure, find a point on the plane and move from it along the normal in both directions to get a point in the two desired parallel planes, then write plane equations),
    78 (find pt on each line (set parameters to zero!), project the 2 point difference vector along the normal to the parallel planes that contain them);
    [optional challenge problem for the curious: 81].

    Ignorable but fascinating worksheet: vector projection takes you to nearest point

  11. F: Quiz 3  on lines and planes (see archive); view figure in color online; previous Quiz answer keys are online;
    12:R (Review problems are not in WebAssign, but in the e-text at Chapter 12 Review, Exercises section, this worksheet link has solutions):
    13 [resolution in the plane],
    14 [torque is position vector crossed into the force vector,  use geometric definition, vectors must have same initial point to compare for included angle, MKS units mean convert to meters],
    22 [distance between point and plane],
    23 [Start by going back to the parametrizations by setting each ratio to the parameter t or s,
           as in  t = (x-1)/(2) -> x = 1+2 t (solve for x)]
    24 [2 intersecting planes],
    25 [find a point on the line of intersection, use the stated normal, write the eqn of the plane],
    26 [4 part problem, use parametric equations for part b)],
    check with bob if you have questions about any of these;
    27 [2 parallel planes]

    in class we work together on 26

    optional: read section 12.6 only for fun, since we will be using quadric surfaces during the course so why not skim through this material quickly?
    for mathematically curious: 38 [what is the distance of a point from any coordinate plane?---what is the distance of the point (x,y,z) from the plane y = 1? [soln is ellipsoid]]

    WEEK 4[-1]: TO EDIT:
  12. M: Lecture Notes 13.1 vector calculus: space curves;  [cubic, cutcylinder]; [eliminate parameter]
    Use Maple to plot spacecurves always;
    trying to figure out what curves look like from their equations is a waste of time here
    ;
    13.1: 1, 3 ,5, 7, 9, 15, 23, 33, 35 [just copy and paste together, but choose t =-10..10 for curve], 39,
    47.

    Maple stuff:
    > F(t):= <t, t2 , t3> : F(t)  creates a Maple function, PlotBuilder will plot it (not the function definition)
    > F '(t)

    > with(plots):
    > spacecurve(F(t), t=0..1,color=red)   or use PlotBuilder

  13. W: Lecture Notes 13.2a vector calculus ops; [twisted cubic: mw];   [integrate velocity, angles: mw2];
    13.2 : [[ignorable  1:  pdf, 2]], 3, 5 [recall: exp(2t) = (exp(t))2, what kind of curve is this?], 7, 10, 15, 17, 19, 23, 27;
     (see Maple video: secantlinevideo.mw; vectorfunctionlinearapprox.mw)

  14. Th: Lecture Notes 13.2b vector calculus and vector ops followup;
    Maple worksheet vectoravg.mw to see how the integral of a vector-valued function can be interpreted visually; rules of differentiation extend from scalar-functions to vector-valued functions easily (example below).
    13.2: 35 [angle between tangent vectors], 37, 43, 49, 50;
    [[product rule examples: 56, use combined dot/cross product differentiation rules 4,5 from this section:
      (a · (b x c)) ' = a' · (b x c))  + a · (b' x c))  + a · (b x c')) ]].
    [product rule holds for all the products involving vector factors as long as you keep the order of factors the same in each resulting term if cross products are involved;  usual sum rules always apply]

    To use dot product from Common Symbols palette:
    >
    with(VectorCalculus):  BasisFormat(false):

    [ignorable template for plotting curve and tan line together]
    [ignorable technology to do vector integrals]

  15. F: Quiz 4  (see archive):
    Lecture Notes 13.3a arclength; [numerical integration
    13.3 (arclength): arclength toy problems require squared length of tangent vector to be a perfect square [or a factorization that makes a u-sub work!] to be integrable usually!
    [ignorable summary arclength and arclength parametrization];
    [integral of speed with respect to time is distance traveled! so the length of a curve is the integral of the magnitude of the tangent vector! easy.]
    3, 5 [perfect square]; 9;
    11 [need to abort Maple's integration, convert to inert form here],
    13 [hint: to parametrize the curve, first express y and then z in terms of x, then let x = t; another perfect square],
    14 [hint: let x = cos(t), y = 2 sin(t) for the ellipse, then solve the plane eqn for z to get the parametrization, for one revolution of this ellipse;
    for the approximate integration, use the circled exclamation mark on the top line center of the  toolbar to stop Maple from going on forever, then make the integral inert selecting it in the input region and using the Format menu, Convert to, Inert Form. Then you can numerically approximate it no problem.]

    WEEK 5[-1]:
  16. M: Lecture Notes 13.3b curvature; [osculating circle play];
    13.3 (curvature): 20, 23,  31 [do not use formula 11: instead use the parametrized curve form r = <t,a t4,0> of the curve y = a x4, then let t = x to enter in WebAssign, adjust for the red number which enters the WebAssign version];
    51 [twisted cubic: perfect square!], 53,
    55 [Same as for problem 31: r = <a cos(t),b sin(t),0> . For osculating circle, normals at axis intercepts are pointing inward along axes, just go distance ρ along axis to get center, write equation of circle with radius of curvature there.]

    > with(Student[VectorCalculus]):
       SpaceCurveTutor(<t,t
    2,0>,t=-1..1)      from the Tools Menu, Tutors, Vector Calculus, Space Curves [choose animate osculating circles]


  17. W:  Lecture Notes 13.4 motion in space;
    13.4 (splitting the acceleration vector, ignore Kepler's laws):
    [[numerical derivatives: read 1 , so you don't waste time entering data]],
    3, 5, 11 [v = exp(t) + exp(-t) since v2 is a perfect square],
    17 [optional: graph; rotating the curve around, see the animation],
    19 (minimize a function when its derivative is zero (critical point)! confirm minimum by plotting function);
    40 [perfect square!],  45 [read this for the answer and visualize it!].

    [ignorable summary on geometry of spacecurves and space curve curvature and acceleration;
     Maple worksheets for these handouts: rescaled twisted cubic, helix]

    Valentine's Day:    but  are better; heart in a box 

  18. Th: suggested review problems: 13.R: [4, 6, 10, 24]; [11: TNB frame]
    in class we play with 24 using this worksheet (without continuous curvature but just continuous value and slope, such piecewise curves joining points are called cubic splines, important in engineering and design);
    [ignore this one 14a: use parametrized curve r = <t, t4 - t2,0> to evaluate the curvature; by symmetry  N(0) easily must point down find osc circle: x2 + (y+1/2)2 = 1/4, 14b*: edit the template and do the zoom plot to see the close match of the circle to the curve, including the separate original plot showing the large scale behavior];

    [highway banking pdf soln]

  19. F: Quiz 5 on line! (see archive) good prep for Test 1;
    Lecture Notes 14.1 real functions of n>1 independent variables [text graphs, plots, Plotbuilder];
    Maple is the appropriate tool to make multivariable plots.
    14.1: 3 [WebAssign graphing? report back, I vetoed the rest], 5, 11, 16, 18, 20,
    35 [let me know if you get hassled for your guesses!], 36, 45,
    81 (common answer here, showing how the data is fit).

    WEEK 6[-1]:
  20. Test thru chapter 13 Thursday. Change: Tuesday 5:30pm review session in OldFal 105
  21. M: Lecture Notes 14.2 multivariable limits; [plot-explore.mw];
    14.2:
     2, 11, 23, 39, 42, 45, 53
    [all the continuous function limits were skipped as trivial]
    [toolbar plot option: contour, or "style=surfacecontour" or  PlotBuilder style "surface with contour"].

  22. 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.

  23. Th: Test 1 thru chapter 13

  24. F: no quiz (test week);
    Lecture Notes 14.3b higher order partial derivatives, etc; [example: decimal formulas]
    14.3: second and higher derivatives (and implicit differentiation!; optional: why partials commute)
    50, 51, 52 [handwork simpler!], 53, 59 [use this example for higher derivatives in Maple],
    72 [sign of fxy? is fx increasing or decreasing in the y direction?],
    73 [Read this problem, but only do parts a and b. Part c has 0 credit to help you avoid doing too many calculations. just average the adjacent secant line slopes on either side of the point where the partial derivative is to be evaluated, as in the opening example: mw, pdf, this is not a testing problem! tedious so I show you how to work through it, in fact no red numbers, this is the solution, just read through it and enter the final numbers],
    83, 85 [nonideal gas; implicit differentiation, each partial derivative assumes the third variable is held constant; it is easier to do this by hand than by Maple!].
    [[read: 81 PDE example]]



    WEEK 7[-1]:
  25. M: Lecture Notes 14.4a linear approximation;
     
    [linear approximation and tangent planes: differentiability illustrated; moving tangent plane, differentiability-zoom]

    summary: linear approximation;
    14.4: 3, 5, 20, 25, 27, 28 [tabular data linear approximation].

  26. W:  14.4:  Lecture Notes 14.4b differential approximation;
    summary: differential approximation and error estimates;
    31, 37, 39, 41, 42 [hint: differential of volume but change in height is twice thickness],
    48 [first solve for P],
    49 [compute differential d (1/R) =-R^(-2) dR etc,  this worksheet explains how a general derivation without specific values answers the question once and for all; hand derivation: pdf].

    Optional examples.
     4x6prints?; [diff approx example]

  27. h: Test 1 back; check BlackBoard grades against those on your paper quizzes and test;
    come discuss with bob if grade not high enough for your tastes;
    14.5:  Lecture Notes 14.5a chain rule; [Laplacian];  (I never use tree diagrams);
    chain rule: 1 [just repeat answer for part 2, don't waste time], 9, 19, 27, 39, 40.

  28. F: Quiz 6 (see archive) due Wednesday after break; more chain rule practice:
     Lecture Notes 14.5b chain rule etc; [2nd derivatives]  [[read wave equation: 51 ]];
    14.5: (related rates, etc):
      [[41: solution explained, read, insert answer]], 42, 44, 45, 
    related rates: 47 [cute animation], 48 [Doppler effect], 51 [wave superposition].


    Spring Break.  

    WEEK 8[-1]:
  29. M: Lecture Notes 14.6a directional derivatives; [visualize it]
    14.6 (directional derivatives; stop at tangent planes to level surfaces): 1 [graphical estimation], 7, 11, 17, 25, 27, 35.
     
    optional: [44 graphical estimation of gradient]

  30. W: Lecture Notes 14.6b gradient and contours;  [visualize it 3d (temperature example)];
    level surface tangent planes; note z = f(x,y) corresponds to F(x,y,z) = z - f(x,y) = 0):
    14.6: 33, 37,
    49, 51  [visualize it 3d],
    55, 60 [set gradient equal to k n, where n is a normal to the given plane and solve for x,y,z , then backsub results into original equation for surface to determine k],
    65 [plug parametrized coordinates into equation of surface to solve for the parameter],  
    71.

    ignorable: visualize it 2d [optional lecture notes example: 67].

     
  31. Th: Lecture Notes 14.7a max-min stuff;
     summary of  2D max-min 2nd derivative test [ignorable derivation];
     14.7 (textbook example 4): 1, 3, 5, 13, 14,
    25  (because of the symmetry, in first quadrant,  y = x, then solve by hand to get critical points, including second derivative test and evaluation of f at critical points, plot x=0..2,y=0..2,z=0..10 with context menu),
    27 (symmetry also here  y = x).
    you can use the plot3d command form context menu to see if it looks right!

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


    No office hour today. The dr is sick.

  32. F: Quiz 7 (see archive);
     Lecture Notes 14.7b more max-min stuff; [visualize it];
    14.7:
    boundaries: 35 [ignorable gene fraction example: mw,  pdf]
    word problems (box): 43 [minimize square of distance],
    51, 55 [soln: pdf, mw].


    WEEK 9[-1]: Test 2 on chapter 14 in class Thursday.
  33. M: work together on 14.7.54 first [solution handouts/s9-14-7-54.mw];
    [answer: height is 2.5 times the square base; obviously cost of materials is not the design factor for normal aquariums, no?],
    14.R: (review problems; note some of the highest numbered problems not chosen here refer to 14.8, which we did not do): some in class if time: 18, 21,
    25
    [tan plane, normal line but F(x,y,z)=z-f(x,y)=0], *** [solution 25, 29]
    29
    [tan plane, normal line but F(x,y,z)=LHS-RHS=0],
    33 [linear approx], 34 a or b [error analysis with differential], ***
    39 [chain rule], 42 [implicit differentiation],
    53 [critical pts analysis]. ***

    useful online summary  derivatives of 2d and 3d functions;
    with important summary of multivariable derivative and differential notation


    Tuesday 5:30pm voluntary problem session room Old Falvey 104 (next to our classroom).

  34. W: Quiz 7 back (see archive); remarks; more review.
    error estimation problem [soution: pdf, mw];
    14.7: 8, 16 [Maple gives the critical points, classify them, check contourplot and plot3d]

  35. Th: Test 2 on chapter 14.

  36. F:  Lecture Notes 15.1a interated and Riemann integrals; [cross-sections]
    See Maple Tools Menu, select Tutors, Calculus Multivariate, Approximate Integration Tutor (midpoint evaluation usually best);
    Only do midpoint part b) in 1, 3:
    15.1: 1 [do this problem using the Maple Approximate Integration Tutor in this template worksheet];
    3, [don't do upper right],
    6 [midpoint sampling: (m,n)=(2,3), x along 20 ft side, y along 30 ft side: answer = 3600, edit worksheet for your red numbers or set up yourself],
    7 [for average value divide integral by area of rectangle, we talk about this next time].

    WEEK 10:
  37. 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)]

  38. W: Quiz 7b due Wednesday after Easter Break: open resource but no discussion;
    Answer key to test 2 (see archive);
    tangent plane to graph of f(x,y) is not a line, normal line to graph of f(x,y) is a line in space;
    when using linear approximation or differentials, all partial derivatives are evaluated at the "simple" point you start with.

    Easter Recess:  

  39. W:  Lecture Notes 15.2 on double integrals, nonrectangular regions
    [lecture ex.4, optional final page lecture example]
    online handout on double integrals [visual];
    15.2: 5, 13 [it is clear you have to integrate in the x direction first],
    17, 25 [make a diagram!], 35 [what is the base of this solid?],
    43, 55, 61.

    No office hour today: blood drive donation.

  40. Th: Lecture Notes 10.3 on polar coordinate grid and curves;
    [detour on polar coordinates for integration in the plane (review from MAT1505)];
    online handout on polar coordinates and polar coordinate integration (page 1, the integration is next time);
     (stop reading 10.3 after polar curves;
    but we only need very simple polar curves r = r (θ) this semester, no limacons or cardiods!);
    10.3:  3, 5 [in both cases make your own diagram first, reference angles are multiples of Pi/4 or Pi/3];
    9, 11, 22,
    25 [substitute directly for r 2 then divide through by r ].


  41. F: Quiz 8  (see template quiz 8 from F22 in archive);
    15.3 (Lecture Notes 15.3 on polar coordinate integration; [lecture examples, pool example, Explore];
    online handout: more polar coordinate integration examples: page 2;
    view this worksheet to understand how to draw an iteration diagram for polar coordinates;
    use Maple to evaluate trig integrals that you cannot do; it is the set up of the limits which is important;
    15.3: 4, 9, 13, 22 [express in polar coordinates, find intersection to get angular limits],
    32 [u-sub!],  35 [find out where the graphs intersect to find radius of circular region of integration!],
     39, 46 [the red number is irrelevant, everyone has the same problem].

    No office hour today: 2 meetings!

    WEEK 11: 4 weeks to go!
  42. M: Check Blackboard grades; Test 2 and Quiz 7b back together(answer keys in archive);
    Lecture Notes 15.4a on centers of mass/centroids [alternative story]
    ["lamina" just means a 2d region, like a flat aluminum piece with boundary where we ignore the thickness];
    15.4 (center of mass, "centroid" = geometric center when constant density): 1, 5, 7, 9, 13,
    17 [same for everyone, but let Maple do the evaluation of the double integral, setting up the limits is the important thing here, not tedious algebra!].

    optional read handout on distributions of stuff (center of mass, centroid, averaging with a weighting function! this is similar to probability distributions)

    we skip moments of inertia--- of course who cares about centers of mass or geometric centers of regions (CENTROIDS!)--- but this is typical of many "distribution" problems, including probability, and we have some intution about where these points should lie so they are good practice in setting up integrals and seeing results which agree with our intuition.

    Don't look at the sun! [without proper glasses] no office hour today, viewing party!

  43. W:  late quizzes?;
    Lecture Notes 15.4b on probability (too ambitious review etc, just read the short textbook section);
    [textbook examples]
    [if you wish to review one variable probability, view the 3 minute video opening section 8.5];
    15.4: 29,
    exponential distributions 31, 32 (just use Maple to evaluate double integrals directly to decimal values).

    if you are curious:  [[normal distribution: 33 (no WebAssign problems!)]].

    We are skipping 15.5 surface area.

  44. Th: Lecture Notes 15.6a triple integrals; read first two sections, stop before applications;
    short version handout: example of iterating triple integral 6 different ways
    [tetrahedrons: tripleintexample.mw, tripleintegralexample2.mw, explore];
    changing iteration to adapt to rotational symmetry 15.6.example3 [pdf];
    15.6: 4 [resist temptation to just plug into Maple, go thru the hand steps at least this once: step by step Maple check],
    9 [diagram suggests y first, then z, then x],
    12 [once x int done, what is projection into y-z plane? draw 2d diagram of outer double integral],
    17 [diagram for out double integral enough],
    21 [use polar coords in the unit circle in the y-z plane for outermost double integral, do the innermost integral in the x direction],
    23, 25.

  45. F:  Quiz 8 answer key posted, please consult first;
    Quiz 9 on double integrals in cartesian and polar coords (variation of Q8 20S in archive);
    Lecture Notes 15.6b on deconstructing triple integrals (changing order of integration)[15.6.38];
    in class exercise: exercise in setting up triple integrals in Cartesian coordinates; [result discussed; pdf];
    then we will try setting up the figure for 35 together;

    15.6: 31, 35,
    37 [setup], [soln];
    39 [plot of solid to help you visualize after you try to draw this yourself].

    WEEK 12:
  46. M: Lecture Notes 15.6c on center of mass; [centroid of hemisphere, wedge of cylinder];
    Friday in class we tried together to sketch the following solid (3d centroid example):
    25: Use a triple integral to find the volume of the wedge in the first octant that is cut from the parabolic cylinder  y = x2  by the planes z = 0, y+z = 1  then use Maple to evaluate the triple integral you set up for its volume; then find the centroid: .mw, pdf;

    15.6: 44, 45 , 46 [These worksheets help you set up the problem.]

  47. W: Lecture Notes 15.7 on cylindrical coordinates;
    [weddingring.mw, snowcone.mw],
    15.7 (cylindrical coords): 1, 3, 6, 7,  15, [22], 25, 31, 32.

  48. Th: Lecture Notes 15.8 on spherical coordinates;
     [snow cone, sharp donut];
    15.8 (spherical coords): 1, 4, 5, 10, 15 [make an r-z diagram!],
    17, 20 [cyl coords NOT!], 24, 27,
    31 [by symmetry: centroid on z axis],  43 [set up so easy also in cyl coords]. <<<< make sure you can do these supporting diagrams for spherical integration.


  49. F: 15.R: Take home test 3 on chapter 15 out, due back a week from Monday on Day 53;
     we try selected problems in class:
    43a [rotational symmetry centroid; draw r-z diagram for the solid of revolution, slope intercept formula for z as functioni of r!],
    48 [cartesian to spherical],
    53 [change cartesian order] ; [see solutions 48, 53];
    18, 20 [switch order!],

    polar coords: 27, 38;
    47 [2d cartesian to polar],
    email bob if you have trouble with any of these.


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


     WEEK 13:
  50. M: Bring your Test 3 materials to class. You will spend the period working on that.

  51. W:  Lecture Notes 16.1 on vector fields!; [fieldplots]
     16.1: 2 [use signs of components and slope to understand direction], 7, 15,
    23 [factor components], 29 [read this worksheet],
    26, 31 [orthogonal to level curves, points towards larger values of function],
    37 [change in position is velocity at given point times increment in time].


  52. Th: Lecture Notes 16.2a  on scalar line integrals; [example mw];
    online handout on scalar line integrals
    (ignore text discussion of "scalar" integrals with respect to dx, dy, dz separately: these are really vector line integrals);
    16.2 ( ∫ f ds scalar line integrals only):
     2 [u-sub will work], 3 [use polar coords, then u-sub],
    11 [write vector eq of connecting line segment, t = 0..1];
    33 [perfect square simplification using cos^2+sin^2=1, ds/dt  independent of sin/cos],
    35 [35, scale up from lecture notes on unit circle], 
    optional 0 credit example of application: 50 (read?).


  53. F: Lecture Notes 16.2b on vector line integrals; [visualize];
    (  ∫ F · dr =   ∫ F · (dr/dt) dt  =  ∫ F · T ds =   ∫ F1 dx +F2 dy + ... ; always use vector notation!):
    16.2:  7 [ C <x+2y, x2> · <dx, dy>; piecewise parametrized curve: x = t or just substitute y=y(x) for y and dy],
    17 [r = r1 + t (r2 - r1), t=0..1],
    19, 23 [u sub works], 42 [let y = t; work = line integral of force vector field], 
    44 [straight line parametrization];
    optional no credit if you are interested otherwise ignore: 47
    [see worksheet for set up],
    53 [notice parallel projections along this line are constant on each line segment, so can multiply it by the length, add two separate results].



    WEEK 14:
  54. M: test 3 due in class; check current grades in BlackBoard;
    Lecture Notes 16.3 on conservative vector fields; [inverse square force line integral 16.3 example 1; counterexample];
    16.3: 3, 11 [find potential function to evaluate],
    22, 30 [find potential to evaluate],
    31 [any circle around the origin has nonzero line integral, no?]
    [Read only if curious 41].
    Optional note: the final section of 16.3 on conservation of energy is really important for physical applications and so it is worth reading even if it is not required for this course.

  55. Tu (F): Lecture Notes 16.4 on Green's theorem; "oint"! [example: 22]
    16.4 (Green's theorem): 3, 7, 9, 13 [this can be done by hand!], 21
    [optional: the line integral technique for integrating areas of regions of the plane is cute but we just don't have time for it so you can ignore it.]

  56. W (M):16.5: divergence and curl [not on final exam]
     Lecture Notes 16.5 on grad, div, curl
     Lecture Notes 16.5b on visualizing div, curl propertiesvisualize;
     [Maxwell's eqns, magnetic field lines, electric field lines]

    no office hour: dentist apptmnt

  57. @ Th: CATS reviews at beginning of class;
     Lecture Notes 16.6-9 on surface integrals and Gauss and Stokes (pages 4 and 5 only);
    parabola and hemisphere scalar and vector line integral examples;
    [flux motivation: sunshine on Earth surface and seasons].

    Friday 2pm Mendel G92 last minute problem session. Test 3 answer key posted online.


    Final Exam
    (not cumulative 16.1-4: scalar and vector line integrals, Green's Thm, potential functions, polar coords):


Come by my office for a 5 minute visit to see where I am when you might need help, especially if you are having any difficulty in the class so far. Discuss Test 1 once back if appropriate.

Tests:
Test 1:  Week 6:
Test 2Week 9-10:
Test 3: Week 12-13:

FINAL EXAM:
 [MWF 11:45 AM]  Sat, May 4     01:30 pm - 04:00 pm
 [MWF 12:50 PM]  Thurs, May 9 02:30 pm - 05:00 pm
[exchanging of exam slots is possible; ask bob]

Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS


3-jan-2024 [course homepage]
[log from last time 22F taught with Stewart Calculus 8e]









does anyone ever scroll down to the end?