This is a tentative calendar for MATH 220-54 in the Fall 2003

The schedule below will be adjusted as needed.

Syllabus. -- E-mail your questions and comments!! --   rossa@xavier.edu

WEEK 1

M 8-25		Gett'n ready (NO CLASS -- yet...)		Maple worksheets		Homework
T 8-26		1.1  Questions				Send me your e-mail address and phone. Finish reading the section and answering the questions in your journal. Sec.1.1, #2,3,7,9,13,16,17,19,23,24,25 Write a brief response to Zucker's article. (Do you agree/disagree? Are you surprised? Is it all obvious?..)
R 8-28		1.1  Distance, Spheres, Planes, Cylinders		Introplots.mws		- Try to finish all exercises from 1.1.  - send me e-mail about which ones still stick.  - Take a good look at Introplots.mws (If you never used Maple: Intro.mws ) Read section 1.2.  Questions for journal.
F 8-29		1.2 Discuss Initial questions. Graphing in Maple (examples from 1.1) I will collect journals today.		3-D graphing		Sec. 1.2 #1,3,4abe,5,6,11 Begin thinking about a parametrization for your initials.

WEEK 2

M 9-1		Labor Day, No class
T 9-2		1.2 - return journals finding parametrizations of a given curve different parametrizations for the same curve parametric equations and Maple		graphing parametric equations with Maple		Use param equations to have Maple write your initials. Submit Maple worksheet by next Monday. Prepare Section 1.3.   Questions
R 9-4		1.3 vectors and basic vector operations				Sec. 1.3 # 4,5,7,10,9,6
F 9-5		1.3 discussion of questions/exercises. 1.4 vector valued functions (and connection       to parametric equations.)				Prepare Section 1.4.   Questions

WEEK 3

M 9-8		1.4 Discussion of prep. questions		vector valued functions (graphs)		Prepare Section 1.5.  Questions
T 9-9		Adressing many of your journal questions... Sec. 1.4 (cont.): Why does the componentwise derivative of a vector valued function produce a vector in the direction of motion whose magnitude equals (inst.) speed??
R 9-11		1.5 summary:  - derivative of position vectors... (sec. 1.4)  - parametric eq's <-> vector language  - arc length (from speed |p'(t)|)     incl. review of Calc I integral (pretend const speeds...) Exercises 12, 16a, 15e		Sec. 1.5 Expl 4		Sec. 1.5 #5,7,9,11,12,13,15,17
F 9-12		1.5 finish  - position/velocity/acceleration in vector context    (discuss issues with HW)  - visualize position, velocity, and acceleration vectors    (time didn't permit this... view Maple worksheet over weekend!)		velocity and acceleration vectors with Maple		Prepare Section 1.6.  Questions (Sec. 1.6. is quite long and full. Please do a good prep job!)

WEEK 4

M 9-15		1.6 Some introductory discussion and exercises				Sec. 1.6 #3,4,5,7,9,14,18,23,26,29,31,33  - italics indicate exercises which were (at least     partially) discussed in class - finish them!  - bold face numbers are most central  - the other exercises are no less important (but     maybe a little more advanced)
T 9-16		15 minute quiz about sec. 1.1-1.6 1.6: Proof of Theorem 4.        pojections        distance from point to line				Sec. 1.6 #25 E-mail me topics/questions from Sec. 1.6 which you hope to see discussed further in class. Prepare Section 1.7.    Questions
R 9-18		1.6 A few words about "work" 1.7 Lines  - vector eq.  - param eq.s  - symmetric scalar equations				Sec. 1.7 #1,4,5,6,7,12,13,15,17,19,21,23,26,31
F 9-19		1.7 discussion of HW problems				Get ready for test on Monday I will look at your journals on Monday.

WEEK 5

M 9-22		Exam 1 (sec. 1.1-1.7) Bring your journals!				Prepare Section 1.8.    Questions (For HW exercises see Questions)
T 9-23		1.8 We discussed algebraic and geometric properties of the cross product, as well as a number of exercises.				Finish problems for 1.8. Let me know what we will have to discuss in class. (e-mail) Preparation for Section 2.1
R 9-25		2.1 highlights:  - how are multivariate function different     or not from Calc.1 or Chapter1...  - level curves  - linear functions (more than one variable)  - representing functions (table, graph, ...)		Missouri Farmland - html - Maple		Prepare Secion 2.2: Questions For initial discussion of Section 2.2, try # 1,4,6,7
F 9-26		A brief Quiz (Sec.1.7, 1.8) Discussion of difficulties 2.1 Graphs and "maps" with Maple Comments				Journal Questions: Sec. 2.1: #13,15,17 and answer the questions: (1) Why are graphs of functions z = f(x,y)       surfaces? (2) Why are linear functions important? For initial discussion of Section 2.2, try Sec. 2.2 # 1,4,6,7

WEEK 6

M 9-29		Bizarre Graph Contest winners: Gaby, Therese, Dan - Congratulations! 2.2: Partial Derivatives: What are they? How do we find them? A first stab at it.		cross-sectional models		Sec. 2.2 #1,4,6,7,9,11,12,13,18
T 9-30		2.2 Partial derivatives Definition and Meaning connection to slope/tangency How to determine partials from  - formula  - table of function values  - contour map the case when both (all) partials are zero		from partials to the tangent plane		Over break:  - Finish Section 2.2 problems.  - Write down your questions.  - Take a real good look at the Maple worksheet. Questions? Prepare Section 2.3. Questions
R 10-2		Fall Holiday - no classes
F 10-3		Fall Holiday - no classes

WEEK 7

M 10-6		2.3 We discussed:  - the idea of linear approximation for functions    of one, two, and more variables - and how the    idea for "one" extends to two and more  - how the formula for this linear approximation    can be used to write down the equation of a tangent    plane to z = f (x,y) at any point (x0,y0,f(x0,y0))  - we took a careful look at the example on page 119    (see Maple) and we did exercise 2...				Section 2.3. # (2,4,5,7,9) 11,15,17 Prepare Section 2.4. Questions
T 10-7		Discssion of journal related items Finish discussion of Section 2.3 Begin discussion of Section 2.4 ... But we spent our time discussing  - level curves, contour maps  - estimating partials from a contour map  - how we get a tangent vector in x-direction    from the slope in x-direction  - stationary points, local max/min, saddles:      properties (partials = 0) and contour map features		picture p.129: gradient field		HW: If you have questions on 2.3 lingering: e-mail me or come to my office Prepare Section 2.4 (Questions) Further Exercises (besides those mentioned in  prep questions): Sec. 2.4: #7,8,11,13,17
R 10-9		2.4 The gradient is a vector in which the partial derivatives are "stored". What are its properties? (i) gradient is perpendicular to contour line (ii) gradient points in direction of steepest incline,     and its length tells us how steep that steepest     incline is. Why? How do we know? It follows from the fact that the directional derivative in the direction of a unit vector u can be calculated as the dot product between u and the gradient... and the fact that this is so follows quickly from the formula the linear approximation L(x,y) to f(x,y) at a point...
F 10-10		Exam 2				Finish all assigned homework for Sec. 2.4. Prepare Section 2.5.  Questions

WEEK 8

M 10-13		2.5 delayed until tomorrow) We disussed some difficulties with Sec. 2.4.  - exercise 8 ("direction of level path...")  - exercise 17 ("direction of steepest descent...")				Take another careful look at Section 2.5. Questions and do Exercises 1,2. We will (probably/hopefully/maybe) begin discussion of Sec. 2.6: Quadratic approxi- mations. In preparation, take a look at Questions
T 10-14		2.5 differentiability vs. existence of partials       why the gradient deserves to be called       "total derivative" - ...but only if the function       at hand is differentiable!!!       What it means to be differentiable       Connections to Calc I idea of derivative       and "differentiable" Also: Exercises 1 and 2, and what they show... 2.6       (we did not get there)		Sec. 2.5 pics		Finish your work on Exercises 1,2 of Section 2.5. If there are further questions, please see me or e-mail me/the class. Read/prepare Section 2.6 Part of this preparation is the review of Taylor polynomials from Calc I. More specific instructions are contained in Questions
R 10-16		2.6 quadratic (cubic,...) Taylor polynomials       for multivariable functions We spent most of our time talking about Taylor polynomials in the single variable case. But with that in mind, I think you should be able to put together the rest yourself...		Sec. 2.6 pics		Sec. 2.6 #Homework Take a peek at Sction 2.7, just in case
F 10-17		Finish 2.6 (1) Convince yourself that the formula for Q(x,y) (p.144 in text) is "correct" (i.e. yields a polynomial which has the same functionvalue, 1st, and 2nd partials at (x0,y0) as f(x,y). (2) Find Q(x,y) at the given point. Compare the graphs of the given function and Q(x,y) near (x0,y0) using Maple.  (a) f(x,y)=sin(x)cos(2y),     (x0,y0)=(0,0)  (b) g(x,y)=x2y2+xy3,           (x0,y0)=(1,1)		Examples from class		Read Sec. 2.7 carefully, so we can finish it on Monday. I would like to focus our class discussion on p.154/155: "The general case". You should attempt to clarify everything up to there. If that's not possible, mark the passages/phrases/arguments/conclusions which are not clear to you.

WEEK 9

M 10-20		2.7 optimizing a multivariable functions f(x,y)       at a stationary point (x0,y0)... The idea: Examining the quad. approximation at the stationary point (x0,y0)... The Key: Understanding the behavior of quadratic functions with no linear terms Since the question came up: The "D-test" to characterize stationary points of functions of two variables (Theorem 3, p.145) requires that both 2nd partials are continuous in some neighborhood of the stationary point (x0,y0).		The general case using pictures (p.154)		Sec. 2.7 #3,4,7,9,11,15,16,20,22,26 Prepare Sec. 2.8 You should be able to do: #2,4 : "function composition"       #3    : Find "derivative matrix" I may post more questions ...
T 10-21		Academic Day - no classes before 4p.m.
R 10-23		Exercises from Section 2.7: Questions? 2.8 The Chain Rule (brief class summary in pdf)				Sec. 2.8: #8,9,10,11,12,15 Read/Prepare Section 4.4
F 10-24		Discuss questions related to Sec. 2.7 and 2.8 (We discussed #20,21 in 2.7, and #9 in 2.8.)				Prepare Section 4.4 Questions

WEEK 10

M 10-27		4.4 Lagrange Multipliers or: "Constrained Optimization".		An Example		Sec. 4.4 # 2,4,6 Prepare Section 3.1 Questions
T 10-28		Finish discussion of 4.4 3.1 Double Integrals / Multiple Integrals  - review of R-integral in Calc I  - the definition via Riemann sums for    functions with two variables		Riemann Sums with Maple		Do Questions if not done yet. Sec. 3.1 # 3,4,9,11 Read all of Section 3.1., including "Intepreting Multiple Integrals"
R 10-30		3.1		Riemann Sums		Sec. 3.1 # 6,7
F 10-31		Exam 3 (focus on Sec. 2.4 - 2.8 and 4.4) There will be one take-home-question (due Monday)				Prepare (parts of) Sec. 3.2 Questions Please bring your Journals. (I will collect them on Monday)

WEEK 11

M 11-3		Brief discussion of test Sec. 3.1: What do double (multiple) integrals calculate?    The definition of multiple integrals as Riemann sums    (...and the meaning determined from that definition...) Review Calc I, once more:   Calculating distance from function which tells   speed at all times... (antideriv. / pretend constant speeds) Connections to multiple integrals 3.2 A brief first Example (region = rectangle, then triangle)
T 11-4		Return tests. Collect remaining journals. Section 3.2: Multiple Integrals: How and Why...    All of the ins and outs with  SS x2+y2 dA first on the rectangle [0,2]x[0,1], and then over a triangle...				Section 3.2 old: #1cd, 2a, 4 new: # 5,6,8,9,10,11,12*     (* may look a bit more challenging...) Prepare Section 3.3 Questions
R 11-6		Discuss difficulties that became visible through homework Finish discussion of Section 3.2 (how to evaluate multiple integrals in cartesian coordinates) We did not get to Section 3.3 as planned. But Sec. 3.2 is now crystal clear. Good.				Finish al omework for Sec. 3.3 Prepare Section 3.3 Questions
F 11-7		3.3 Multiple integrals in polar coordinates  - questions about polar coordinates in general  - converting a function from cartesian to polar coordinates    (and vice versa)  - the polar rectangle and its area				Sec. 3.3 #6,7 Prepare Section 3.4 for initial discussion Questions

WEEK 12

M 11-10		3.3 Exercise 6 and related issues				Prepare Section 3.4 for initial discussion Questions
T 11-11		3.4 Cylindrical coordinates  - what's the "shadow region"?  - where does the "r" come from...  - Exercise 1b, and the need to visualize...				Tomorrow we will talk about spherical coordinates, and triple integrals in spherical coordinates. The question has been asked: "How can you tell which coordinate system system should be used?" Whaddoyouthink? HW Sec. 3.4: 8,9,13,15. Also: Let G be the solid in the first octant bounded by the sphere x2+y2+z2=4 and the coordinate planes. Evaluate the triple integral over G of f(x,y,z)=xyz in (a) rectangular, (b) cylindrical, (c) spherical coordinates. Finally: The 25 point extra credit problem I handed out is a really good one at this point. You may want to use technology for part (a) after you set up the integral!
R 11-13		3.4 Spherical coordinates  - We spent most of our time on the HW    problem marked "Also"				Homework: Exercise 15 in section 3.4 Part (c) of the HW problem marked "Also" Finish all other problems from Sec. 3.4
F 11-14		3.4 Finish  - The xyz integral in spherical cooedinates  - Maple and multiple integrals... 3.5 A few words about this section...      (If time permits)				- Finish all problems in Chapter 3.  - Take a look at section 3.5.     (I will show you a few things including Examples on Monday)  - Begin to read Sec. 5.1

WEEK 13

M 11-17		3.5 Change of coordinate systems in multiple integrals ... Some remarks, and an Example (spherical coordinates)				Prepare Section 5.1 Questions
T 11-18		5.1 Line integrals  - Vector fields  - Integrating a vector field along           an oriented curve  - What does the value of a line         integral tell us? (it's not area)		class example (Maple)		HW Sec. 5.1 #1,3  (For #1 you may need to review parametrizations of straight line segments) For the integrals in #3, use Maple to draw the vector field (integrand) and the path along which the vector field is integrated. After calculating these integrals you will hopefully ask the question: What do these integrals calculate? Many of them turned out to be zero! Why is that? Can one see/guess that from the pictures? Read/prepare for discussion of Section 5.2 Questions
R 11-20		5.1 Revisiting above questions via Homework discussion 5.2 - interpreting line integrals       (guessing value from picture)		Template to draw vector field and path
F 11-21		5.2  Line integrals  - Algebraic properties (statements    and rationale)  - independence of parametrization    (why it is not "obvious" and a 1st     attempt to prove it)

WEEK 14

M 11-24		Exam 4
T 11-25		5.2 A more detailed (and corrected) proof    of line integrals' independence of parametrization  - Fundamental Theorem for Line Integrals     statement and proof  - the question: how can we tell if a vector field is    a gradient of some function F(x,y): R2 --> R, and    if "yes", how do we find F(x,y)				HW: Sec. 5.2: #1,3,
R 11-27		Thanksgiving Holiday
F 11-28		Thanksgiving Holiday

WEEK 15

M 12-1		Project due Test improvements due (optional) 5.2 How to decide if a vector field  is a gradient (of some F(x,y)), and  if "yes", how do we find F(x,y).				Sec. 5.2: #5,6,8 Read Section 5.3 Questions
T 12-2		5.3 Green's Teorem  - statement and Proof (didn't get to proof)				Sec. 5.3: #1,3,5*,6*,9 *interesting applications ... esp. for physicists Read Section 5.4 Questions
R 12-4		5.3 The proof of Green's Theorem In how far Green's Theorem is a bit like the "Fundamental Theorem of Calculus" ...and a look ahead... (Divergence Theorem, Stokes Theorem are other theorems like FTC) But first need to invent Surface integrals as generalization of double integrals just like Line Integrals are generalization of CalcI type integrals...				Read Section 5.4 Questions
F 12-5		5.4/5.5 Line integral of scalar valued function... Surface integral of scalar valued function as generalization of Line integral...  - the theoretical development... We will look at an example on Monday.				Try reading section 5.4 again... as well as section 5.5. Sec 5.4 #1,2. You will have to find parametrizations of the cone and a planar piece... (we talked about the cone in class, and a planar piece is just the graph of a function... so (u,v,f(u,v)) is a parametrization of the surface) Sec 5.5 #1. Also: identify the surface represented by the parametrization given. (e.g. graph with Maple)

WEEK 16

M 12-8		5.4/5.5 surface integrals (of scalar functions) Examples: based on homework .I think we'll focus on the cone and go through the various stages of the theory presented on Friday, leading all the way up to a surface integral of a scalar valued function on the cone...				HW: Integrate the function f(x,y,z) = x+y+z  over the cone z = sqrt(x^2+y^2). I recommend to do this twice: Once using each of the parametrizations from class. (One is just (u,v,sqrt(u^2+v^2)) with D = unit disc, and the other parametrization is using u = r and v = theta (i.e. polar coordinates) with D = [0,1]x[0,2Pi]) Read section 5.6. (all). Questions Physicists: How does the definition of divergence and curl allow the common interpretation...?
T 12-9		Most time was spent on HW problem. 5.6 Surface integrals of vector fields (in R3)
R 12-11		5.6 Surface integrals of vector fields (brief intro) 5.6 divergence and curl:       derivatives of 3D vector fields (brief: definition       and interpretation) 5.7 The Divergence Theorem (statement)
F 12-12		5.7 Stokes' Theorem (?) good byes...

Final Exam: Monday, Dec. 15, 1-2:50 p.m.