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Class Diary for M241, Fall 2018,
Jochen Denzler
Class Diary for M241, Fall 2018, Jochen Denzler
does not include recitation sessions (Thu), which will usually discuss
homework and questions. If specific material is assigned into recitation
sessions, then I will include their time slot here.
Wed Aug 22:
Intro and sneak preview to multi-variable calculus. Points and vectors, and
their coordinates. Addition, scalar multiplication, and dot product of
vectors. Notes as pdf file. (Bear with me
until I get Canvas up)
Thu Aug 23:
Recitation session will go through examples and details of dot
product and start cross product
Fri Aug 24:
Cross product of vectors. (Do read chapter 12.4 as
well) Notes as pdf file.
Hwk: Ch 12.3, ## 46,50,78,82; Ch 12.4, ##11,44 (each is worth 5 points).
Ch 12.4, #73 (worth 3+3+3 points). Turn in to recitation leader on
Thursday
Mon Aug 27:
Scalar Triple Product and volume; Planes in space repr'd as ax+by+cz=d.
Notes as pdf (but also consult book
12.4,5).Hwk: see here(5pts each) also due in
recitation on Thursday
Wed Aug 29:
Vector valued functions; paths (parametrized curves) vs curves; simple
examples; derivative of a vector valued function.
Notes as pdf (also see book 13.1,2).
Hwk: 13.1, ##4,5,8,10,34
Fri Aug 31:
Product rules and their proof; chain rule. Examples; tangent lines and
velocity. Notes as pdf (also see book Ch
13.2). Hwk: Ch 13.2, ##1,4 (2 points each),22,40,57,58,63 (5 points
each)-- (For #40, you need to read the sec in the book)
Mon Sep 03:
LABOR DAY (no class)
Wed Sep 05:
Example: cycloid; arclength; reparametrization by arclength; curvature.
Notes as pdf (also refer to pertinent
chapters of book).
Hwk: Ch 13.3: ##4,7,15 (5pts each), 37 (3+3 points). Then Sec 11.2#29,
followed by Sec 13.3#40 (5pts each). Hwk Ch 13.4: ##6,10,24 (5 pts
each). All due Thu *next* week
Fri Sep 07:
Functions of several variables. Slicing into SV functions; level
sets. Examples. (See 14.1 for many more
examples) Notes as pdf.
Hwk due Thu: The cardioid(6 pts). From Ch 14.1:
Preliminary Q3, Ex. 2,12,18(explain!) (3pts each); ##30,31,46,47 (4pts each);
#56(only for t>0, and in (b) only c>0) (6pts),
#57 (5pts).
Mon Sep 10:
an example, and cylindrical coordinates; Limits: their definition (and what it
means), and key propreties. Hwk: none to hand in this time, but do look
carefully at Examples 5-7 in Sec 14.2. Partial derivatives just
started. Notes as pdf
Wed Sep 12:
Partial derivatives and directional derivatives (defined and explained).
Notes as pdf. [Note: this is only the first
half of Sec 14.3, plus the half-page after example 5 in Sec 14.5.] Hwk
Ch14.3, ## 2 (2pts), 7 (4pts), 9-12 (2pts each), 41
(3pts). Also this pblm on directional
derivatives (5pts). These are due Thu *next* week)
Fri Sep 14:
Examples where partial derivatives do not determine directional
derivatives. The role of existence of a tangent plane (intuitive).
A brief leftover: exchange of order in partial derivatives.
Notes as pdf. Hwk: Ch 14.3, ##
25,34,64,73,74,76 (4pts each) due Thu
Mon Sep 17: Differentiability; Linear Approximation; the differential;
Notes as pdf
Hwk: Ch 14.4 ##4,13,25 (5pts each) due Thu
Wed Sep 19:
The gradient. Geometric interpretation and calculational rules.
Notes as pdf Hwk: Ch 14.5, ##2, 3, 6, 21, 32,
43, 58. This hwk is intended to be done by Monday!! It will not be collected
for grading. But it will be exam relevant, so please do it as seriously as if
it were collected for grading. Sample
solutions will be offered between Monday and the exam.
Fri Sep 21:
The multi-variable chain rule. Notes as
pdf Hwk Ch 14.6, ##2,4,20-23,26,28. Same policy (no turn-in) as for Wednesday's
homework will apply. Expect solutions to be posted Tuesday night.
Mon Sep 24:
Comments on ambiguities of partial-derivative notation and how to avoid
confusion. Local and global extrema and first derivative necessary condition for local
extrema. Notes as pdf.
Hwk Ch 14.7, ## 1,3,5ab (not 5c yet)
Wed Sep 26:
Second derivative test for minima, maxima, saddle points: (a) The ready-made (but
mysterious-looking) version for 2-variables, (b) the intuitive (but not so
ready-made) version for any number of variables, and (c) how you get to the
ready-made version from the intuitive one in 2
variables. Notes in pdf. (You only find (a)
and (c) in the textbook, with (c) buried in a proof at the end of the chapter that you'll
probably not read, so do refer to the notes.) Hwk Ch14.7: now do 5c as
well. This one won't be graded. -- More hwk for grading to be posted shortly
Fri Sep 28:
EXAM 1
Mon Oct 01:
How *not* to distinguish min from max; existence of global min and max for continuous
functions on closed and bounded sets. Constrained minima and maxima via
Lagrange mulipliers (more on that Wed). Notes as
pdf Hwk: Ch. 14.7: ##8, 20 (5pts each),26 (2+3+3 pts), 35 (3x3pts), 54
(5pts). These are due Thu after fall
break.
Wed Oct 03:
Lagrange multipliers finished up. Notes as
pdf (a few comments appended after class) Hwk: Redo #54 from Ch 14.7,
but this time using Lagrange multipliers (5pts). From Ch. 14.8: ## 21,26,28
(5pts each) 49 (2+3+1pts), 53 (6pts). HINTS on Canvas for these. They are due
Thu after fall break
Fri Oct 05:
FALL BREAK
Mon Oct 08:
The Riemann integral in 2D, over
rectangles. Notes. Hwk: Ch
15.1, ##6,34,40 (5pts each) due Thu
Wed Oct 10:
Same for higher dimension; Riemann integral in 2D over other bounded domains.
Notes Will post Hwk soon; it will be due
Thu next week
Fri Oct 12:
Some comments on Exam 1 (more on Canvas); examples for limits of integration in
2dim and 3dim integrals. Notes as pdf.
Hwk from Ch 15.2+3; these will be due Thursday next week: Ch15.2
prelim'questions 1 and 4 (3pts each), Exerc 4,7,16, 34,43,44,56
(5pts each) A few hints found on canvas. Ch 15.3 prelim'question 2 (3pts) and Exercises 6,13,23 (5pts
each). 15.2-JD1 integral over crescent (3+2+1+2 pts)
Mon Oct 15:
another example; Change of variables; overview, and especially the example of
polar coordinates. Notes as pdf.
Hwk Ch 15.4: prelim Question 1 (3pts), Exercises 3,6,18 (5pts each) due Thursday
Wed Oct 17:
Change of variables into cylinder and spherical coordinates (incl introduction
of spherical coordinates in the first place - see Ch 12.7 for those).
Notes. Hwk Ch 15.4: ##21,22,26,38,42,43. These will not be
collected for grading but will be exam relevant. Solutions to be posted next
week on Canvas
Fri Oct 19:
one example; dA for the general change of coordinates in 2
dim. Notes as pdf -- Hwk: no more hwk to be
assigned from today
Mon Oct 22:
(JV substituted - no notes posted) Change of Variables (Ch 15.6)
Wed Oct 24:
(JV substituted - no notes posted) Change of Variables finished; some
applications. Some Review for exam.
Fri Oct 26:
EXAM 2
Mon Oct 29:
Some more applications (moment of inertia); scalar line integrals.
Notes. -- Hwk due Thursday: Ch 15.5,
##8,13,26,45 (5pts each). That's all. FYI: some hwk from Ch 15.6 will be
assigned as part of the set due NEXT week
Wed Oct 31:
Applications for scalar line integrals. Work, and vector line integrals.
Notes. Hwk (due NEXT Thu): delayed Hwk
from Ch 15.6: #21,37,39 (5pts each); from Ch 16.2: Prelim'Questions 2,3 (6pts
each), Exerc 9,16,24,28,36,61 (5pts each).
Fri Nov 02:
Vector line integrals over gradient fields don't depend on the
path. Conservative vector fields, and how to find an antiderivative of them.
Notes. Hwk will be assigned on Monday
Mon Nov 05:
conservative field equivalent path independent line integral equivalent
circulation 0. Conservative implies cross-derivatives match, but converse only
in simply connected domains. Notes. Hwk
from Ch 16.3 (due Thu): Prelim Q's 1,2,3 (3,7,4 pts respectively); Exercises
4,7,11,16,21 (5pts each)
Wed Nov 07:
Curl of a vector field; example of nonconservative vector field with vanishing
cross derivatives in punctured plane; potential energy in
physics. Notes. New Hwk will be posted on Friday
Fri Nov 09:
More examples of vector fields and how to graph them. Their curl, and their
div. Notes. Hwk: Ch 16.1, ##13-16 (8pts
total);
23,28,29 (5 pts each); 33-36 (4pts each). Due Thursday
Mon Nov 12:
prelim' geometric interpretations of curl and div. (Sneak preview of pertinent
integral theorems). Parametrized surfaces
started. Tangent vectors. Notes. No new
homework today.
Wed Nov 14:
normal vectors; area element; surface area, scalar surface integrals.
Notes. Hwk: Ch 16.4, #1,10,14,22, and
my own problem. These will not be turned
in, but are exam relevant. (Today's material is the last to be covered on
Monday's exam.)
Fri Nov 16:
Flux Integrals. Notes. No hwk today
Mon Nov 19:
EXAM 3
Wed Nov 21:
Review flux integrals; application: Law of induction in electrodynamics; flux
integrals through curves in the plane: a different version of vector line
integrals (see Example 9 in Sec
16.2). Notes.
Hwk Ch 16.5, #7,17,25-27.(Don't take the word `river' seriously in these
last problems. Real water in a river satisfies div v = 0, which the given v
doesn't). Also Ch 16.2, #65,67.
Pretend these to be due Thu after Thanksgiving, but we won't collect
them because there would be no time to return them
Fri Nov 23:
THANKSGIVING BREAK
Mon Nov 26:
Green's Theorem. Notes. Hwk: Ch 17.1,
## 2,3,11,16,21,25. Won't be collected for grading b/c there would be no
time to return them.
Wed Nov 28:
Area by means of Green's theorem. Stokes' Thm and Divergence Thm stated and
their connection to Green's thm revealed: Green is special case of Stokes for
surfaces contained in the plane, and is also equivalent to the 2dim version of
the divergence theorem. Notes. Hwk is
posted along with Friday's.
Fri Nov 30:
Stokes Theorem proved. Vector Potential.
Notes. Hwk: from Ch 17.1: #36 yet (also
think *why* we would deem it preferable to use Green rather than calculating
the circulation directly); from
17.2: Prelim'Q #1 and Exerc ##1,5,15,20,31.
Mon Dec 03:
Divergence Theorem. Notes. Hwk: Ch
17.3, #7, 20,22,27
Wed Dec 05: STUDY DAY: Open office hour in regular classroom available
from 8-9am.
Wed Dec 12: FINAL EXAM 08:00-10:00
(scheduled by university policy)
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