Spring 2015 Math 233H schedule and homework from Stewart 7 (in the
past, I've used Marsden & Tromba _Vector Calculus_ and may choose some
material from there) due each Wednesday - the *ed problems are for you
to ponder (assignments are always subject to change as necessary):
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********************** The semester at a glance *********************
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Week Sections Events
1/19 Intro, 12.1-12.3 Extra session Wed eve, MLKJr Mon!
1/26 12.4, 12.5, 12.6 Quiz #1, Extra session Wed eve
2/2 10.1, 13.1, 13.2 Extra session Wed eve, snow Mon!
2/9 13.3, 13.4 Quiz #2 cancelled, snow Mon!!
2/16 14.1, 14.2 Tue is Mon, Tue eve review! Exam 1: Wed 6pm
2/23 14.3, 14.4 No Mon class!
3/2 14.5, 14.6 Last day to drop with a "W": Mon
3/9 14.7, 14.8 Quiz #3: Mon in class!
3/23 15.1, 15.2 Extra session Wed eve
3/30 15.3, 10.3 Extra session Wed eve
4/6 15.4, 15.5* Exam 2: Wed 4/8 @ 6pm (LGRT 1334)
4/13 15.9* Extra session Wed eve, no Mon class!
4/20 16.1, 16.2 Wed is a UMass Mon, no Mon class!
4/27 16.3, 16.4 Extra session Wed eve! Thu is Mon!
5/4 FINAL EXAM Tue 8am (5/5) in our classroom (LGRT 202)
The following sections of Stewart 7 will be (un)covered. Some* sections
may not be lectured on, but you should read them on your own. This
list may change as the semester evolves:
12.1 Three-dimensional coordinate systems
12.2 Vectors
12.3 The dot product
12.4 The cross product
12.5 Equations of lines and planes
12.6 Cylinders and quadric surfaces
10.1 Curves defined by parametric equations
13.1 Vector functions and space curves
13.2 Derivatives and integrals of vector functions
13.3 Arc length and curvature
13.4 Motion in space, velocity and acceleration
14.1 Functions of several variables
14.2 Limits and continuity
14.3 Partial derivatives
14.4 Tangent planes and linear approximations
14.5 The chain rule
14.6 Directional derivatives and the gradient vector
14.7 Maximum and minimum values
14.8 Lagrange multipliers
15.1 Double integrals over rectangles
15.2 Iterated integrals
15.3 Double integrals over general regions
10.3 Polar coordinates
15.4 Double integrals in polar coordinates
15.5 Applications of double integrals*
15.7 Triple Integrals
15.8 Triple Integrals in Cylindrical Coordinates
15.9 Triple Integrals in Spherical Coordinates*
16.1 Vector fields
16.2 Line integrals
16.3 The fundamental theorem for line integrals
16.4 Green's theorem
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1/28 Consider a quadrilateral Q with vertices A,B,C,D. What can
you say about the new quadrilateral P with vertices X=(A+B)/2,
Y=(B+C)/2,Z=(C+D)/2,W=(D+A)/2 at the midpoints of Q?
Read Chapter 12!
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2/4 Chapter 12 Exercises: 3,4(a,b,c,d,f,h),5,6,7,8*,9*,11,12,14
For a tetrahedron (a pyramid with triangular base), show that
the sum of the outer unit normal vectors weighted by the areas
of the faces must vanish (also true for any polyhedron). [This
turns out to be a "Discovery Project" problem in Chapter 12!]
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2/11 Prove the "bac-cab" relation for cross product:
a x (b x c) = b(a.c) - c(a.b)
[so called because the parentheses go "in the back" - this
is a more memorable way to state 12.4.11 Theorem 6].
Chapter 12 Exercises: 15,17,19,24,29,31,35,38*
Chapter 12 Problems Plus: 1*,8*
Explore the quadric surfaces in section 12.6 and try to figure
out how some surfaces are really limiting cases of others: e.g.
planes are limits of spheres (and many other surfaces), cylinders
are limits of ellipsoids and hyperboloids, etc.
Read Chapter 13!
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2/18 For the helix X(t)= compute the velocity X'(t)
and acceleration X"(t). Compute their dot product X'(t).X"(t)
- could you have predicted this from the fact that the speed
|X'(t)| is constant? [Hint: consider |X'(t)|^2=X'(t).X'(t) and
use the product (Leibniz) rule. Cf. Chapter 13 Exercises 1,9]
Chapter 13 Exercises: 2,5,6,9,11,18,21*,23*,24*
Chapter 13 Problems Plus: 3[assume gravitational acceleration is
<0,0,-g>, g=32feet/(sec)^2],6*[experiment with rope],7*[Math235]
****** Review @ 5:30PM on Tuesday 2/17 in 1334 LGRT ******
!!!!!!!!!!! Midterm #1 @ 6PM in 1322/1334 LGRT !!!!!!!!!!!
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2/25 Recall the special case of "bac-cab" when a = c is a unit vector:
the parentheses can go in either place, and the vector a x b x a
is projection of b to the plane perpendicular to a (we checked this
algebraically for a = i, but the general case follows from the right-
hand rule). Use this to show that the curvature vector of any (not
necessarily unit speed) parametrized curve X(t) is
K(t) = X'(t) x X"(t) x X'(t)/|X'(t)|^4
and thus its length (the curvature function) is
k(t) = |K(t)| = |X'(t) x X"(t)|/|X'(t)|^3.
[Hint: expand acceleration into tangent and normal parts.]
[Remark: from the special baa-aab case of bac-cab when a.a=1,
we get a x b x a = b - a(a.b), which subtracts from b the
component of b along a, i.e. it projects b to the plane
perpendicular to a.]
[Note: this problem can be turned in next Monday in class!]
Read Chapter 14!
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3/4 Chapter 14 Exercises: 2,4,5,7,8,10
Chapter 14 Problems Plus: 4
*We observed in class that the function f(x,y) = xy
has this property: through each point on
its graph pass 2 distinct straight lines which entirely
lie on this graph. What functions have 3 such lines?*
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3/11 Chapter 14 Exercises: 13,16,19,21[compare mixed 2nd partials],
25,28,31*
*In class we discussed f(x,y)=2xy/(x^2+y^2). Explain why there
is no single limit value L for f(x,y) as tends to <0,0>,
but that instead, the closed interval I=[-1,1] is a "limit set"
in the follwing sense: for any neighborhood V of I in R, we can
find a neigborhood U of <0,0> in R^2 with f(U) contained in V.*
Chapter 14 Problems Plus: 4,5*
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*************************** SPRING BREAK ******************************
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3/25 Chapter 14 Exercises: 11,33,34,35,37,40*
Chapter 14 Problems Plus: 6*
*In class we discussed that a function f(x,y) with partials
f_x and f_y defined at =<0,0> might fail to be differtiable
at <0,0> (i.e. fail to have good linear approximation). Try to
write a formula for such a function (perhaps cosult Wikipedia,
or other sources - and if you do, please cite your sources).*
*Similarly, we discussed Clairut's Theorem about the equality of
of mixed partials. Try to write a formula for a function f(x,y)
for which f_xy and f_yx are both defined at <0,0>, but are NOT
equal (of course, by the hypotheses of Clairut's Theorem, f_xy and
f_yx, as functions of , can't both be continuous at <0,0>).*
We explored the chain rule f_t = f_x x_t + f_y y_t for the
t-derivative of the composite function f(x(t),y(t)) of a curve
in a plane domain D followed by a function f(x,y)
on D (using arrows: I \subset R --> D \subset R^2 -f-> R),
but ran out of time at the end to check: if the curve is a contour
(or level set) for f, then \grad f is perpendicular to the curve,
i.e. its dot-product (grad f). with the velocity
must vanish. Please finish the argument [hint: since f=const on
the contour, f_t=?]
Start reading Chapter 15!
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4/1 Chapter 14 Exercises: 44,47,49,50,52
In class (and at last Wed eve session) we discussed the "egg carton"
surface, the graph of f(x,y)=(sin x)(sin y). Since f(x,y) is a
"periodic" function (it has the same values if either x or y are
translated by 2\pi) we can completely describe its behavior on a
"fundamental domain" such as the square [-\pi,\pi]\times[-\pi,\pi]
R^2. We found all the critical points for f(x,y) on this domain and
conjectured which were local maxima, minima and saddles. Please work
out the details, and check our claim that #maxima+#minima-#saddles=0
for this example. What happens to this (signed) count of critical
points if we use a bigger domain, like [-m\pi,m\pi]\times[-n\pi,n\pi]
for some integers m,n>0? (My egg carton was square and held 36 eggs,
in which case m=? and n=?) [For more on this sort of thing, please
google "Morse Theory" - and (while you're at it) see if you can find
the classic short animated film "Peaks, Pits and Passes"!?!]
Continue reading Chapter 15!!
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4/8 Remark on the very last question ("My egg carton was square and held
36 eggs, in which case m=? and n=?"): some of you figured out that for
each [-\pi,\pi]\times[-\pi,\pi] square, there were 2 minima (the eggs
nestle there) among the 8 critical points, so we'd need 18 of these
to hold 36 eggs - how could that be (as 18 is not a perfect square)?!
[Hint: in each square, there's a pair of smaller squares (rotated 45
degrees), each with half the area, and each with half as many critical
points - in particular 1 minimum - as in the original square; my egg
carton is really a 6 x 6 array of those smaller squares.]
Chapter 14 Exercises: 55,56,59,61,64,66*
Chapter 14 Problems Plus: 1,8*
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4/15 Chapter 15 Exercises: Take a break and enjoy the sunshine...
Chapter 15 Problems Plus: ... and warm weather!!!
Finish reading Chapter 15 and start reading Chapter 16!
Not a problem, but a little project for before Wednesday's class:
How would you measure the volume of potato?
OK, you could be clever, like Archimedes 2300 years ago, and find
instead the displaced volume of fluid. But suppose you were on a
rocky desert planet, with no fluids (like water or even sand) at
your disposal, and you had to figure it out by slicing and dicing.
The practical aspect of the project is this: can you (collectively)
please procure (perhaps from a DC, or a farmer or grocer) about 1
potato per person so that we can do some experiments on Wednesday
afternoon? I can supply the knives, but remember (or beware!) it
was the Ides of March - and not the Ides of April - when Caesar's
generals sliced and diced him!
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4/22 Thanks to those of you who DID procure a potato today! Unfortunately,
there weren't enough to go around; nor did I have the sharp knives or
measuring tools that we might use to carry out the experiment I had in
mind carefully, and so it became a "thought" experimemt in class. But
we heard some good methods (I particularly liked the idea of mashing
the potato: in order to get around the lack of fluids on our desert
island, we turn the potato into a fluid).
So here's part 1 of this weekend's homework (for those of you with a
potato, please use your potato; for those without, please use a fruit
substitute, something about the same size):
(i) procure (e.g. borrow) length and volume measuring devices with metric
units (please stick to CGS: cm for length, and (cm)^3=cc=ml for volume);
(ii) first try to estimate carefully the volume of your potato (or fruit
substitute) by a method like the Archimedes displacement method - if instead
you want to use the Matt mashing method, do step (iii) before (ii);
(iii) slice your potato (or fruit substitute) into 1 cm thick slabs (home
fries?), then slice those again into 1 (cm)^2 cross-section strips (freedom
fries?) whose lengths you can measure to estimate the volume; you could go
even further and dice those into 1 (cm)^3 cubes (tater tots?) and simply
count, but (as we saw today in class) that makes it hard to reassemble into
a potato (the boundary "skin" help, just like with a puzzle).
How close are your two ways of measuring volume? Keep notes and photographic
evidence of your experiment! (You may work alone or in teams of at most 3.)
*Think about what would account for the difference - besides measurement
errors, there's a boundary effect - what feature of the boundary is being
measured?*
Chapter 15 Exercises: 1,2,3,4
Chapter 15 Problems Plus: 1,2*
Continue reading Chapter 16!!
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4/29 [We made the next stuff due "infinitely" later.... ;-]
Chapter 15 Exercises: 5,7,8,10,11,15,49*
Chapter 15 Problems Plus: 4*,5*,6*
*Consider the solid region defined by drilling a cylindrical hole
through the center of a ball
<--??-->
/| |\
|| ||
\| |/
<----??---->
so the resulting height is 2H (the "cored apple" described in class).
What is its volume?*
Chapter 16 Exercises: 1,2,7,9,11,12,18,20
Chapter 16 Problems Plus: 1*,3,5
In the last two classes (and the evening session between) we discussed
grad, curl and div [since I cannot easily make a "nabla" in ASCII, I'll
use these monosyllables instead] for smooth functions and vectorfields
on R^3. Recall that the geometric interpretation of grad f is the
vectorfield whose direction is that for which f increases most rapidly
and whose length is the rate of increase. Another way to say this is
the "flow" along grad f is (locally) how we translate as f increases.
(If v=v(X) is a vectorfield, where X is position and t is time, then v
has flow F(t,X)=X+tv(X)+O(t^2) which gives the path of a point starting
at X(0) moving to X(t)=F(t,X) after time t - this involves solving a
differential equation, but for small time we can approximate the flow
by X(t)=F(t,X)=X+tv(X) - the point has initial velocity X'(0)=v(X).)
We also saw curl v gives the tendency of the flow of v to rotate: it
measures the circulation (path integral) of w around a loop, as made
precise with Stokes Theorem:
The double integral of (curl w).N over any surface
(with unit normal N) equals the path integral of w
around the boundary loop (the right-hand rule gives
the correct sign: if the thumb points along N, then
the fingers indicate the way we follow the loop).
We checked (using the the equality of mixed 2nd partials) the
important identity:
curl(grad f)=0
and we saw how this means curl v is the "obstruction" to v
being grad f: path integrate v along any loop and find that
its circulation must vanish, then recover f (up to constant).
What about the divergence div w of a vectorfield w? At the
very end of class we hurried to see it measures the volume
expansion under the w-flow X(t)=X+tw(X)+O(t^2):
If we write X(t)=, w= and ignore
O(t^2) terms in what follows, then the chain rule gives
dX(t)=dX+tdv=
where the "..." are the analogous compnents involvling w2 and w3.
Using the "^" here to denote the way we multiply the differentials
used to compute area or volume (skew symmtric like cross product
or determinant) we get (after some algebra) volume element
dx^dy^dz(t)=dx^dy^dz(1+t(w1_x + w2_y + w3_z))
since lots of terms cancel or are O(t^2). Therefore at t=0
d/dt[dx^dy^dz(t)] = (w1_x + w2_y + w3_z) = div w
The analog of Stokes Theorem for divergence is sometimes called
the Gauss Law or Divergence Theorem:
The triple integral of div w over a region
of R^3 equals the double integral of w.N
over the boundary surface with (outerward)
unit normal N.
The latter is called the "flux" of w across the boundary surface
and if this flux vanishes for all closed surfaces, then w is
"volume preserving" and (equivalently) has div w = 0.
For example, if you apply this to position vector field w=X with
div X = 3 (recall we did that calculation in class) then you can
quickly see that the area of the unit sphere is exactly 3 times
the volume of the unit ball (4\pi = 3 (4\pi/3)) or more generally
for the area and volume for the sphere and ball of radius R.
Analogous to the important identity above, you should check:
div(curl v)=0
but the vanishing of div w doesn't guarantee that w = curl v.
(That leads to deeper features of vectorfields - like harmonic
functions and forms - which may be treated in Math 425 or 704,
and are the foundation of "Calculus on Manifolds" or "Differential
Topology"!
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See you all at 8AM Tuesday 5 May 2015 in our regular classroom
for the Math 233H final exam (mainly on chapters 15 and 16)...
... and have a great summer (and beyond)!!!!!!!!!!!!!!!!!!!!
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******************** Old HW problems (M & T, etc.) ********************
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Take a triangle in the plane with vertices P, Q and R. The
midpoints of its edges form a new triangle with vertices
(P+Q)/2, (Q+R)/2 and (R+P)/2. What happens when we iterate this
process? To what point in the plane do the successive triangles
limit to? And what is the limiting shape of these triangles?
The answer to the analogous questions for a quadrilateral is related
to the "parallelogram conjecture" mentioned in class.
What happens for a general n-gon?
Read Chapter 1 of M & T
1.1#8,14,16,22,26
1.2#8,12(for 8 only),16,26
1.3#3,4,5,17,26,30
1.4#2,6,14(may be useful at Jiffy-Lube ;)
Consider the outer normals to the triangular faces a tetrahedron,
each with length equal to the area of the corresponding face; use
vector methods to show the sum these normal vectors is zero.
Read Chapter 2 of M & T
2.1#2,10,18,30,32
2.2#9
2.3#1,13,8,14,20
2.4#12,18
2.5#4,8,12,14,20,24
2.6#3,6b,16c
Review problems from Chapter 2 of M & T
Read Chapter 3 of M & T
3.1#2,4,8,16
3.2#4,7*
3.3#4,10,18,24,27,36*,42
3.4#10,12
Here is what the topographic map is for: find a contour which
loops around and encloses an interesting region of Amherst;
highlight this contour; now try to locate all critical points
inside (the maxima should be easy, minima are probably under
water, and the saddles are tricky); use the relation
#maxima + #minima - #saddles = 1
to ensure that you have found all the saddles!
Read Chapter 4 of M & T
4.1#6,7,14,18
4.2#7,12,13,14*,15*
4.3#3,7,9,18*
4.4#2,8,13,21,26
Read Chapter 5 of M & T
5.1#1,5
5.2#2,3,4
5.3#1,11,12,16
5.4#1,10,14
5.6#3,12,15,24
Read Chapter 6 of M & T
6.1#1,2
6.2#1,2,6,7,8,16
6.3#4,8,13,14,17
Read Chapter 7 of M & T
7.1#11
7.2#6,14
These are sample topics from a previous 233H final exam
about which you may wish to think:
a) derivatives and integrals of the function
f(x,y) = cos((x+y)/(x-y))
over various domains in the (x,y) plane;
b) the change of variables T: R^2 -> R^2 defined by
T(u,v) = (x(u,v), y(u,v))
where
x(u,v) = e^u cos v
[only u in the exponent]
y(u,v) = e^u sin v;
c) properties of the solid region defined by drilling a
cylindrical hole of radius 8 through the center of a ball of
<--16-->
/| |\
|| ||
\| |/
<----20---->
radius 10 (this was the "cored apple" described in class).