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Here
is information about the final exam
which is on Monday, June 8th, from 8am11am.
0) Review Session with Frank is on Friday, 23:30pm
in McH 1240.
Details about the test are here:
1) The
gradient is the hub of many things. More
specifically, you need to understand the meaning of
the gradient and how it is used to
compute directional derivatives,
compute equations of tangent planes to level
surfaces (the gradient is normal to this plane)
compute equations of tangent lines to curves of
intersection of surfaces
find directions where the rate of change is maximal,
minimal or a particular value,
find critical points.
calculate the divergence
and curl of a vector field
2) Limits/Differentiation: You need to understand the
idea behind the derivative as a linear map and as the
linear approximation to a function. You need to
understand that the derivative is represented as a
matrix or a vector (in case the function is
realvalued). You need to be able to use the chain
rule to compute the derivative of a composition of
functions. You need to know the relationship between
the existence of the partials and the
differentiability of the function. You need to know
under what conditions the mixed partials are equal.
3) Taylor Polynomials: You need to understand what
the Taylor polynomial for a function of a certain
degree represents. You need to be able to compute the
Taylor polynomial of degree one or two, but you do
not need to be able to compute the error term.
4) Maximum/Minimum Problems: you need to understand
the relation between local extrema and the gradient,
and you need to be able to find and classify critical
points. You will need to be able to find global
extrema and justify their existence. This also
includes the ability to find extrema with
constraints. Particularly Lagrange multipliers as a
method to optimize functions subject
to one (but not more) constraints.
5) Paths/Curves: you need to understand the idea of a
path as a vector valued function and be able to
describe the curve that is the graph of a path. You
need to be able to find the velocity and acceleration
vectors and compute the speed, assuming that the path
c(t) represents the position
of a particle at time t as it's traveling on the
curve parametrized by c(t). You need to be able to
set up (but not evaluate) an integral that represents
the length of a curve.
6) Directional Derivatives; you need to understand
the geometry of the directional derivative and how to
compute it. You also need to be able to incorporate
the chain rule into the mix when it comes to actual
questions about these rates of change at a point. The
issue being that there is a difference between the
slope at a point in a certain direction and the
change in value of the function in that direction if
one moves at a speed other than unit speed.
7) Vector fields. You need to understand what a
vector field is; how to verify that a curve is a flow
line of a given field; you need to be able to
determine whether a vector field is a gradient vector
field; you need to be able to compute the curl and
the divergence of a vector field.
8) Tangent lines/Tangent planes: You need to be able
to compute those under the various scenarios of
defining/obtaining them. This also includes being
able to find points on a curve/surface so that the
tangent space has a given property.
9) Visualization/Classification of Surfaces and
curves. You need to know the names and shapes of the
main curves and surfaces as well as their algebraic
representations.
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Specific
practice materials can be found here:
Here is a pdf
with practice questions, and practice
final 1 (Ignore Problem 6). Solutions to the
practice problems are here
and here
(Thanks to former TA's Chris and Wyatt for writing
those).
Here is practice
final 2. Solutions are here.
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Midterm
was Friday, May 1st and covered Chapter 1 and
2.12.3. The tests are graded (many thanks to the
TA's for such a quick turnaround) and will be
returned in sections this week. Solutions are here.
The statistics for the test are as follows:
Average:
83%
Median: 84 (out of 100)
Standard Deviation: 14
A score of 65% or higher is considered passing on
this test. Please count your points and feel free to
speak with us (your TA or myself) should you have any
questions in regards to the grading.
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We
discuss the cycloid in class today. Here is a link to
a visualization of the brachistochrome problem:
http://curvebank.calstatela.edu/brach/CubicNickForever.gif
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This
site also has interesting links to other important
curves: http://curvebank.calstatela.edu/home/home.htm
and so does this site using Richard Palais' software
3DExplor: http://virtualmathmuseum.org/galleryPC.html
(there are some stunning images and applets to be
found on that site)
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Here
is a link to a wolfram demonstration project to
demonstrate the isochronous property of a cycloidal
pendulum (following Huygen's):
http://demonstrations.wolfram.com/CycloidalPendulum/
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Midterm practice
materials (that might come in handy for final):
Here is the pdf
with practice questions for material from Ch1 and
2.12.3. Solutions to these practice problems are here.
For #14, the answers are no,no,yes,yes. The
negative answers for (a) and (b) can be obtained from
the example I gave in class yesterday, and the
positive answers in (c) and (d) follow from theorems
we stated in class (and whose proofs are in the
internet supplement to the book at http://bcs.whfreeman.com/marsdenvc6e/#767470__774716__)
Actual
tests I gave in this class in Winter 12 are here (we
had two midterms then:
Midterm 1, Winter 2012 is here
and solutions are here.
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Not relevant for
the midterm, but useful for the final and further
study of Chapter 2 and 3 material. For practice
materials for Section 2.4, consider the first problems
from this pdf
with additional practice questions (Solutions to
these practice problems are here)
Midterm 2, Winter 2012 is here
solutions are here.
For your midterm, consider only problem #1, which is
related to Section 2.4. The remaining problems will
be good practice for the final.
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Here
is the link to Yusuf's cool applets to visualize
surfaces: http://people.ucsc.edu/~ygoren/math23sandbox/main.html
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Syllabus is
available by following this link for a pdf.
The syllabus contains lots of useful information such
as a tentative lecture schedule, final exam day and
time, grading policy and homework access. Read it carefully.
You may want to print it out for quick reference.
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Class meets in
Humanities Lecture Hall 206, MWF 89:10AM
DETAILS FOR
HOMEWORK ASSIGNMENTS:
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Homework 1 is due in
section, 4/8 or 4/9, and here are the assigned
problems:
Section 1.1: 4,5,9,15,18,21,27,37
Section 1.2: 1,3,10,12,13,15,20,25,27,29,30
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Homework 2 is due in
section, 4/15 or 4/16, and here are the assigned
problems:
Section 1.3: 1,2,3,4,6,8,11,15ac,16a,19,20,26,30,39
Section 1.4: 1,3,6ac,7ab,10,21
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Homework 3 is due in
section, 4/22 or 4/23) and here are the assigned
problems:
Section 1.5: 1,4,7,8,10,11,24
Section 2.1: 1,3,5a,6,9,13,18,
27,30,40
Section 2.2: 1,2,3,6,8,11,25,26 (you can hand this in
this week or next)
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Homework 4 is due in
section, 4/29 or 4/30) and here are the assigned
problems:
Section 2.3: 1,3,8ab,9,14,16a,19ab,28
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Homework 5 is due in
section, 5/6 or 5/7) and here are the assigned
problems:
Section 2.4: 1,3,5,6,8,11,14,17,22
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Homework 6 is due in
section, 5/13 or 5/14) and here are the assigned problems:
Section 2.5: 3ab,5,7,8,13,15,16,22,26,35
Section 2.6: 2ad,3bc,4,5,7,9a,10ab,20,25,26,27 (you
can hand this in this week or next)
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Homework 7 is due in
section, 5/20 or 5/21) and here are the assigned
problems:
Section 3.1: 6,7,12,13a,14,18,21c,31,32
Section 3.2: 1,3,6,9,10
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Homework 8 is due in
section, 5/27 or 5/28) and here are the assigned
problems:
Section 3.3: 5,14,21,22,23,25,28,29,38,40,41,52
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Homework 9 is due in
section, 6/3 or 6/4) and here are the assigned
problems:
Section 3.4: 2,4,10,11,12,20,23,28
Section 4.1: 1,6,7,16,20,21,22,23
Section 4.2: 3,6,7,8,9
Here is the list of the homework problems from the
remaining sections. These need not be turned in.
Section 4.3: 1,5,7,9,10,12,15,20
Section 4.4: 2,11,14,25,26,27,30,33,39
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Here is the list of
sections:
01A

LBS

61491

MW

10:00AM11:45AM

Suzana
Milea

McHenry
Clrm 1257

44

45

1

999

Open

01B

LBS

61492

MW

02:00PM03:45PM

Connor
Jackman

McHenry
Clrm 1257

41

45

2

999

Open

01C

LBS

61493

TuTh

08:00PM09:45PM

Jonathan
Chi

McHenry
Clrm 1257

45

45

0

999

Closed

01D

LBS

61494

TuTh

12:00PM01:45PM

Suzana
Milea

McHenry
Clrm 1240

48

48

0

999

Closed

01E

LBS

61495

TuTh

06:00PM07:45PM

Jonathan
Chi

McHenry
Clrm 1240

19

48

1

999

Open

01F

LBS

61496

TuTh

02:00PM03:45PM

Connor
Jackman

McHenry
Clrm 1240

35

48

0

999

Open

Here is the list of
TA's, plus their contact
and OH:
TA
Name

EMail

Office
Hours & Location

Suzana
Milea

smilea@ucsc.edu

Mon Tue Thu
2:00pm  3:00pm
in McHenry 4112.

Jonathan
Chi

jbchi@ucsc.edu

Tues 24, Wed 2:304:30
in McHenry 4112.

Connor
Jackman

cfjackma@ucsc.edu

Mon &
Wed 1011:30am
in
McHenry 1261.


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Here
is the link to the page with the equations and
graphs of the quadric
surfaces.
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Here
is a link to a free
linear algebra text for those of you who want
to learn/review more linear algebra. Note: We will
not need to use linear algebra beyond what is
covered in Chapter 1 in our textbook by
Marsden/Tromba.


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Links
to the vector applets are here:
1) vector
sum
2) dot or
scalar product
3) What
good are dot products?
4) Parallelepiped
(its volume is computed by the scalar triple
product). Another look at this is here.


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I
recommend you download and experiment with DPGraph,
a free utility for visualizing surfaces. We will
use this in class and it can help you for
selfstudy and the homework. Get it at http://dpgraph.com/,
click on "List of Site Licensees" in the
box near the top, and find UCSC in the list. Here
are some examples of a few surfaces:


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Graphs
of planes, cylinders and quadric surfaces
a) The Right
Circular Cylinder x^2+y^2=1
b) The Parabolic
Cylinder z=x^2
c) Ellipsoid
d) Elliptic
Paraboloid
e) Hyperbolic
Paraboloid
f) Cone
g) Hyperboloid
of one sheet
h) Hyperboloid
of two sheets
i) Moving
from one sheet through a cone to two sheets.
k) intersection of the two
planes
x+4y3z=1 and 3x+6y+7z=0 ; The picture is here.
l)The parametric equation
of the line (which is this intersection) is
x= 46t+ 1/3, y=2t+1/6, z= 18t ; The picture is here.


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Polar Coordinates/Polar Curves:
a)Flower
b)Henri's
Butterfly, again
c) Oscar's
Butterfly
d) Lemniscate
e) Archimedean
Spiral.
Cylindrical
Coordinates/Spherical Coordinates:
a)Sphere
in spherical coordinates.
b)Vase
in spherical coordinates, on
table
c)Hyperboloid
of two sheets, cylindrical coordinates
d)Cone
in cylindrical coordinates.
e)Cylinder
with cylindrical coordinates
f) Ellipsoid
with cylindrical coordinates.
Guess
What?
a)guess?


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Stuff
about differentiability:
a) Example
of a function with partials at (0,0)
but that is not differentiable at (0,0). It's
crinkled.
b) Example
of a function that is differentiable, but it's partials are not continuous.


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a)
No
limit at the origin
b) Limit
exists at origin.
c) Paraboloid with tangent plane.


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Paths
and curves:
a) Helix c(t)=<cos t, sin t, t> in
box.
b)
Helix c(t)=<cos t, sin t, t>
with x,y,z coordinate system
c)
Semicubical
Parabola
c(t)=<t^3, t^2> in the box
d) Semicubical
Parabola
c(t)=<t^3, t^2> with x,y,z
axis
e) Twisted
Cubic c(t)=<t,
t^2, t^3> in the box
f) Twisted
Cubic c(t)=<t,
t^2, t^3> together with the x,y,z axis (sans
labels though)
g) Cylinder y=x^2 intersecting the cylinder z=x^3
has the twisted cubic as its Error! Hyperlink reference not
valid..
h) applet for
cycloid: http://www.ies.co.jp/math/java/calc/cycloid/cycloid.html
i) The cycloid
in dpgraph.
j) Twisted
Cubic pointwise
k) Twisted
Cubic with tangent line.


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The
National Map viewer: http://nmviewogc.cr.usgs.gov/viewer.htm
Santa Cruz has coordinates Lat: 36.97417 degrees,
Long: 122.02972 degrees


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More
topomaps at: www.topozone.com


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Investigations
into the directional derivaive with DP Graph files:
The surface
f(x,y) = ln (x^2+y^2)
The surface
f(x,y) = ln (x^2+y^2) and its tangent plane at
(x,y)=(3,1)
The surface
f(x,y) = ln (x^2+y^2), its tangent plane at
(x,y)=(3,1) and the vertical plane through
(x,y)=(3,1) that points in the direction of
(1,1).
Just the tangent
plane and the vertical plane.
Just the tangent
plane and the vertical plane. The vertical
plane can be rotated about the line l(t) = (3,1, 0) + t(0,0,1) by
using the scrollbar and varying the values of a.
Same but with
the surface.

Here
is a link to a cool Java applet we will use in class
to analyze vector
fields.
