WELCOME TO MATH 1-2-3

 

The web site for the on-the-ground Mathematics courses of Frank Buerle at UCSC.

 

What's New/What's True June 4, 2015

 

It's not new, but it's pretty (awesome): e^(pi * i)+1=0. This is called Euler's identity and is arguably the most amazing and beautiful mathematical formula. Find out more about it here or here. More info on the great Swiss mathematician Leonhard Euler can be found here. Not long ago was the 300th (!) anniversary of his birth.

A link for the adventurous Calculus Student: http://www.math.tamu.edu/~tvogel/gallery/gallery.htm

 

Math 23A Spring 2015:

 

      Here is information about the final exam which is on Monday, June 8th, from 8am-11am.
0) Review Session with Frank is on Friday, 2-3:30pm 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 real-valued). 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.

      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.

      Midterm was Friday, May 1st and covered Chapter 1 and 2.1-2.3. The tests are graded (many thanks to the TA's for such a quick turn-around) 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.

      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

      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)

      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/

      Midterm practice materials (that might come in handy for final):

Here is the pdf with practice questions for material from Ch1 and 2.1-2.3. 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.

      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.


      Here is the link to Yusuf's cool applets to visualize surfaces: http://people.ucsc.edu/~ygoren/math23sandbox/main.html

      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.

      Class meets in Humanities Lecture Hall 206, MWF 8-9:10AM


DETAILS FOR HOMEWORK ASSIGNMENTS:

      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

      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

      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)

      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

      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

      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)

      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

      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

      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

      Here is the list of sections:

 

01A

LBS

61491

MW

10:00AM-11:45AM

Suzana Milea

McHenry Clrm 1257

44

45

1

999

Description: pen Open

01B

LBS

61492

MW

02:00PM-03:45PM

Connor Jackman

McHenry Clrm 1257

41

45

2

999

Description: pen Open

01C

LBS

61493

TuTh

08:00PM-09:45PM

Jonathan Chi

McHenry Clrm 1257

45

45

0

999

Description: losedClosed

01D

LBS

61494

TuTh

12:00PM-01:45PM

Suzana Milea

McHenry Clrm 1240

48

48

0

999

Description: losedClosed

01E

LBS

61495

TuTh

06:00PM-07:45PM

Jonathan Chi

McHenry Clrm 1240

19

48

1

999

Description: pen Open

01F

LBS

61496

TuTh

02:00PM-03:45PM

Connor Jackman

McHenry Clrm 1240

35

48

0

999

Description: pen Open



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


TA Name

E-Mail

Office Hours & Location

Suzana Milea

smilea@ucsc.edu

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

Jonathan Chi

jbchi@ucsc.edu

Tues 2-4, Wed 2:30-4:30
in McHenry 4112.

Connor Jackman

cfjackma@ucsc.edu

Mon & Wed 10-11:30am
 in McHenry 1261.

 

 

      Here is the link to the page with the equations and graphs of the quadric surfaces.

      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.

      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.

      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 self-study 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:

      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+4y-3z=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.

      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?

      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.

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

      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.
 

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

      More topo-maps atwww.topozone.com

      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.

 

General Stuff:

 

Links on Global Climate Change:
MSRI workshop:http://www.msri.org/calendar/workshops/WorkshopInfo/432/show_workshop
KQED forum: Global Climate Change in Local Context: http://www.kqed.org/epArchive/R704111000
IPPC: web site: http://www.ipcc.ch/, The scientific basis:http://www.grida.no/climate/ipcc_tar/wg1/index.htm
ClimatePrediction.Net:  http://climateprediction.net/

 

 

 

Some good web sites for info or help on the mathematics are:

 

https://www.wolframalpha.com/
http://mathworld.wolfram.com/

http://tutorial.math.lamar.edu/

http://mathforum.org

 

Not doing as well on tests as you expected? Having trouble studying effectively? Check out the improved sections on Study Tips and Test-Taking (with a section on Math & Test Anxiety) under Resources-Help.

Trying to keep up with world events? I find good info on the radio by listening to NPR, PRI, BBC on FM 88.1, 88.5, 88.9, 90.3 and 91.9. For those among you that speak german and can use some alternate perspective, a decent news and analysis  site is at http://www.spiegel.de