MATH-348 Advanced Engineering Mathematics - Spring 2010

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{{PDF Table Item|filename=Math348_syllabus.spring2010.pdf | title=MATH348.Spring2010.Syllabus}}
 
{{PDF Table Item|filename=Math348_syllabus.spring2010.pdf | title=MATH348.Spring2010.Syllabus}}
 +
{{PDF Table Item|filename=Math235_hw2Edit1.pdf | title=MATH348.Spring2010.Syllabus}}
 
==Lecture Slides==
 
==Lecture Slides==
 +
{{PDF Table Item|filename=Math348_s11.hw7.pdf | title=00.LS.Introduction }}
 +
{{PDF Table Item|filename=Math348_s11.hw7Soln.pdf | title=00.LS.Introduction }}
 +
{{PDF Table Item|filename=Math348_s11.hw8e.pdf | title=00.LS.Introduction }}
 +
{{PDF Table Item|filename=Math348_s11.hw8Soln.pdf | title=00.LS.Introduction }}
 
{{PDF Table Item|filename=Math348_LS.00.pdf | title=00.LS.Introduction }}
 
{{PDF Table Item|filename=Math348_LS.00.pdf | title=00.LS.Introduction }}
 
{{PDF Table Item|filename=Math348_LS.01.pdf | title=01.LS.Classical Vector Spaces}}
 
{{PDF Table Item|filename=Math348_LS.01.pdf | title=01.LS.Classical Vector Spaces}}
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{{PDF Table Item|filename=Math348_LS.04.pdf | title=04.LS.Abstract Vector Spaces}}
 
{{PDF Table Item|filename=Math348_LS.04.pdf | title=04.LS.Abstract Vector Spaces}}
−
{{PDF Table Item|filename=Math348_LS.05.pdf | title=04.LS.Fourier Series to Fourier Integral to Fourier Transform}}
+
{{PDF Table Item|filename=Math348_LS.05.edit.pdf | title=05.LS.Fourier Series to Fourier Integral to Fourier Transform - Update 4/5/2010}}
 +
{{PDF Table Item|filename=Math348_LS.06.pdf | title=06.LS.1D Heat Equation-Separation of Variables}}
 +
{{PDF Table Item|filename=Math348_LS.07.pdf | title=07.LS.The Acoustic Approximation and Wave Equations}}
  
 
==Lecture Notes==
 
==Lecture Notes==
 +
{{PDF Table Item|filename=Math235_PE.pdf | title=??}}
 
{{PDF Table Item|filename=Math348_LN.00.pdf | title=00.LN.Overview And Outline}}
 
{{PDF Table Item|filename=Math348_LN.00.pdf | title=00.LN.Overview And Outline}}
 
{{PDF Table Item|filename=Math348_LN.01.update.pdf | title=01.LN.LinearDefinitions : Updated 1.27.2010. Footnotes have been added referencing locations in the text where these definitions can be found. }}
 
{{PDF Table Item|filename=Math348_LN.01.update.pdf | title=01.LN.LinearDefinitions : Updated 1.27.2010. Footnotes have been added referencing locations in the text where these definitions can be found. }}
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{{PDF Table Item|filename=Math348_LN.11.pdf | title=11.LN.Fourier Integral to Fourier Transform }}
 
{{PDF Table Item|filename=Math348_LN.11.pdf | title=11.LN.Fourier Integral to Fourier Transform }}
 
{{PDF Table Item|filename=Math348_LN.12.pdf | title=12.LN.Fourier Transform }}
 
{{PDF Table Item|filename=Math348_LN.12.pdf | title=12.LN.Fourier Transform }}
 +
{{PDF Table Item|filename=Math348_LN.13e.pdf | title=13.LN.IntroToPDE }}
 +
{{PDF Table Item|filename=Math348_LN.14.pdf | title=14.LN.HeatEquation }}
 +
{{PDF Table Item|filename=Math348_LN.15.pdf | title=15.LN.WaveEquation }}
  
 
==Assignments==
 
==Assignments==
 
{{PDF Table Item|filename=Math348_HW.00.pdf | title=Homework0 - Due Jan. 18th by 5:00pm}}
 
{{PDF Table Item|filename=Math348_HW.00.pdf | title=Homework0 - Due Jan. 18th by 5:00pm}}
−
{{PDF Table Item|filename=Math348_HW.00.Soln.pdf | title=Homework0 - Solutions}}
+
{{PDF Table Item|filename=MHW2_edit2.pdf | title=Homework0 - Solutions}}
 
{{PDF Table Item|filename=Math348_HW.01.edit1.pdf | title=Homework1 - Due Feb. 3rd by 5:00pm - Note: Updated 1/19/2010, fixed a typo in problem 2 matrix 2, a_{22} = -3 }}
 
{{PDF Table Item|filename=Math348_HW.01.edit1.pdf | title=Homework1 - Due Feb. 3rd by 5:00pm - Note: Updated 1/19/2010, fixed a typo in problem 2 matrix 2, a_{22} = -3 }}
 
{{PDF Table Item|filename=Math348_HW.01.Soln.pdf | title=Homework1 - Solutions }}
 
{{PDF Table Item|filename=Math348_HW.01.Soln.pdf | title=Homework1 - Solutions }}
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       [http://ticc.mines.edu/csm/wiki/images/a/a4/Math348_HW.1.4.2.2.lsp.Plot.pdf Geometry of Least Squares Problem of Problem 4  Set 2]
 
       [http://ticc.mines.edu/csm/wiki/images/a/a4/Math348_HW.1.4.2.2.lsp.Plot.pdf Geometry of Least Squares Problem of Problem 4  Set 2]
 
       [http://ticc.mines.edu/csm/wiki/images/8/86/Math348_HW.1.4.2.3.Plot.pdf Interpolated Parabolas of Problem 4  Set 3]
 
       [http://ticc.mines.edu/csm/wiki/images/8/86/Math348_HW.1.4.2.3.Plot.pdf Interpolated Parabolas of Problem 4  Set 3]
−
 
+
Fourier Transform
 
{{PDF Table Item|filename=Math348_HW.02.edit2.pdf | title=Homework2 - Due Feb. 12th by 5:00pm : 1) Header Box Updated 2) Problem 4.2 \lambda = n^2}}
 
{{PDF Table Item|filename=Math348_HW.02.edit2.pdf | title=Homework2 - Due Feb. 12th by 5:00pm : 1) Header Box Updated 2) Problem 4.2 \lambda = n^2}}
 
{{PDF Table Item|filename=Math348_HW.02..edit2.Soln.pdf | title=Homework2 - Solutions : Update - There were a couple of typos, nothing major, corrected. 2/8/2010 : Updated again - One of the typos I corrected last time was not a typo at all (1.4). I have put it back in its place.}}
 
{{PDF Table Item|filename=Math348_HW.02..edit2.Soln.pdf | title=Homework2 - Solutions : Update - There were a couple of typos, nothing major, corrected. 2/8/2010 : Updated again - One of the typos I corrected last time was not a typo at all (1.4). I have put it back in its place.}}
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{{PDF Table Item|filename=Math348_HW.05.e2.pdf | title=Homework5 - Due: March 31st }}
 
{{PDF Table Item|filename=Math348_HW.05.e2.pdf | title=Homework5 - Due: March 31st }}
 
{{PDF Table Item|filename=Math348_HW.05.Soln.pdf | title=Homework5 - Solutions}}
 
{{PDF Table Item|filename=Math348_HW.05.Soln.pdf | title=Homework5 - Solutions}}
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{{PDF Table Item|filename=Math348_HW.06.pdf | title=Homework6 - Due: April 5th }}
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{{PDF Table Item|filename=Math348_HW.06.pdf | title=Homework6 - Due: April 12th }}
 
{{PDF Table Item|filename=Math348_HW.06.soln.pdf | title=Homework6 - Solutions }}
 
{{PDF Table Item|filename=Math348_HW.06.soln.pdf | title=Homework6 - Solutions }}
 +
{{PDF Table Item|filename=Math348_HW.07e3.pdf | title=Homework7 - Due April 28 - Last edits (minor) at 4:31pm. }}
 +
{{PDF Table Item|filename=Math348_HW.07e2.soln.pdf | title=Homework7 - Solutions - Last edits (minor) at 4:31pm  : Note at a step in problem 1.1 I use quantities like I1 and I2. By these I mean l1 and l2. }}
 +
{{PDF Table Item|filename=Math348_HW.08.pdf | title=Homework8 - Due May 3 }}
 +
{{PDF Table Item|filename=Math348_HW.08.soln.pdf | title=Homework8 - Solutions }}
  
 
==Exams==
 
==Exams==
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===Exam II===
 
===Exam II===
−
Exam II will be held on April 7th in class. There will be no notecards or calculators. The exam will have five required questions and contain material outlined in the following review:
+
Exam II will be held on April 16th in class. There will be no notecards or calculators. The exam will have five required questions and contain material outlined in the following review:
  
 
{{PDF Table Item|filename=Math348_e2review.pdf|title=Exam 2 - Review Sheet}}
 
{{PDF Table Item|filename=Math348_e2review.pdf|title=Exam 2 - Review Sheet}}
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{{PDF Table Item|filename=Math348_e2.summer2009.pdf|title=Exam II - Summer2009 }}
 
{{PDF Table Item|filename=Math348_e2.summer2009.pdf|title=Exam II - Summer2009 }}
 
{{PDF Table Item|filename=Math348_e2Soln.summer2009.pdf|title=Exam II - Summer2009 Solutions}}
 
{{PDF Table Item|filename=Math348_e2Soln.summer2009.pdf|title=Exam II - Summer2009 Solutions}}
 +
{{PDF Table Item|filename=Math348_e2.fall2009.pdf|title=Exam II - Fall2009 }}
 +
{{PDF Table Item|filename=Math348_e2Soln.Fall2009.pdf|title=Exam II - Fall2009 Solutions - Graphs Included}}
 +
  
 
     Exam II -  Statistics
 
     Exam II -  Statistics
−
     Mean = xx
+
     Mean = 36.5 (72.5%)
−
     Median = xx
+
     Median = 37.5 (75%)
−
     A's = x, B's = x, C's = x, D's = x, F's = x, Total Number of Exams  = xx
+
     A's = 9, B's = 32, C's = 38, D's = 19, F's = 16, Total Number of Exams  = 114
 +
    A's = 8%, B's = 28%, C's = 33%, D's = 17%, F's = 14 %
 +
 
 +
{{PDF Table Item|filename=Math348_e2.spring2010.pdf|title=Exam II - Spring2010 }}
 +
{{PDF Table Item|filename=Math348_e2Soln.Spring2010.pdf|title=Exam I - Spring2010 Solutions}}
 +
 
 +
===Final Exam===
 +
The final exam will be held Saturday May 8th from 7:00pm-9:00pm. The classes will be testing in the following rooms:
 +
    Class    : Meeting Time    : Testing Room  : Proctor
 +
    MATH348B : 11:00am Section : Petroleum Hall : Jennifer Strong
 +
    MATH348C : 1:00pm Section  : CT 102        : Scott Strong
 +
    MATH348D : 2:00pm Section  : CO209          : Doug Poole
 +
Since we will be in different rooms it is very important that you go to the room associated with your section.
 +
 
 +
 
 +
There will be no notecards or calculators. The exam will have ten required questions and contain material outlined in the following review:
 +
 
 +
{{PDF Table Item|filename=Math235_hw2.pdf|title=Final Exam - Review Sheet}}
 +
 
 +
The following is an old 50 minute PDE exam, which should give you some idea of the content and structure of the PDE portion of the exam.
 +
 
 +
{{PDF Table Item|filename=Math348_ePDE.old.pdf|title=OLD PDE EXAM - See Soln for the graph in problem 1}}
 +
{{PDF Table Item|filename=Math348_ePDESOLN.old.pdf|title=OLD PDE EXAM - SOLN}}
  
 
=Other Materials=
 
=Other Materials=
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===Three Planes in Space===
 
===Three Planes in Space===
 
{{PDF Table Item|filename=Math348_planes.summer2009.pdf | title=Three Planes in Space - Four Different Ways}}
 
{{PDF Table Item|filename=Math348_planes.summer2009.pdf | title=Three Planes in Space - Four Different Ways}}
 +
{{PDF Table Item|filename=MATH235-EI-Reading2.pdf | title=Three Planes in Space - Four Different Ways}}
 
   '''Legend for the Animations'''
 
   '''Legend for the Animations'''
 
     Red = First Plane Equation
 
     Red = First Plane Equation
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===Ordinary Differential Equations===
 
===Ordinary Differential Equations===
 
{{PDF Table Item|filename=Math348_ODEReview.fall2009.pdf|title= Review of Ordinary Differential Equations (DRAFT - 11/16/09)}}
 
{{PDF Table Item|filename=Math348_ODEReview.fall2009.pdf|title= Review of Ordinary Differential Equations (DRAFT - 11/16/09)}}
−
{{End PDF Table}}
+
[http://en.wikipedia.org/wiki/Millennium_Bridge_(London) Millennium Bridge - Wikipedia]
 +
 
 +
[http://www.youtube.com/watch?v=eAXVa__XWZ8 You Tube Video - Millennium Bridge Resonance]
 +
 
 +
===Heat Equation===
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/V.mp4 Heat Movie 1 - abs(x)]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/parabola.mp4 Heat Movie 2 - parabola]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/doubleV.mp4 Heat Movie 3 - Double V]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Heat_Dirch_Force.avi Heat Movie 4 - Forced Heat Equation with B.C. u(0,t)=u(L,t)=0 ]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Heat_Neumman_Forcing.avi Heat Movie 5 - Forced Heat Equation with B.C. u_{x}(0,t)=u_{x}(L,t)=0 ]
 +
 
 +
===Wave Equation===
 +
 
 +
====1D Wave Equation====
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Wave_Dirch_Triangle1.divx.ff.avi  Wave on a 1-D Sting with Fixed Endpoints]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Wave_Dirch_Triangle2.divx.ff.avi  Wave on a 1-D Sting with Fixed Endpoints - Animated with first 5 Fourier Modes (Fundamental Mode in Red)]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Wave_Neumman_Triangle2.divx.ff.avi  Wave on a 1-D Sting with FLAT Endpoints from HW10]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Wave_Neumman_Triangle1.divx.ff.avi  Wave on a 1-D Sting with FLAT Endpoints from HW10 - Animated with first 5 Fourier Modes (Fundamental Mode in Red)]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Traveling%20Waves.divx.ff.avi  Traveling Wave :<math>u_{0}(x) = -\tanh(x)</math>: Red = Right Traveling, Blue=Left Traveling, Black = Superposition]
 +
 
 +
====2D Wave Equation Rectangular and Polar====
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/2DWave_Rec_BookEx1a.avi Rectangular Membrane Movie 1 -Text Example pg577]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/2DWave_Rec_BookEx1b.avi Rectangular Membrane 2 -Text Example pg577]
 +
 
 +
[http://www.falstad.com/membrane/ Applet - Pretty Cool]
 +
 
 +
[http://hyperphysics.phy-astr.gsu.edu/Hbase/music/recmem.html Rectangular Membrane Modes]
 +
 
 +
[http://www.kettering.edu/~drussell/Demos/MembraneSquare/Square.html Animations of Rectangular Membrane Modes - Pretty Good]
 +
 
 +
[http://www.kettering.edu/~drussell/demos.html Animations done by Dr. Russell - All sorts of stuff! ]
 +
 
 +
 
 +
[http://music.nebrwesleyan.edu/wtt/index.html The Well-Tempered Timpani By Richard K. Jones]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Membrane1.mp4 Vibrating Membrane1 - 12.9.1 Example]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/membrane2.mp4 Vibrating Membrane2 - 12.9.1 Example]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Mambrane3.mp4 Vibrating Membrane3 - 12.9.1 Example]
 +
 
 +
[http://www.mines.edu/~sstrong/AEM_Materials/Mambrane4.mp4 Vibrating Membrane4 - 12.9.1 Example]
 +
 
 +
 
 +
 
 +
====Nonlinear Wave Phenomenon====
 +
 
 +
[http://en.wikipedia.org/wiki/Shock_wave  Wikipedia Article on Shock Waves]
 +
 
 +
[http://physics-animations.com/Physics/English/swa_txt.htm  Animation of Shock Wave Formation in Pressure Field]
 +
 
 +
[http://www.youtube.com/watch?v=jZ3Hhdr8EjI&feature=PlayList&p=2AE42B05690BA5AD&playnext=1&playnext_from=PL&index=2  Shock Wave (Plane) - You Tube 1]
 +
 
 +
[http://www.youtube.com/watch?v=QX04ySm4TTk&feature=PlayList&p=2AE42B05690BA5AD&index=3&playnext=2&playnext_from=PL Shock Wave (Plane) - You Tube 2]
 +
 
 +
[http://www.youtube.com/watch?v=9r_2l5_FviU Shock Wave (Explosion) - You Tube 3]
 +
 +
[http://www.youtube.com/watch?v=GvVQEl_q2ZM Shock Wave (Explosion) - You Tube 4 : Ignore The the cartoon bubble]
 +
 
 +
[http://www.youtube.com/watch?v=pz7lcHN7GxQ Shock Wave (Simulation) - You Tube 5 : Notice the distortion of the expanding wave-front]
 +
 
 +
[http://www.break.com/index/explosion-shock-waves-in-slow-motion.html Shockwave Slowmo]
 +
 
 +
[http://exploration.grc.nasa.gov/education/rocket/shock.html NASA - Shock Wave Simulator]
 +
 
 +
[http://www.tfarchive.com/cartoons/wallpapers/shockwave17.jpg Shockwave :)]

Latest revision as of 14:24, 20 April 2011

Main Page > Mathematical and Computer Sciences Course Wikis

Contents

Course Information

MATH348: Advanced Engineering Mathematics - Introduction to partial differential equations, with applications to physical phenomena. Fourier series. Linear algebra, with emphasis on sets of simultaneous equations. Prerequisite: MATH 225 or equivalent.

Instructor Information

Instructor : Scott Strong

Office : Chauvenet Hall 266

Office Phone : 303.384.2446

email : math348.spring2010@gmail.com

Course Calendar

Classes Begin : January 13th, 2010

Lecture Days : Monday, Wednesday, Friday

Course Sections : 

  B : 11:00am-11:50am - Location: Coolbaugh Hall 131
  C : 1:00pm-1:50pm - Location: Green Center 211
  D : 2:00pm-2:50pm - Location: Alderson Hall 430

Last Day to Drop Without a W : January 28th

Last Day to Withdraw : March 30th

Classes End : May 14th, 2010

Important Dates : 

   February 14th : No Classes
   March 15th-19th : Spring Break
   April 8th-10th : E-Days
   May 3th-7th : Dead Week
   May 7th : Dead Day

Office Hours

Fixed Office Hours : 

 MWF : 12:00pm-12:50pm
 Monday : 3:00pm-5:00pm

If you cannot meet during the previous office hours then please contact me to schedule another meeting time. Please see this google calender to see the times I am unavailable.

Textbook Information

    Textbook : Advanced Engineering Mathematics - Erwin Kreyszig, ISBN 978-0-471-48885-9
    9th Edition Amazon : Advanced Engineering Mathematics - Erwin Kreyszig, ISBN 978-0-471-48885-9
    8th Edition Amazon (Used) : Advanced Engineering Mathematics - Erwin Kreyszig, ISBN 978-0-471-48885-9

Course Materials

Pdf.png These downloads require Adobe Acrobat Reader
MATH348.Spring2010.Syllabus
MATH348.Spring2010.Syllabus

Lecture Slides

00.LS.Introduction
00.LS.Introduction
00.LS.Introduction
00.LS.Introduction
00.LS.Introduction
01.LS.Classical Vector Spaces
02.LS.Geometry in R^n
03.LS.KinematicsAndDynamics
03.LS.KinematicsAndDynamics - Those Evil Natured Robots

Linear Algebra and its Applicationsby Peter D. Lax

04.LS.Abstract Vector Spaces
05.LS.Fourier Series to Fourier Integral to Fourier Transform - Update 4/5/2010
06.LS.1D Heat Equation-Separation of Variables
07.LS.The Acoustic Approximation and Wave Equations

Lecture Notes

??
00.LN.Overview And Outline
01.LN.LinearDefinitions : Updated 1.27.2010. Footnotes have been added referencing locations in the text where these definitions can be found.
02.LN.Introduction To Linear Equations
03.LN.Solving Linear Systems
04.LN.Square Systems - Determinants and Matrix Inversion
05.LN.Introduction to Linear Vector Spaces
06.LN.Chapter 7 - Wrap Up
07.LN.Eigenproblems
08.LN.Diagonalization
09.LN.Introduction to Fourier Series : Review of Periodic and Symmetric Functions
10.LN.Complex Fourier Series
11.LN.Fourier Integral to Fourier Transform
12.LN.Fourier Transform
13.LN.IntroToPDE
14.LN.HeatEquation
15.LN.WaveEquation

Assignments

Homework0 - Due Jan. 18th by 5:00pm
Homework0 - Solutions
Homework1 - Due Feb. 3rd by 5:00pm - Note: Updated 1/19/2010, fixed a typo in problem 2 matrix 2, a_{22} = -3
Homework1 - Solutions 
     Graphics for Homework 1
     Geometry of Problem 2 System 1
     Geometry of Problem 2 System 2
     Geometry of Problem 2 System 3
     Geometry of Problem 2 System 4
     Geometry of Problem 2 System 5
     Interpolated Parabolas of Problem 4  Set 1
     Interpolated Parabolas of Problem 4  Set 2
     Geometry of Least Squares Problem of Problem 4  Set 2
     Interpolated Parabolas of Problem 4  Set 3

Fourier Transform

Homework2 - Due Feb. 12th by 5:00pm : 1) Header Box Updated 2) Problem 4.2 \lambda = n^2
Homework2 - Solutions : Update - There were a couple of typos, nothing major, corrected. 2/8/2010 : Updated again - One of the typos I corrected last time was not a typo at all (1.4). I have put it back in its place.


Homework 3 - Note : I have just noticed a pesky typo. Equation (2) from the assignment, (26) from the solutions, should read l_1 u(a) + k_1 u'(a) = 0 and NOT l_1 u(a) + k_1 u'(b) = 0

Homework3 - Due Feb. 22th by 5:00pm : Update - There were multiple things going on here. Once I updated the assignment with an old copy that was missing problems.... Ugh, it's all fixed up now. :)
Homework3 - Solutions
Homework4 - Due: March 12th
Homework4 - Solutions
Homework5 - Due: March 31st
Homework5 - Solutions
Homework6 - Due: April 12th
Homework6 - Solutions
Homework7 - Due April 28 - Last edits (minor) at 4:31pm.
Homework7 - Solutions - Last edits (minor) at 4:31pm  : Note at a step in problem 1.1 I use quantities like I1 and I2. By these I mean l1 and l2.
Homework8 - Due May 3
Homework8 - Solutions

Exams

Exam I

Exam I will be held on March 1st in class. There will be no notecards or calculators. The exam will have five required questions and contain material outlined in the following review:

Exam 1 - Review Sheet

The following exams with solutions are posted for your review.

Exam 1 - Fall2008
Exam 1 - Fall2008 Solutions
Exam I - Spring2009
Exam I - Spring2009 Solutions
Exam I - Summer2009
Exam I - Summer2009 Solutions
Exam I - Fall2009
Exam I - Fall2009 Solutions 


    Exam I -  Statistics
    Mean = 37.15 (74,31%)
    Median = 38 (76%)
    Mode = 47 (94%)
    A's = 34, B's = 17, C's = 24, D's = 18, F's = 25, Total Number of Exams  = 118
    A's = 29%, B's = 14%, C's = 20%, D's = 15%, F's = 21 %
Exam I - Spring2010
Exam I - Spring2010 Solutions

Exam II

Exam II will be held on April 16th in class. There will be no notecards or calculators. The exam will have five required questions and contain material outlined in the following review:

Exam 2 - Review Sheet

The following are the results of Q+A's from previous semesters:

Exam 2 - Spring2009 Q + A
Exam 2 - Fall2008 Q + A

The following exams with solutions are posted for your review.

Exam II - Spring2009 See Soln for problem 3 graph.
Exam II - Spring2009 Solutions
Exam 2 - Fall2008
Exam 2 - Fall2008 Solutions
Exam II - Summer2009
Exam II - Summer2009 Solutions
Exam II - Fall2009
Exam II - Fall2009 Solutions - Graphs Included 


    Exam II -  Statistics
    Mean = 36.5 (72.5%)
    Median = 37.5 (75%)
    A's = 9, B's = 32, C's = 38, D's = 19, F's = 16, Total Number of Exams  = 114
    A's = 8%, B's = 28%, C's = 33%, D's = 17%, F's = 14 %
Exam II - Spring2010
Exam I - Spring2010 Solutions

Final Exam

The final exam will be held Saturday May 8th from 7:00pm-9:00pm. The classes will be testing in the following rooms: 

   Class    : Meeting Time    : Testing Room   : Proctor 
   MATH348B : 11:00am Section : Petroleum Hall : Jennifer Strong
   MATH348C : 1:00pm Section  : CT 102         : Scott Strong
   MATH348D : 2:00pm Section  : CO209          : Doug Poole

Since we will be in different rooms it is very important that you go to the room associated with your section.


There will be no notecards or calculators. The exam will have ten required questions and contain material outlined in the following review:

Final Exam - Review Sheet

The following is an old 50 minute PDE exam, which should give you some idea of the content and structure of the PDE portion of the exam.

OLD PDE EXAM - See Soln for the graph in problem 1
OLD PDE EXAM - SOLN

Other Materials

Linear Algebra

Three Planes in Space

Three Planes in Space - Four Different Ways
Three Planes in Space - Four Different Ways 
 Legend for the Animations
   Red = First Plane Equation
   Orange = Second Plane Equation
   Yellow = Third Plane Equation
   Green = Column Space of A (AKA the set of all linear combination of the pivot columns of A)
   Blue = Right Hand Side for non-homogeneous problem.

Animation : Ax=0 with oo-many solutions that form a line in space.

Animation : Ax=b with oo-many solutions that form a line in space.

Animation : Ax=b with a single solution

Animation : Ax=b with no solutions

Linear Algebra Software

Linear Algebra Toolkit

Fourier Methods

Review of Functions

Special Angles and the Unit Circle
A61.TrigIdentities

Odd and Even Functions (Wikipedia) : (see Also 09.LN)

Periodic Functions (Wikipedia) : (See Also 09.LN)

Fourier Series

FS for f(x}=x, x \in (-\pi,\pi)
FS for f(x}=Exp(Abs(x)), x \in (-\pi,\pi)

Fourier Series - Wikipedia Gibbs Phenomenon - Wikipedia

Fourier Transform

Fourier Transform - Wikipedia Wikipedia - Sinc Function Mathworld - Sinc Function Wikipedia - Nyquist-Shannon Sampling Theorem

Mathworld - Convolution (Animation)

Convolution and Diffraction (Animations)

Convolution and Diffraction (Animations)

Wikipedia - Convolution (Animation)

Green's Function - Wikipedia

Frequency Response Graph for a Harmonic Oscillator m=k=1, Gamma = {1,.5,.25,.125}

Partial Differential Equations

Ordinary Differential Equations

Review of Ordinary Differential Equations (DRAFT - 11/16/09)

Millennium Bridge - Wikipedia

You Tube Video - Millennium Bridge Resonance

Heat Equation

Heat Movie 1 - abs(x)

Heat Movie 2 - parabola

Heat Movie 3 - Double V

Heat Movie 4 - Forced Heat Equation with B.C. u(0,t)=u(L,t)=0

Heat Movie 5 - Forced Heat Equation with B.C. u_{x}(0,t)=u_{x}(L,t)=0

Wave Equation

1D Wave Equation

Wave on a 1-D Sting with Fixed Endpoints

Wave on a 1-D Sting with Fixed Endpoints - Animated with first 5 Fourier Modes (Fundamental Mode in Red)

Wave on a 1-D Sting with FLAT Endpoints from HW10

Wave on a 1-D Sting with FLAT Endpoints from HW10 - Animated with first 5 Fourier Modes (Fundamental Mode in Red)

Traveling Wave :u0(x) = − tanh(x): Red = Right Traveling, Blue=Left Traveling, Black = Superposition

2D Wave Equation Rectangular and Polar

Rectangular Membrane Movie 1 -Text Example pg577

Rectangular Membrane 2 -Text Example pg577

Applet - Pretty Cool

Rectangular Membrane Modes

Animations of Rectangular Membrane Modes - Pretty Good

Animations done by Dr. Russell - All sorts of stuff!


The Well-Tempered Timpani By Richard K. Jones

Vibrating Membrane1 - 12.9.1 Example

Vibrating Membrane2 - 12.9.1 Example

Vibrating Membrane3 - 12.9.1 Example

Vibrating Membrane4 - 12.9.1 Example


Nonlinear Wave Phenomenon

Wikipedia Article on Shock Waves

Animation of Shock Wave Formation in Pressure Field

Shock Wave (Plane) - You Tube 1

Shock Wave (Plane) - You Tube 2

Shock Wave (Explosion) - You Tube 3

Shock Wave (Explosion) - You Tube 4 : Ignore The the cartoon bubble

Shock Wave (Simulation) - You Tube 5 : Notice the distortion of the expanding wave-front

Shockwave Slowmo

NASA - Shock Wave Simulator

Shockwave :)

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