ÿþ<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 12"> <meta name=Originator content="Microsoft Word 12"> <link rel=File-List href="applied-math-methods-S12_files/filelist.xml"> <link rel=Edit-Time-Data href="applied-math-methods-S12_files/editdata.mso"> <title>methods of Applied Mathematics II - 2011</title> <link rel=themeData href="applied-math-methods-S12_files/themedata.thmx"> <link rel=colorSchemeMapping href="applied-math-methods-S12_files/colorschememapping.xml"> </style> <meta name=author content="Irene Gamba"> <meta name=CREATED content="20110118;7140000"> <meta name=CHANGEDBY content=Irene> <meta name=CHANGED content="20110118;7210000"> <meta name="Info 1" content=""> <meta name="Info 2" content=""> <meta name="Info 3" content=""> <meta name="Info 4" content=""> </head> <body lang=EN-US link=blue vlink=blue style='tab-interval:.5in'> <div class=WordSection1> <h2 align=center style='margin-top:0in;text-align:center'><span style='mso-fareast-font-family:"Times New Roman"'>Spring 2012 -- Methods of Applied Mathematics II <br> <br> </span><span style='font-size:13.5pt;mso-fareast-font-family:"Times New Roman"'>M 383D (56070) and CAM 385D (64425) </span><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></h2> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Instructor: </span><u><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"; color:blue'><a href="http://www.ma.utexas.edu/users/gamba/">Prof. Irene M. Gamba</a></span></u><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><br> Office: RLM 10.166, Phone: 471-7150<br> E-Mail: </span><u><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"; color:blue'><a href="mailto:gamba@math.utexas.edu">gamba@math.utexas.edu</a></span></u><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><br> Office hours: by appointment<o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Teaching Assistant: <span class=SpellE>Chenglong</span> Zhang<o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Office hours: Tuesdays from 5:00 to 6:30pm  RLM 12.166<o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Office<span class=GramE>: RLM 13.152 </span> <br> E-Mail: <a href="mailto:chenglong@ices.utexas.edu">chenglong@ices.utexas.edu</a><o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span><br> <br> Meeting Hours: M  W 3:00-4:30pm RLM 11.176 <br> <br> </span><span class=GramE><span style='font-size:12.0pt;mso-fareast-font-family: "Times New Roman";font-weight:normal'> <span style='font-size: 13.0pt;mso-fareast-font-family:"Times New Roman";font-weight:normal'> A weekly discussion session will be held at RLM 12.166 from 5:00pm to 6:30pm.</span><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'> <o:p></o:p></span></h3> <p class=western style='margin-bottom:0in;margin-bottom:.0001pt;mso-outline-level: 4'><b><u><span style='color:blue'><a href="http://www.ma.utexas.edu/users/gamba/S12/HW-AMMII-S12">Homework Problems and due dates: </a></span></u><o:p></o:p></b></p> <h3 style='margin-top:0in;margin-right:0in;margin-bottom:13.0pt;margin-left: 0in'><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Textbook</span><span style='font-size:12.0pt;mso-fareast-font-family:"Times New Roman"'>: </span><span class=SpellE><span style='font-size:12.0pt;mso-fareast-font-family:"Times New Roman"; font-weight:normal'>Arbogast</span></span><span style='font-size:12.0pt; mso-fareast-font-family:"Times New Roman";font-weight:normal'>-Bona notes (2011 version).</span><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"; font-weight:normal'> </span><span style='font-size:13.0pt;mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Homework, Exams, and Grades</span><span style='font-size:12.0pt;mso-fareast-font-family: "Times New Roman";font-weight:normal'>: Homework will be assigned regularly. Students are encouraged to work in groups; however, each student must write up his or her own work. Two mid-term exams will be given. The final exam will be comprehensive and given during finals week (location and schedule TBA). The final grade will be based on the homework and the three exams. </span><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h3> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Course Description:<o:p></o:p></span></h3> <p class=western style='margin-bottom:0in;margin-bottom:.0001pt;mso-outline-level: 4'><b>This is the second semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CAM. <o:p></o:p></b></p> <h3><span class=GramE><span style='font-size:13.0pt;mso-fareast-font-family: "Times New Roman"'>Semester I.</span></span><span style='font-size:13.0pt; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h3> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l5 level1 lfo1;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>Preliminaries (topology and <span class=SpellE>Lebesgue</span> integration) <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l5 level1 lfo1;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><span class=SpellE><b>Banach</b></span><b> Spaces <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l5 level1 lfo1;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>Hilbert Spaces <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l5 level1 lfo1;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>Spectral Theory <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:.5in;text-indent:-.25in;mso-outline-level:4;mso-list:l5 level1 lfo1; tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family: "Times New Roman"'><span style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>Distributions<o:p></o:p></b></p> <h3><span class=GramE><span style='font-size:13.0pt;mso-fareast-font-family: "Times New Roman"'>Semester II.</span></span><span style='font-size:13.0pt; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h3> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l4 level1 lfo2;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>The Fourier Transform (3 weeks) <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The Schwartz space and tempered distributions. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The Fourier <span class=GramE>transform</span>. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The <span class=SpellE>Plancherel</span> Theorem. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Convolutions. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:1.0in;text-indent:-.25in;mso-outline-level:4;mso-list:l4 level2 lfo2; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family: "Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Fundamental solutions of PDE's. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l1 level1 lfo3;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><span class=SpellE><b>Sobolev</b></span><b> spaces (3 weeks) <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l1 level2 lfo3;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Basic Definitions. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l1 level2 lfo3;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><b>Extention</b></span><b> Theorems. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l1 level2 lfo3;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Imbedding Theorems. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:1.0in;text-indent:-.25in;mso-outline-level:4;mso-list:l1 level2 lfo3; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family: "Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The Trace Theorem. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l0 level1 lfo4;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>8.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><span class=SpellE><b>Variational</b></span><b> Boundary Value Problems (BVP) (3 weeks) <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l0 level2 lfo4;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Weak solutions to elliptic BVP's. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l0 level2 lfo4;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><b>Variational</b></span><b> forms. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l0 level2 lfo4;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Lax-<span class=SpellE>Milgram</span> Theorem. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l0 level2 lfo4;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><b>Galerkin</b></span><b> approximations. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:1.0in;text-indent:-.25in;mso-outline-level:4;mso-list:l0 level2 lfo4; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family: "Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Green's functions. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level1 lfo5;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>9.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>Differential Calculus in <span class=SpellE>Banach</span> Spaces and Calculus of Variations (4 weeks) <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The <span class=SpellE>Frechet</span> derivatives. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The Chain Rule and Mean Value Theorems. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Higher order derivatives and Taylor's Theorem. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><b>Banach's</b></span><b> Contraction Mapping Theorem and Newton's Method. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Inverse and Implicit Function Theorems, and applications to nonlinear functional equations. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><b>Extremum</b></span><b> problems, Lagrange multipliers, and problems with constraints. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:1.0in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l3 level2 lfo5;tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>The Euler-Lagrange equation. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:1.0in;text-indent:-.25in;mso-outline-level:4;mso-list:l3 level2 lfo5; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family: "Courier New";mso-bidi-font-weight:bold'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><b>Applications to classical mechanics and geometry. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:.5in;text-indent:-.25in;mso-outline-level:4;mso-list:l2 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family: "Times New Roman"'><span style='mso-list:Ignore'>10.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>Some Applications (if time permits) <o:p></o:p></b></p> <h3><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'>Some references:<o:p></o:p></span></h3> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>R. A. Adams, <span class=SpellE><i>Sobolev</i></span><i> Spaces</i>, Academic Press, 1975. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>J.-P. <span class=SpellE>Aubin</span>, Applied Functional Analysis, Wiley, 1979. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>C. <span class=SpellE>Caratheodory</span>, <i>Calculus of Variations and Partial Differential Equations of the First Order</i>, 1982. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>E.W. Cheney and H.A. Koch, <i>Notes on Applied Mathematics</i>, Department of Mathematics, University of Texas at Austin. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>L. <span class=SpellE>Debnath</span> and P. <span class=SpellE>Mikusinski</span>, <i>Introduction to Hilbert Spaces with Applications</i>, Academic Press, 1990. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>G.B. <span class=SpellE>Folland</span>, <i>Introduction to Partial Differential Equations</i>, Princeton, 1976. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>I.M. <span class=SpellE>Gelfand</span> and S.V. <span class=SpellE>Fomin</span>, <i>Calculus of Variations</i>, Prentice-Hall, 1963; reprinted by Dover Publications. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>8.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>J. <span class=SpellE>Jost</span> and X. Li-<span class=SpellE>Jost</span>, Calculus of Variations, Cambridge, 1998, <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>9.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]><b>A.N. <span class=SpellE>Kolmogorov</span> and S.V. <span class=SpellE>Fomin</span>, <i>Introductory Real Analysis,</i> Dover Publications, 1970 <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>10.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>E. <span class=SpellE>Kreyszig</span>, <i>Introductory Functional Analysis with Applications</i>, Wiley, 1978. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>11.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>E.H. <span class=SpellE>Lieb</span> and M. Loss, <i>Analysis</i>, AMS, 1997. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>12.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>J.T. <span class=SpellE>Oden</span> &amp; L.F. <span class=SpellE>Demkowicz</span>, <i>Applied Functional Analysis</i>, CRC Press, 1996. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>13.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>F.W.J. <span class=SpellE>Olver</span>, <span class=SpellE><i>Asymptotics</i></span><i> and Special Functions</i>, Academic Press, 1974. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>14.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>M. Reed &amp; B. Simon, <i>Methods of Modern Physics, Vol. 1, Functional analysis</i>. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>15.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>W. <span class=SpellE>Rudin</span>, <i>Functional Analysis</i>, McGraw Hill, 1991. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>16.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>W. <span class=SpellE>Rudin</span>, <i>Real and Complex Analysis</i>, 3<i>rd</i> Ed., McGraw Hill, 1987. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>17.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>H. Sagan, Introduction to the Calculus of Variations, Dover, 1969. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>18.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>R.E. Showalter, <i>Hilbert Space Methods for Partial Differential Equations</i>, available at World Wide Web address </b><b><u><span style='color:blue'><a href="http://ejde.math.txstate.edu/mono-toc.html">http://ejde.math.txstate.edu//mono-toc.html.</a></span></u> <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in; margin-left:.5in;margin-bottom:.0001pt;text-indent:-.25in;mso-outline-level: 4;mso-list:l6 level1 lfo7;tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-list:Ignore'>19.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>E. Stein and G. Weiss, <i>Introduction to Fourier <span class=GramE>Analysis</span> on Euclidean Spaces</i>, Princeton, 1971. <o:p></o:p></b></p> <p class=western style='margin-top:5.0pt;margin-right:0in;margin-bottom:13.7pt; margin-left:.5in;text-indent:-.25in;mso-outline-level:4;mso-list:l6 level1 lfo7; tab-stops:list .5in'><![if !supportLists]><b><span style='mso-fareast-font-family: "Times New Roman"'><span style='mso-list:Ignore'>20.<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span></b><![endif]><b>K. <span class=SpellE>Yosida</span>, <i>Functional Analysis</i>, Springer-<span class=SpellE>Verlag</span>, 1980. <o:p></o:p></b></p> <p class=western style='margin-bottom:0in;margin-bottom:.0001pt;mso-outline-level: 4'><b style='mso-bidi-font-weight:normal'><span style='mso-no-proof:yes'><img border=0 width=4 height=4 id="_x0000_i1025" src="applied-math-methods-S12_html_4921951b.gif"></span></b><b><o:p></o:p></b></p> <p class=western style='margin-bottom:12.0pt;mso-outline-level:4'><b><o:p>&nbsp;</o:p></b></p> <h3><span style='font-size:12.0pt;mso-fareast-font-family:"Times New Roman"'>The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.</span><span style='font-size:13.0pt;mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h3> </div> </body> </html>