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Course: W 1003, Spring 2011
School: Columbia
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Algebra Name: College Quiz 2 January 30, 2007 x+1 1. Solve the nonlinear equality 2 < x3 . Express the solution using interval notation and graph the solution set. 2. A father is four times as old as his daughter. In 6 years, he will three times as as old she is. How old is the daughter now? 3. Two cyclists, 90 mi apart, start riding toward each other at the same time both at constant speed. One cylces twice as fast as the other. If they meet 2 h later, at what speed did each cyclist travel? 1
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Columbia - W - 1003
Name:College Algebra Quiz 3January 30, 20071. Find the slope and y-intercept of the line 3x 5y + 30 = 0 and draw its graph.2. Consider the function3x,if x &lt; 0;f (x) = x + 1,if 0 x 2;2(x 2) , if x &gt; 2;a) Evaluate f (5), f (0), f (1), f (2), f (5
Columbia - W - 1003
Name:College Algebra Quiz 4February 13, 2007x21. Given the functions f (x) = x and g (x) = x+2 nd f g and g g and their domains.2. Sketch the graph of the function f (x) = 3 2 x 1, not by plotting points, but bystarting with the graph of a standard
Columbia - W - 1003
Name:College Algebra Quiz 5February 20, 20071. Given f (x) = x + 1,(a) Sketch the graph of f .(b) Use the graph of f to sketch the graph of f 1 .(c) Find f 1 . What is its domain?2. Factor the polynomial x6 2x3 + 1 and use the factored form to nd t
Columbia - W - 1003
Name:College Algebra Quiz 6March 22, 2007Write all answers in exact form; ie in terms of e, ln, log, etc.1. Solve the equation e2x ex 6 = 02. Solve the equation ln(x 1) + ln(x + 2) = 1.3. Find the annual percentage yield for an investment that earns
Columbia - W - 1003
Name:College Algebra Quiz 7April 24, 20071. Find all possible triangles that satisfy the given conditions b = 73, c = 82, B = 58 .cos sin csc =.2. Verify the identity1 sin cos cot 3. Given sec x = 3 , 270 &lt; x &lt; 360 , nd sin 2x and cos x .221
Columbia - W - 1003
Precalc Review Problems for First ExamThis practice exam is longer than the actual exam will bethe actual exam will consist ofroughly 100 points of similar problems to those listed below.1.(9 points) Simplify the expression and eliminate any negative e
Columbia - W - 1003
Solutions to Precalc Review Problems for First ExamPlease let me know if you cant get any of these answersthey may not be correct!d7x3 y 15, (c)c6z32. (a) (7 2y )(7 + 2y ), (b) (a 1)(a + 1)(b 2)(b + 2), (c) (3x + 5)(x 3 2)(x2 + 3 2x + 3 4)x2 + 1x
Columbia - W - 1003
Precalc Review Problems for Second ExamThis practice exam is longer than the actual exam will bethe actual exam will consist ofroughly 100 points of similar problems to those listed below. Please leave all answers in exact forminvolving e, ln, log, etc.
Columbia - W - 1003
Solutions to Precalc Review Problems for Second ExamPlease let me know if you cant get any of these answersthey may not be correct!1. Ill go over these in or before classI still havent gured out how to post images.122. x3 + x2 + 3x + 5 +.x25x 23.
Columbia - W - 1003
Name:College Algebra PretestJanuary 16, 20071. Solve the equation 12 2(x 14) = 6(x + 5).2. Solve the system of linear equations4x + 3y = 45x 7y = 383.4.5.6.SolveSolveSolveSolvethethethethequadratic equation (3x 12)2 = 144.quadratic equ
Columbia - W - 1003
Name:College Algebra Quiz 1January 23, 200721. Simplify the expression and eliminate any negative exponent(s)x2 x 6 x3 + x2.x2 +2x x2 2x32. Simplify3. Solve the quadratic equation x2 + 8x + 12 = 0.1(2x3 ) (3x4 )( x3 ) 4.
Columbia - S - 2010D
Linear Algebra S2010D Sec.1, Summer 2006Exam 1Name:June 9, 2006Do all problems, in any order.Show your work. An answer alone may not receive full credit.No notes, texts, or calculators may be used on this exam.ProblemPossible PointsPoints Earned
Columbia - S - 2010D
Linear Algebra Extra Credit ProblemsMay 31, 2006The following problems may be done for extra credit on the rst exam. Normally I encourage everyone to work together on homeworks but since this isfor extra credit Id prefer if you work on your own. This a
Columbia - S - 2010D
Name:Linear Algebra Extra Credit for Final ExamJune 13, 2006The following problems may be done for extra credit on the nal exam. More will be addedas time goes on. These problems will be due on Friday June 23 at 5PM. Same policy as lasttimeplease han
Columbia - S - 2010D
Linear Algebra S2010D Sec.1, Summer 2006Final ExamName:June 27, 2006Do all problems, in any order.Show your work. An answer alone may not receive full credit.No notes, texts, or calculators may be used on this exam.Problem Possible PointsPoints Ea
Columbia - S - 2010D
Linear Algebra Homework 1 Selected SolutionsAny mistakes are the sole responsibility of the author-T.Peters1.1.6gRepresent as a matrix and row reduce:132321 2122122 45 20120 00 10120 005120 100120 100100 100321251321
Columbia - S - 2010D
Linear Algebra Homework 2 Selected SolutionsAny mistakes are the sole resposibility of the authorT.Peters1312.3.1c A = 2 1 1 .2 2 1212111= (1 2) 3(2 + 2) + (4 + 2) = 3+13(i) det A = 1 2 22 12 1T212111 2 12 2 2 1311 311(ii) adj
Columbia - S - 2010D
Linear Algebra Homework 3 Selected SolutionsNote: Any mistakes are the sole responsibility of the authorT.Peters3.6.12 Let A be a 4 5 matrix. If a1 , a2 , a4 are linearly independent anda3 = a1 + 2a2 and a5 = 2a1 a2 + 3a4 determine RREF(A).Heres an ex
Columbia - S - 2010D
Name:Linear Algebra Quiz 4.2June 20, 20061.(20 points)4 211(a) Compute the characteristic polynomial of A.(b) Compute the eigenvalues of A.(c) Find a basis for R2 consisting of eigenvalues of A.(d) Find an invertible matrix P and a diagonal matri
Columbia - S - 2010D
Ok check it out. I think this is right:d1 (ej )(ek , el ) = ek ej (el ) + ej (ek )el + ej (ek el )iiii(using Einstein notation)m= ek il ej + ik ej el + ej (Tkl em )iinn= il Tkj en + ik Tjl en + Tkl ej(without Einstein)jjinn= n=j (il Tk
Columbia - S - 2010D
Name:Linear Algebra Quiz 2June 5, 2006 4111.(10 points) Is the set cfw_ 2 , 3 , 6 linearly independent or dependent? Ex221plain.2.(10 points) Consider the set S of all 3 3the formab0 d00upper triangular matrices, i.e., matrices ofcefS
Columbia - S - 2010D
Name:Linear Algebra Quiz 3June 13, 20061.2132, and = cfw_,.1111(a)(4 points) Find the transition matrices from -coordinates to standard coordinates, coordinates to standard coordinates, and -coordinates to -coordinates, i.e., nd [id]STD ,
Columbia - S - 2010D
Name:Linear Algebra Quiz 4June 19, 20061.(20 points)220Given A = 2 2 0004(a) Compute the characteristic polynomial of A.(b) Compute the eigenvalues of A.(c) Find a basis for R3 consisting of eigenvalues of A.(d) Find an invertible matrix P and a
Columbia - S - 2010D
Linear Algebra Review Problems for First ExamNote: This practice test is longer than the actual examThe actual exam will total 100 pts,with roughly the same values as assigned here. Hopefully actually this isnt too much review.1.(12 points) Find all so
Columbia - S - 2010D
Linear Algebra Review Problems for Final ExamNote: The nal will consist of 100 points with roughly equivalent point assignments.1.(20 points)101Given A = 0 1 0111(a) Compute the characteristic polynomial of A.(b) Find the eigenvalues of A.(c) Find
Columbia - S - 2010D
Linear Algebra Review Problems for First ExamSolutions1.(12 points) Find all solutions to the following system of linear equations 3x1 2 12 1 1 y = 1 0z75 2Solution: Represent as a matrix and row reduce:1 2 12 1 175 21 2 10 3 30 9 91 2 1
Columbia - PHYSICS - 1401
F IRST EXAM - P HYSICS C1401XP ROFESSOR SCIULLISEPTEMBER 29, 2003ATTENTION:1. Read this FRONT page while waiting for the exam to begin.Do NOT open either this exam book or the blue book untilyou have been given permission by the proctor.2. The proc
Columbia - PHYSICS - 1401
Columbia - PHYSICS - 1401
Columbia - PHYSICS - 1401
Columbia - PHYSICS - 1401
Physics 1401(1) From To No. Acc Frac 0 0 0 0.0% 1 2 0 0 0.0% 3 4 0 0 0.0% 5 6 0 0 0.0% 7 8 0 0 0.0% 9 10 0 0 0.0% 11 12 1 1 1.1% 13 14 1 2 2.2% 15 16 0 2 2.2% 17 18 3 5 5.6% 19 20 1 6 6.7% 21 22 1 7 7.9% 23 24 1 8 9.0% 25 26 1 9 10.1% 27 28 0 9 10.1% 29 3
Columbia - PHYSICS - 1401
From 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99To 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
Columbia - PHYSICS - 1401
From 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196To No. Cum. 0 0 5 0 0 10 0 0 15 0 0 20 0 0 25 0 0 30 0 0 35 1 1 40 2 3 45 0 3 50 0 3 55 1 4 60 0 4 65 1 5 70 2 7
Columbia - APPH - 4200
Experiments in Fluids 34 (2003) 566571DOI 10.1007/s00348-002-0582-9Blasius: A life in research and educationW.H. Hager566Abstract The Blasius boundary layer solution is a basicfeature of uid mechanics, and the rst application ofPrandtls boundary la
Columbia - APPH - 4200
Blasius Boundary LayerAPPH 4200 Physics of Fluids Columbia UniversitySimilarity EquationIn[1]:=eq = f@hD D@f@hD, 8h, 2&lt;D 2 + D@f@hD, 8h, 3&lt;D 0 1 f@hD f @hD + fH3L @hD 0Out[1]=2In[2]:= In[3]:= Out[3]=hBig = 25.0; bc = 8f '@hBigD 1, f@0D 0, f '@0D 0
Columbia - APPH - 4200
67tIl0'!L-vi -. f (CO_TIl-vfl)' Trz t A1 Lt I' S F,1 I: . S vi fC,l R E L t2 ':S C? r: .l-r, AJ c. l) .f~ LrIt (~i) C (D&quot; 0) ,_ THle ~r /r-c L I&quot;'t) L ') r-r tATIlIt7Tifc-\)ll;- -.LEfJT (~(-tt)&quot; y(-rJ) =- (0,1 0) 0HUlrlL ( := ( &quot; r7 4- r( E . 5 n. C)
Columbia - APPH - 4200
APPH 4200Physics of FluidsIntroductionSeptember 6, 20111Info Course website:http://www.apam.columbia.edu/courses/apph4200x/ Instructor:Prof. Mike Mauel, mauel@columbia.edu TA:Ken Zhao &lt;kz2179@columbia.edu&gt;Email us with questions/comments/help
Columbia - APPH - 4200
APPH 4200Physics of FluidsCartesian Tensors (Ch. 2)September 8, 2011 Last Lecture: Review Example Fluid Flow Problems Geometric Identities Vector Calculus1Hydrostatics of a Sphere2Examples: Fluid FlowContinuity (mass conservation)Euler equati
Columbia - APPH - 4200
APPH 4200Physics of FluidsCartesian Tensors (Ch. 2)September 8, 20111. Geometric Identities2. Vector Calculus1Scalars, Vectors, &amp;Tensors Scalars: mass density (), temperature(T), concentration (S), charge density (q) Vectors: ow (U), force (F),
Columbia - APPH - 4200
APPH 4200Physics of FluidsCartesian Tensors (Ch. 2)September 13, 20111. Geometric Identities2. Vector Calculus1Scalars, Vectors, &amp;Tensors Scalars: mass density (), temperature(T), concentration (S), charge density (q)Vectors: ow (U), force (F),
Columbia - APPH - 4200
APPH 4200Physics of FluidsKinematics: Describing Fluid Flow (Ch. 3)September 15, 20111. Quick Review2. Lagrangian (material) &amp; Eulerian (eld)3. Streamlines &amp; Pathlines4. Point Deformations: Stretching, Pinching, and Rotating5. Principal Axes (Cauc
Columbia - APPH - 4200
APPH 4200Physics of FluidsReview (Ch. 3) &amp; Fluid Equations of Motion (Ch. 4)September 20, 20111.! Chapter 3 (more notes)2. ! Vorticity and Circulation3.! Navier-Stokes Equation1Summary:Cauchy-Stokes Decomposition2Velocity Gradient Tensor3Mate
Columbia - APPH - 4200
APPH 4200Physics of FluidsFluid Equations of Motion (Ch. 4)September 22, 20111.!!Conservation of Mass2.!Navier-Stokes Equation (Force-Momentum)3.!Mechanical and Thermal Energy4.!Entropy5.!Some examples1Equations of Fluid Dynamics(Conserva
Columbia - APPH - 4200
APPH 4200Physics of FluidsFluid Equations of Motion (Ch. 4)September 27, 20111.!!Integral E.O.M.s Moving/Fixed Volumes2.!Internal Energy3.!Bernoullis Principle4.!Equations in co-rotating frames5.!Examples1Stokesian Fluid2NS Properties(i
Columbia - APPH - 4200
APPH 4200Physics of FluidsProblem Solving and Vorticity (Ch. 5)October 4, 20111.!!Quick Review2.!Vorticity3.!Kelvins Theorem4.!Examples1How to solve uid problems?How To S'L-vl rLUi, Pflc!IJLErl5?d)(A1 lJ:A-! /1- C:E ( AJ 0. oI '&quot;~-r 1 dic
Columbia - APPH - 4200
APPH 4200Physics of FluidsRotating Fluid FlowOctober 6, 20111.!!Hydrostatics of a Rotating Water Bucket (again)2.!Bath Tub Vortex3.!Ch. 5: Problem Solving1Key Denitions &amp; ConceptsUCylindrical coordinates (Appendix B)When = 0 (irrotational)
Columbia - APPH - 4200
APPH 4200Physics of FluidsPotential Flow (Ch. 6)2D Irrotational, Incompressible FlowOct 11, 20111.!!Irrotational Flow2.!Potential Flow in 2D (incompressible)3.!Complex Variables (quick review)4.!Examples1Irrotational Flow (Ch.6) Potential
Columbia - APPH - 4200
APPH 4200Physics of FluidsMore 2D Potential FlowChapter 61.!More examples from Chapter 6: 2D,Inviscid, Irrotational ow2.!Blasius Theorem (Lift and Drag)3.!(Easy) CFD: potential ow in 2D1C ~ :J~x.U. -.(/Il~J-\ rL~T'3-~)(~\JJ'J
Columbia - APPH - 4200
APPH 4200Physics of FluidsSimilarity (Ch. 8)October 19, 20101.!!Dimensional analysis1Dynamic Similarity(or nding the key dimensionless parameters) Wind tunnels (powerful method inexperimental uid mechanics!) Physical insights (what governsdyn
Columbia - APPH - 4200
APPH 4200Physics of FluidsLaminar Flow (Ch. 9)October 19, 20101.!!Dimensional analysis (again)2.!Laminar ow (when viscosity exceeds advection)3.!Examples: Poiseuilles steady ow through a pipe1Q~.r&quot;tJ~~IJ.\)lIICLAJ~(&quot;~XL~-t
Columbia - APPH - 4200
APPH 4200Physics of FluidsStokes Flow (Ch. 9)October 21, 20101.!!Viscous Decay of a Line Vortex2.!Vortex sheet3.!Stokes Solution for Viscous Flow around a Sphere1~~~..'f--i~-:~~y.~&quot;Q~.(.-.l\A0:l.t..~ci1:1 Jc:~~&quot;)
Columbia - APPH - 4200
APPH 4200Physics of FluidsReviewOctober 21, 20101.!!Review2.!Problems from old Midterms3.!Midterm 20081Review Introduction Tensors, vectors, symmetric andantisymmetric tensors, vector calculus,Gauss and Stokes Theorems, Streamlines, path
Columbia - APPH - 4200
APPH 4200Physics of FluidsWaves (Ch. 7)October 28, 20101.!!Waves2.!Linear gravity waves at a free surface1~)&quot;\&quot;.)~&quot;X'-~cli:'0~.~1~l-.Wave (Helmholtz) Equation'41 N~ 'J&quot;t' r&quot;VII(' l&quot;n~'-I'&quot;'P&quot;,tJltX'-4Ii,.~~
Columbia - APPH - 4200
APPH 4200Physics of FluidsInternal Gravity Waves (Ch. 7)November 4, 20101.!!Review of Surface Gravity Waves2.!Linear gravity waves within a continuouslystratied uid (Buoyancy!)1Ripples2~~to~J~~r.~.J\'IrtoJ,J\:~~~\uWave Equ
Columbia - APPH - 4200
APPH 4200Physics of FluidsMore Fluid Waves/Instabilities (Ch. 7)November 18, 20101.!!Tides2.!Jeans instability1TidesThirty days of tide heights at Bridgeport Connecticut U.S.A. as calculatedfrom the Harmonic Constituent data aligned with 0h Su
Columbia - APPH - 4200
APPH 4200Physics of FluidsMore Fluid Waves (Ch. 7)November 16, 20101.!!Phasors2.!Surface Tension3.!Capillary Waves4.!Rayleigh-Taylor Instability5.!Sound6.!Organ Pipes7.!HW 41e&quot;~&quot;J(l~,~\f~'l~ 'I~J.J!1~Jlq:~Q,.I.~.~
Columbia - APPH - 4200
APPH 4200Physics of FluidsFluid Instabilities (Ch. 12)November 23, 20101.!!Introduction2.!Bnard Thermal Instability1~.~v-J-&quot;'~lU'lQ~~t-~~~i'0't~.i.Equilibria can be Stable or Unstable,-.'* 'u. '0~~l&quot;'&quot; '- .t&quot;&quot;\U
Columbia - APPH - 4200
APPH 4200 Physics of FluidsFluid Instabilities (Ch. 12) November 24, 20091.! ! 2.!Kelvin Helmholtz Instability Centrifugal InstabilityTuesday, November 24, 20091Kelvin-Helmholtz InstabilityTuesday, November 24, 20092More BeautyTuesday, November
Columbia - APPH - 4200
APPH 4200 Physics of FluidsA Few More Fluid Instabilities (Ch. 12) Turbulence (Ch. 13) CD December 1, 2009i-.~ .~_~_~m_.-.D_~(c._~.-~.E.-~-l-s.-c:-~-,-/jQ-lL~a.lJ-l'- C-~,F.1.! !Viscous boundary layer and waves Stability of Parallel Flows2.!t-3.!.
Columbia - APPH - 4200
APPH 4200Physics of FluidsBoundary Layers (Ch. 10)December 7, 2010Answer: A pioneer in the mathematical development ofaerodynamics who conceived the idea of a uidboundary layer, considered by many as thegreatest single discovery in uid dynamics.Qu
Columbia - APPH - 4200
APPH 4200 Physics of FluidsReview December 10, 200923 Lectures &gt; 700 pages of text1Lecture 3 Velocity gradient tensor, strain, rotation2Deformation and Flow: Translation, Stretching, Pinching, and Rotating3Simple Comments about Velocity Gradient