A COURSE OUTLINE (September 2001)
|
|
|
- Corey McLaughlin
- 5 years ago
- Views:
Transcription
1 A COURSE OUTLINE (September 2001) 1 Topic I. Line integrals: weeks 1.1 Parametric curves Review of parametrization for lines and circles. Paths and curves. Differentiation and integration of vector-valued functions. Velocity and arc length. Integration of a scalar function on a curve against arc length. Centroid and center of mass. 1. Adams: (third edition) 3.4, 7.1, 7.2, 8.3; (second edition) 1.4, 5.2, Edwards & Penney : ( fifth edition) 10.4, 12.4, 12.5, 12.6, Larson, Hostetler, Edwards: (fifth edition) 10.2, 11.5, Marsden & Tromba: (fourth edition) 1.1, 4.1, 4.2, 7.1; (third edition) 1.1, 3.1, 3.2, Salas & Hille: (seventh edition) 12.5, 12.6, 13.3, 13.4; (eighth edition) 12.6, , 7. Stewart: (fourth edition) 11.1, 13.5, 14.1, 14.2, 14.3, Thomas & Finney: (ninth edition) 10.5, 11.1, 11.3, Vector fields The concept of a vector field. Gradient fields. Examples. Work and flux. Conservative vector fields. 1. Adams: (third edition) 8.1, 8.2; (second edition) 6.1, Edwards & Penney : ( fifth edition) 15.1, 15.2, Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth edition) 4.3; (third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) 17.1, 17.2 (not much on work or flux). 8. Thomas & Finney: (ninth edition)
2 1.3 Line integral I Definition of line integral. Various formulations and notations. Basic properties and importance of orientation, in particular: C 1+C 2 = C 1 + C 2, C =. Independence of parametrization. C 1. Adams: (third edition) 8.4; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) Line integral II Review of the chain rule. C ϕ d r = ϕ(p 1) ϕ(p 0 ). When is a vector field conservative? The condition P y = Q x. The disturbing example of F = ( y x x 2 +y, 2 x 2 +y ) and arctan( y 2 x ). Independence of path and conservativity of the vector field. Conservation of energy. 1. Adams:(third edition) 8.4, 8.2 Example 5; (second edition) 6.4, 6.2 Example Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) 7.2, 8.3 Exercise Salas & Hille: (seventh edition) 17.2, Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) 14.2, Line integral III The interpretation of C F d r as the flux across C of the vector field obtained from F by clockwise rotation through 90 degrees. 1. Adams:(second edition) 7.3 Example 6, 7.4; (third edition) 9.3 Example 6, Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) 4. Marsden & Tromba: (fourth edition) p. 475; (third edition) p Salas & Hille: (seventh edition) 7. Stewart: (fourth edition) (see 17.5 pp for related material.) 8. Thomas & Finney: (ninth edition)
3 2 Topic II. Green s Theorem and double integrals: 2 weeks 2.1 Review Double integrals. Regions of type I,II. 1. Adams: (third edition) 6.1; (second edition) Edwards & Penney : ( fifth edition) 14.1, 14.2, 14.3, 3. Larson, Hostetler, Edwards: (fifth edition) 14.1, Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) 16.2, Stewart: (fourth edition) Thomas & Finney: (ninth edition) Green s theorem Statement. Examples. Proof for a good region (i.e., simultaneously type I & II). Combination of regions. Singularities. Independence of path, simple connectivity, and P y = Q x. Computation of area. 1. Adams: (third edition) 9.4 ; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) The flux form of Green s theorem The line source log(x 2 + y 2 ). Applications: harmonic functions; the equation of continuity; the 2-dimensional heat equation. 1. Adams: (third edition) 9.3 Example 6, 9.5, Exercises 4.4 and 9.5; (second edition) 7.3 Example Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth edition) p. 475, 8.5 ; (third edition) p Salas & Hille: (seventh edition) Stewart: (fourth edition) see 17.5 (pp ). 8. Thomas & Finney: (ninth edition)
4 3 Topic III. Differentiation revisited: Lagrange multipliers, implicit functions: weeks 3.1 Jacobians Jacobian matrix ( derivative) and Jacobian. Matrix chain rule (statement). 1. Adams: (third edition) 4.6 (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) 13.5, Marsden & Tromba: (fourth & third edition) 6.3. Also (fourth edition) p. 119 and (third edition) p. 125, p. 134, p Salas & Hille: (seventh edition) Stewart: (fourth edition) 8. Thomas & Finney: (ninth edition) 12.5, Change of variables in double integrals In particular, formulas for linear transformations and polar coordinates. 1. Adams: (third edition) 6.4; (second edition) Edwards & Penney : ( fifth edition) 14.4, Larson, Hostetler, Edwards: (fifth edition) 10.4, Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Stewart: (fourth edition) Thomas & Finney: (ninth edition) Critical points and the Lagrange multiplier A brief review of critical points. The Lagrange multiplier: a curve in a level surface F = c is orthogonal to F ; Maximizing G( r(t)) when the curve lies on F = c. 1. Adams: (third edition) 5.3; (second edition) Edwards & Penney : ( fifth edition) 13.5, Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth edition) 3.3; (third edition) Salas & Hille: (seventh edition) Stewart: (fourth edition) Thomas & Finney: (ninth edition)
5 3.4 Implicit differentiation Recall of Gaussian elimination. The simple case of F (x, y) = c. A matrix formulation of the general case. 1. Adams: (third edition) 4.8; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth edition) 3.5; (third edition) Salas & Hille: (seventh edition) Stewart: (fourth edition) 15.5 (pp ). 7. Schaum s outline: 8. Thomas & Finney: (ninth edition) CLASS TEST 4 Topic IV. Surface integrals and Stokes theorem: 3 weeks 4.1 Vector analysis The basic differential operators of grad (i.e., ), curl (i.e., ), and div (i.e., ). The basic relation: functions vector fields vector fields functions. Important fact: two in succession always gives zero. Inverse square law and ( 1 r ) = Adams: (third edition) 9.2; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth edition) 4.4; (third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) 17.1 (p. 1079), Thomas & Finney: (ninth edition) 12.7, 14.4, 14.7, Surface integration Parametric surfaces. The standard case of the graph of a function z = f(x, y). Integration of a scalar function. The flux of a vector field across a parametric surface. The case of a cylinder over a plane curve (recall flux across a curve). Example of the sphere. Orientable surfaces. 1. Adams: (third edition) 8.6; (second edition) Edwards & Penney : ( fifth edition) 14.8, Larson, Hostetler, Edwards: (fifth edition) 15.5, Marsden & Tromba: (fourth & third edition) 7.3, 7.4, 7.5,
6 5. Salas & Hille: (seventh edition) 17.6, Chapter Stewart: (fourth edition) 17.6, Thomas & Finney: (ninth edition) 14.5, Flux and circulation: Stokes theorem Statement. Examples. Gluing patches together: Stokes theorem for an orientable surface. Simple computations over the surface of a cube (say) or a sphere. Independence of surface for S curl G d S. 1. Adams: (third edition) 9.4; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) 15.6, Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) Stokes theorem 2 Outline of proof for a parametric surface: reduction to Green s theorem. Green s theorem is a special case. Conservative fields in R Adams: (third edition) 9.4; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) Topic V. Triple integrals and the divergence (or Gauss) theorem: 3 weeks. 5.1 Review of triple integrals Reduction to double integrals for the regions of type I,II, and III. The formulas for change of variables. Cylindrical and spherical coordinates. 1. Adams: (third edition) 6.5, 6.6; (second edition) 4.4, Edwards & Penney : ( fifth edition) 14.6, 14.7,
7 3. Larson, Hostetler, Edwards: (fifth edition) 14.6, Marsden & Tromba: (fourth & third edition) 6.1, Salas & Hille: (seventh edition) 16.7, 16.8, Chapter Stewart: (fourth edition) 16.7, Thomas & Finney: (ninth edition) 13.4, 13.6, Flux of a vector field across the boundary of a solid Example of the sphere and curl F using Stokes theorem and change of orientation. The general result: the divergence theorem (statement). 1. Adams: (third edition) 9.3; (second edition) Edwards & Penney : ( fifth edition) 15.5, Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) Thomas & Finney: (ninth edition) The divergence theorem Harmonic functions. Green s first identity. The flux of 1 r across a sphere. The flux of 1 r across the boundaries of other solids. 1. Adams: (third edition) 9.3; (second edition) Edwards & Penney : ( fifth edition) Larson, Hostetler, Edwards: (fifth edition) Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Stewart: (fourth edition) Thomas & Finney: (ninth edition)
8 5.4 Applications (Optional: up to instructor) Circulation. Heat flow and the continuity equation. Magnetostatics and electric flux (see Adams). Maxwell s equations. 1. Adams: (third edition) 9.5 ; (second edition) Edwards & Penney : ( fifth edition) 3. Larson, Hostetler, Edwards: (fifth edition) 4. Marsden & Tromba: (fourth & third edition) Salas & Hille: (seventh edition) Chapter Stewart: (fourth edition) 8. Thomas & Finney: (ninth edition)
AP Physics B Summer Homework (Show work)
#1 NAME: AP Physics B Summer Homework (Show work) #2 Fill in the radian conversion of each angle and the trigonometric value at each angle on the chart. Degree 0 o 30 o 45 o 60 o 90 o 180 o 270 o 360 o
Bézier Curves and Splines
CS-C3100 Computer Graphics Bézier Curves and Splines Majority of slides from Frédo Durand vectorportal.com CS-C3100 Fall 2016 Lehtinen Before We Begin Anything on your mind concerning Assignment 1? Linux
Minimum Mean-Square Error (MMSE) and Linear MMSE (LMMSE) Estimation
Minimum Mean-Square Error (MMSE) and Linear MMSE (LMMSE) Estimation Outline: MMSE estimation, Linear MMSE (LMMSE) estimation, Geometric formulation of LMMSE estimation and orthogonality principle. Reading:
Math 4. Unit 1: Conic Sections Lesson 1.1: What Is a Conic Section?
Unit 1: Conic Sections Lesson 1.1: What Is a Conic Section? 1.1.1: Study - What is a Conic Section? Duration: 50 min 1.1.2: Quiz - What is a Conic Section? Duration: 25 min / 18 Lesson 1.2: Geometry of
Algebra I: A Fresh Approach. By Christy Walters
Algebra I: A Fresh Approach By Christy Walters 2005 A+ Education Services All rights reserved. No part of this publication may be reproduced, distributed, stored in a retrieval system, or transmitted,
Windcube FCR measurements
Windcube FCR measurements Principles, performance and recommendations for use of the Flow Complexity Recognition (FCR) algorithm for the Windcube ground-based Lidar Summary: As with any remote sensor,
Besides the reported poor performance of the candidates there were a number of mistakes observed on the assessment tool itself outlined as follows:
MATHEMATICS (309/1) REPORT The 2013 Mathematics (309/1) paper was of average standard. The paper covered a wide range of the syllabus. It was neither gender bias nor culture bias. It did not have language
Algebra I: A Fresh Approach. By Christy Walters
Algebra I: A Fresh Approach By Christy Walters 2016 A+ Education Services All rights reserved. No part of this publication may be reproduced, distributed, stored in a retrieval system, or transmitted,
Mathematics. Leaving Certificate Examination Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30
2016. M29 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2016 Mathematics Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30 300 marks Examination number
Lesson 22: Average Rate of Change
Student Outcomes Students know how to compute the average rate of change in the height of water level when water is poured into a conical container at a constant rate. MP.1 Lesson Notes This lesson focuses
The purpose of this experiment is to find this acceleration for a puck moving on an inclined air table.
Experiment : Motion in an Inclined Plane PURPOSE The purpose of this experiment is to find this acceleration for a puck moving on an inclined air table. GENERAL In Experiment-1 you were concerned with
End of Chapter Exercises
End of Chapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. While on an airplane, you take a drink from your water
Numerical Simulation of Wave Loads on Static Offshore Structures
Numerical Simulation of Wave Loads on Static Offshore Structures Hrvoje Jasak, Inno Gatin, Vuko Vukčević Wikki Ltd, United Kingdom Faculty of Mechanical Engineering and Naval Architecture University of
B.U.G. Newsletter. Full Steam Ahead! September Dr. Brown
B.U.G. Newsletter September 2014 THIS NEWSLETTER IS A SERVICE THAT WAS FUNDED BY "NO CHILD LEFT BEHIND" TITLE II PART A HIGHER EDUCATION IMPROVING TEACHER QUALITY HIGHER EDUCATION GRANT ADMINISTERED THROUGH
Using Markov Chains to Analyze a Volleyball Rally
1 Introduction Using Markov Chains to Analyze a Volleyball Rally Spencer Best Carthage College sbest@carthage.edu November 3, 212 Abstract We examine a volleyball rally between two volleyball teams. Using
LOW PRESSURE EFFUSION OF GASES adapted by Luke Hanley and Mike Trenary
ADH 1/7/014 LOW PRESSURE EFFUSION OF GASES adapted by Luke Hanley and Mike Trenary This experiment will introduce you to the kinetic properties of low-pressure gases. You will make observations on the
CHEMISTRY 102D ASSIGNMENTS. WEEK 1 (August 23-27) Introduction, Classification of Matter, Significant Figures, Dimensional Analysis
CHEMISTRY 102D ASSIGNMENTS WEEK 1 (August 23-27) Topics: Introduction, Classification of Matter, Significant Figures, Dimensional Analysis Reading: Zumdahl*, Chapter 1.1-1.3, 1.5-1.7, 1.9, Appendix A1.1
Math 116 Practice for Exam 1
Math 116 Practice for Exam 1 Generated September 3, 215 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 4 questions. Note that the problems are not of equal difficulty, so you may want to
Vector Synoptic Maps at NSO. Luca Bertello
Oulu (Finland) workshop, January 23-26, 217 Vector Synoptic Maps at NSO Luca Bertello National Solar Observatory Boulder, Colorado Limitations Affecting Quality of the Vector Maps Oulu (Finland) workshop,
For example, the velocity at t = 10 is given by the gradient of the curve at t = 10, 10 t
R15 INTERPRET THE GRADIENT AT A POINT ON A CURVE AS THE INSTANTANEOUS RATE OF CHANGE; APPLY THE CONCEPTS OF AVERAGE AND INSTANTANEOUS RATE OF CHANGE (GRADIENTS OF CHORDS AND TANGENTS) IN NUMERICAL, ALGEBRAIC
COMPLEX NUMBERS. Powers of j. Consider the quadratic equation; It has no solutions in the real number system since. j 1 ie. j 2 1
COMPLEX NUMBERS Consider the quadratic equation; x 0 It has no solutions in the real number system since x or x j j ie. j Similarly x 6 0 gives x 6 or x 6 j Powers of j j j j j. j j j j.j jj j j 5 j j
Boyle s Law: Pressure-Volume. Relationship in Gases
Boyle s Law: Pressure-Volume Relationship in Gases The primary objective of this experiment is to determine the relationship between the pressure and volume of a confined gas. The gas we use will be air,
End of Chapter Exercises
End of Chapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. While on an airplane, you take a drink from your water
REAL LIFE GRAPHS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Real Life Graphs Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier REAL LIFE GRAPHS Version: 2.1 Date: 20-10-2015 Mathematics Revision Guides
Chapter 12 Practice Test
Chapter 12 Practice Test 1. Which of the following is not one of the conditions that must be satisfied in order to perform inference about the slope of a least-squares regression line? (a) For each value
Where are you right now? How fast are you moving? To answer these questions precisely, you
4.1 Position, Speed, and Velocity Where are you right now? How fast are you moving? To answer these questions precisely, you need to use the concepts of position, speed, and velocity. These ideas apply
Jasmin Smajic 1, Christian Hafner 2, Jürg Leuthold 2, March 16, 2015 Introduction to Finite Element Method (FEM) Part 1 (2-D FEM)
Jasmin Smajic 1, Christian Hafner 2, Jürg Leuthold 2, March 16, 2015 Introduction to Finite Element Method (FEM) Part 1 (2-D FEM) 1 HSR - University of Applied Sciences of Eastern Switzerland Institute
LOW PRESSURE EFFUSION OF GASES revised by Igor Bolotin 03/05/12
LOW PRESSURE EFFUSION OF GASES revised by Igor Bolotin 03/05/ This experiment will introduce you to the kinetic properties of low-pressure gases. You will make observations on the rates with which selected
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York DEPARTMENT: Mathematics COURSE: MAT 1475/MA 475 TITLE: Calculus I DESCRIPTION: Topics include functions, limits, differentiation, tangent
FC-CIV HIDRCANA: Channel Hydraulics Flow Mechanics Review Fluid Statics
FC-CIV HIDRCANA: Channel Hydraulics Flow Mechanics Review Fluid Statics Civil Engineering Program, San Ignacio de Loyola University Objective Calculate the forces exerted by a fluid at rest on plane or
Mesoscale Atmospheric Systems. Upper-level fronts. 13 and 20 March 2018 Heini Wernli. 13 March 2018 H. Wernli 1
Mesoscale Atmospheric Systems Upper-level fronts 13 and 20 March 2018 Heini Wernli 13 March 2018 H. Wernli 1 Upper-level fronts Formation of fronts is favored by presence of quasi-horizontal boundaries:
Waves. harmonic wave wave equation one dimensional wave equation principle of wave fronts plane waves law of reflection
Waves Vocabulary mechanical wave pulse continuous periodic wave amplitude wavelength period frequency wave velocity phase transverse wave longitudinal wave intensity displacement wave number phase velocity
Week 8, Lesson 1 1. Warm up 2. ICA Scavanger Hunt 3. Notes Arithmetic Series
CAN WE ADD AN ARITHMETIC SEQUENCE? Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Week 8, Lesson 1 1. Warm up 2. ICA
The Beginner's Guide to Mathematica Version 4
The Beginner's Guide to Mathematica Version 4 Jerry Glynn MathWare Urbana, IL Theodore Gray Wolfram Research, Inc Urbana, IL. m CAMBRIDGE UNIVERSITY PRESS v Preface Parti: The Basics Chapter 1: What do
The Diver Returns Circular Functions, Vector Components, and Complex Numbers
Co n t e n t s The Diver Returns Circular Functions, Vector Components, and Complex Numbers Back to the Circus 3 The Circus Act 4 As the Ferris Wheel Turns 6 Graphing the Ferris Wheel 8 Distance with Changing
Air in Motion. Anthes, Chapter 4, pp
Air in Motion Anthes, Chapter 4, pp. 71-85 Newtonʼs Laws of Motion 1. Law of Inertia. A body at rest remains at rest, a body in motion remains in motion, unless acted upon by an outside force. 2. F = ma.
ANDRA Benchmark Workshop (Paris-Orly, April 2007)
ANDRA Benchmark Workshop (Paris-Orly, April 27) Javeri (GRS, Köln, Germany): 2-Dim. Case 1B Code TOUGH2/EOS7 developed by Pruess et al, Lawrence Berkeley National Laboratories, California, USA (199), Since
ZIN Technologies PHi Engineering Support. PHi-RPT CFD Analysis of Large Bubble Mixing. June 26, 2006
ZIN Technologies PHi Engineering Support PHi-RPT-0002 CFD Analysis of Large Bubble Mixing Proprietary ZIN Technologies, Inc. For nearly five decades, ZIN Technologies has provided integrated products and
International Journal of Technical Research and Applications e-issn: , Volume 4, Issue 3 (May-June, 2016), PP.
DESIGN AND ANALYSIS OF FEED CHECK VALVE AS CONTROL VALVE USING CFD SOFTWARE R.Nikhil M.Tech Student Industrial & Production Engineering National Institute of Engineering Mysuru, Karnataka, India -570008
Sea and Land Breezes METR 4433, Mesoscale Meteorology Spring 2006 (some of the material in this section came from ZMAG)
Sea and Land Breezes METR 4433, Mesoscale Meteorology Spring 2006 (some of the material in this section came from ZMAG) 1 Definitions: The sea breeze is a local, thermally direct circulation arising from
CT4510: Computer Graphics. Transformation BOCHANG MOON
CT4510: Computer Graphics Transformation BOCHANG MOON 2D Translation Transformations such as rotation and scale can be represented using a matrix M ee. gg., MM = SSSS xx = mm 11 xx + mm 12 yy yy = mm 21
Jeddah Knowledge International School. Science Revision Pack Answer Key Quarter 3 Grade 10
Jeddah Knowledge International School Science Revision Pack Answer Key 2016-2017 Quarter 3 Grade 10 Name: Section: ANSWER KEY- SCIENCE GRADE 10, QUARTER 3 1 1. What are the units for mass? A Kilograms
Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope
There Is Only One Line Passing Through a Given Point with a Given Slope Classwork Opening Exercise Examine each of the graphs and their equations. Identify the coordinates of the point where the line intersects
INSTRUMENT INSTRUMENTAL ERROR (of full scale) INSTRUMENTAL RESOLUTION. Tutorial simulation. Tutorial simulation
Lab 1 Standing Waves on a String Learning Goals: To distinguish between traveling and standing waves To recognize how the wavelength of a standing wave is measured To recognize the necessary conditions
m r [B] mn n> 20r [C] mn n< 8r [D] mn n< 20r
![m r [B] mn n> 20r [C] mn n< 8r [D] mn n< 20r](/thumbs/81/83311830.jpg "m r [B] mn n> 20r [C] mn n< 8r [D] mn n< 20r")
Solve. 5. x = 7 0 6 7 5 4 5 [D] 7. x + x 9x 8 = 0, ±,, 9 [D] ±,. x+ 9 = x 0 5 5, 7 [D] 7 4. Solve the equation. = b 8 [D] 7 5. Solve: 5 x x =0 6, 4 4, 4 4, 5 [D], 4 4 6. Solve: 6+ n < 4 Solve. m r mn n>
Chapter 16. Waves-I Types of Waves
Chapter 16 Waves-I 16.2 Types of Waves 1. Mechanical waves. These waves have two central features: They are governed by Newton s laws, and they can exist only within a material medium, such as water, air,
Module 3: Hydrostatic forces on submerged bodies Lecture 7: Calculation of horizontal component, buoyancy. Forces on submerged bodies (continued)
Forces on submerged bodies (continued) Buoyancy file:///d /Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture7/7_1.htm[5/9/2012 3:05:26 PM] Forces on submerged bodies (continued)
Kinematics of Vorticity
Kinematics of Vorticity Vorticity Ω Ω= V 2 circumferentially averaged angular velocity of the fluid particles Sum of rotation rates of perpendicular fluid lines Non-zero vorticity doesn t imply spin.ω=0.
NHL & NHLPA Future Goals Program Hockey Scholar TM
Curriculum Guide NHL & NHLPA Future Goals Program Hockey Scholar TM Your local NHL team has made it all the way to the Stanley Cup Final and now you just need to win 4 games to bring home the cup! You
The Intrinsic Value of a Batted Ball Technical Details
The Intrinsic Value of a Batted Ball Technical Details Glenn Healey, EECS Department University of California, Irvine, CA 9617 Given a set of observed batted balls and their outcomes, we develop a method
Grade 7 & 8 Math Circles Pair-O -Dice: The Game April 2/3, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Instructions Set Up Grade 7 & 8 Math Circles Pair-O -Dice: The Game April 2/3, 2013 Get into a group of 3 to 5 players. Each group needs 2 different dice
EQ: GPE.7 How do I find the perimeter and area of polygons?
EQ: GPE.7 How do I find the perimeter and area of polygons? Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Week 14,
Introduction to Matlab for Engineers
Introduction to Matlab for Engineers Instructor: Anh Thai Nhan Math 111, Ohlone Introduction to Matlab for Engineers Ohlone, Fremont 1/23 Reading materials Chapters 1, 2, and 3, Moore s textbook Introduction
Vectors. Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction?
Physics R Scalar: Vector: Vectors Date: Examples of scalars and vectors: Scalars Vectors Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction? Magnitude: Direction:
CHEMISTRY - CLUTCH CH.5 - GASES.
!! www.clutchprep.com CONCEPT: UNITS OF PRESSURE Pressure is defined as the force exerted per unit of surface area. Pressure = Force Area The SI unit for Pressure is the, which has the units of. The SI
Atmospheric Circulation. Recall Vertical Circulation
Today s topics: Atmospheric circulation: generation of wind patterns on a rotating Earth Seasonal patterns of climate: Monsoons and Sea Breezes Tropical Cyclones: Hurricanes and typhoons Atmospheric Circulation
CS 4649/7649 Robot Intelligence: Planning
CS 4649/7649 Robot Intelligence: Planning Differential Kinematics, Probabilistic Roadmaps Sungmoon Joo School of Interactive Computing College of Computing Georgia Institute of Technology S. Joo (sungmoon.joo@cc.gatech.edu)
Unit 3 Lesson 2 Properties of Waves. Copyright Houghton Mifflin Harcourt Publishing Company
Florida Benchmarks SC.7.P.10.3 Recognize that light waves, sound waves, and other waves move at different speeds in different materials. MA.6.A.3.6 Construct and analyze tables, graphs, and equations to
BIOMECHANICAL MOVEMENT
SECTION PART 5 5 CHAPTER 12 13 CHAPTER 12: Biomechanical movement Practice questions - text book pages 169-172 1) For which of the following is the athlete s centre of mass most likely to lie outside of
Flow and Mixing in the Liquid between Bubbles
Excerpt from the Proceedings of the COMSOL Conference 2009 Boston Flow and Mixing in the Liquid between Bubbles Bruce A. Finlayson, Professor Emeritus of Chemical Engineering Department of Chemical Engineering,
MODELLING OF WATER FLOW ON SMALL VESSEL S DECK
Monika Warmowska, Jan Jankowski, Polski Rejestr Statków S.A., al. gen. Józefa Hallera 126, Poland, Gdańsk, 80-416 MODELLING OF WATER FLOW ON SMALL VESSEL S DECK Summary Green water moving on deck of small
Tension Cracks. Topics Covered. Tension crack boundaries Tension crack depth Query slice data Thrust line Sensitivity analysis.
Tension Cracks 16-1 Tension Cracks In slope stability analyses with cohesive soils, tension forces may be observed in the upper part of the slope. In general, soils cannot support tension so the results
Boyle s Law: Pressure-Volume Relationship in Gases
Boyle s Law: Pressure-Volume Relationship in Gases Computer 6 The primary objective of this experiment is to determine the relationship between the pressure and volume of a confined gas. The gas we use
OPTIMIZATION OF A WAVE CANCELLATION MULTIHULL SHIP USING CFD TOOLS
OPTIMIZATION OF A AVE CANCELLATION MULTIHULL SHIP USING CFD TOOLS C. Yang, R. Löhner and O. Soto School of Computational Sciences, George Mason University Fairfax VA 030-4444, USA ABSTRACT A simple CFD
COURSE NUMBER: ME 321 Fluid Mechanics I Fluid statics. Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET
COURSE NUMBER: ME 321 Fluid Mechanics I Fluid statics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Fluid statics Fluid statics is the study of fluids in
SHOT ON GOAL. Name: Football scoring a goal and trigonometry Ian Edwards Luther College Teachers Teaching with Technology
SHOT ON GOAL Name: Football scoring a goal and trigonometry 2006 Ian Edwards Luther College Teachers Teaching with Technology Shot on Goal Trigonometry page 2 THE TASKS You are an assistant coach with
Chapter 15 Wave Motion. Copyright 2009 Pearson Education, Inc.
Chapter 15 Wave Motion 15-1 Characteristics of Wave Motion All types of traveling waves transport energy. Study of a single wave pulse shows that it is begun with a vibration and is transmitted through
Mixture Models & EM. Nicholas Ruozzi University of Texas at Dallas. based on the slides of Vibhav Gogate
Mixture Models & EM Nicholas Ruozzi University of Texas at Dallas based on the slides of Vibhav Gogate Previously We looked at -means and hierarchical clustering as mechanisms for unsupervised learning
EE 364B: Wind Farm Layout Optimization via Sequential Convex Programming
EE 364B: Wind Farm Layout Optimization via Sequential Convex Programming Jinkyoo Park 1 Introduction In a wind farm, the wakes formed by upstream wind turbines decrease the power outputs of downstream
WAVE MECHANICS FOR OCEAN ENGINEERING
Elsevier Oceanography Series, 64 WAVE MECHANICS FOR OCEAN ENGINEERING P. Boccotti Faculty of Engineering University of Reggio-Calabria Feo di Vito 1-89060 Reggio-Calabria Italy 2000 ELSEVIER Amsterdam
Exercise 3. Power Versus Wind Speed EXERCISE OBJECTIVE DISCUSSION OUTLINE. Air density DISCUSSION
Exercise 3 Power Versus Wind Speed EXERCISE OBJECTIVE When you have completed this exercise, you will know how to calculate the power contained in the wind, and how wind power varies with wind speed. You
Ozobot Bit Classroom Application: Boyle s Law Simulation
OZO AP P EAM TR T S BO RO VE D Ozobot Bit Classroom Application: Boyle s Law Simulation Created by Richard Born Associate Professor Emeritus Northern Illinois University richb@rborn.org Topics Chemistry,
Math 1070 Sample Final Exam Spring 2016
University of Connecticut Department of Mathematics Spring 2016 Name: Instructor Name: Section: Read This First! This is a closed notes, closed book exam. You cannot receive aid on this exam from anyone
Lecture 1: Knot Theory
Math 7H Professor: Padraic Bartlett Lecture 1: Knot Theory Week 1 UCSB 015 1 Introduction Outside of mathematics, knots are ways to loop a single piece of string around itself: In mathematics, we mean
77.1 Apply the Pythagorean Theorem
Right Triangles and Trigonometry 77.1 Apply the Pythagorean Theorem 7.2 Use the Converse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 Apply the Tangent Ratio
Bubble Dynamics in a Vibrating Liquid. By: James Wymer, Jaggar Henzerling, Aaron Kilgallon, Michael McIntire, Mohammed Ghallab
Bubble Dynamics in a Vibrating Liquid By: James Wymer, Jaggar Henzerling, Aaron Kilgallon, Michael McIntire, Mohammed Ghallab Background In the early 1960 s a series of rocket failures plagued leading
Parsimonious Linear Fingerprinting for Time Series
Parsimonious Linear Fingerprinting for Time Series Lei Li, B. Aditya Prakash, Christos Faloutsos School of Computer Science Carnegie Mellon University VLDB 2010 1 L. Li, 2010 VLDB2010, 36 th International
1. Five more than three times a number x. 3. Seventeen percent of a number x.
Pre-Calculus: 1.7.1 Mastery Practice Name: Date Period Write a mathematical expression for the quantity described verbally: 1. Five more than three times a number x. 3. Seventeen percent of a number x.
CHAPTER 8 WIND AND WEATHER MULTIPLE CHOICE QUESTIONS
CHAPTER 8 WIND AND WEATHER MULTIPLE CHOICE QUESTIONS 1. is the movement of air measured relative to the Earth's surface. a. Gravity b. The pressure gradient force c. The Coriolis Effect d. The centripetal
Mark Scheme (Results) Summer 2009
Mark Scheme (Results) Summer 2009 GCSE GCSE Mathematics (Linear) - 1380 Paper: 1380_2F 2 NOTES ON MARKING PRINCIPLES 1 Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional
Incompressible Flow over Airfoils
Road map for Chap. 4 Incompressible Flow over Airfoils Aerodynamics 2015 fall - 1 - < 4.1 Introduction > Incompressible Flow over Airfoils Incompressible flow over airfoils Prandtl (20C 초 ) Airfoil (2D)
Are Advanced Wind Flow Models More Accurate? A Test of Four Models
Are Advanced Wind Flow Models More Accurate? A Test of Four Models Philippe Beaucage, PhD Senior Research Scientist Michael C. Brower, PhD Chief Technical Officer Brazil Wind Power Conference 2012 Albany
DESIGN AND ANALYSIS OF ALGORITHMS (DAA 2017)
DESIGN AND ANALYSIS OF ALGORITHMS (DAA 2017) Veli Mäkinen 12/05/2017 1 COURSE STRUCTURE 7 weeks: video lecture -> demo lecture -> study group -> exercise Video lecture: Overview, main concepts, algorithm
A Resource for Free-standing Mathematics Units. Graph showing Pressure plotted against Volume for a sample of air in a Boyle s law experiment
Data An experiment to investigate Boyle s law is carried out with the apparatus shown in the diagram. The pressure and volume of the gas (air) trapped in the closed end can be varied by raising or lowering
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York DEPARTMENT: Mathematics COURSE: MAT 1475 TITLE: DESCRIPTION: Calculus I Topics include functions, limits, differentiation, and tangent
Lecture 10. Support Vector Machines (cont.)
Lecture 10. Support Vector Machines (cont.) COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Andrey Kan Copyright: University of Melbourne This lecture Soft margin SVM Intuition and problem
CHAPTER 10: LINEAR KINEMATICS OF HUMAN MOVEMENT
CHAPTER 10: LINEAR KINEMATICS OF HUMAN MOVEMENT 1. Vector mechanics apply to which of the following? A. displacement B. velocity C. speed D. both displacement and velocity 2. If velocity is constant, then
4.3 Oblique Shocks and Expansions Fans: The Supercritical Marine Layer.
4.3 Oblique Shocks and Expansions Fans: The Supercritical Marine Layer. The marine layer is a relatively dense and well-mixed layer of moist air that lies above the sea surface and is often capped by a
Special Topics: Data Science
Special Topics: Data Science L Linear Methods for Prediction Dr. Vidhyasaharan Sethu School of Electrical Engineering & Telecommunications University of New South Wales Sydney, Australia V. Sethu 1 Topics
Calibration and validation of an air quality finite element model around an electric power plant in Gran Canaria island
Calibration and validation of an air quality finite element model around an electric power plant in Gran Canaria island J. Ramírez (1), A. Oliver (2), J.E. González (3), R. Montenegro (1) (1) Instituto
Ranking teams in partially-disjoint tournaments
Ranking teams in partially-disjoint tournaments Alex Choy Mentor: Chris Jones September 16, 2013 1 Introduction Throughout sports, whether it is professional or collegiate sports, teams are ranked. In
Sea State Analysis. Topics. Module 7 Sea State Analysis 2/22/2016. CE A676 Coastal Engineering Orson P. Smith, PE, Ph.D.
Sea State Analysis Module 7 Orson P. Smith, PE, Ph.D. Professor Emeritus Module 7 Sea State Analysis Topics Wave height distribution Wave energy spectra Wind wave generation Directional spectra Hindcasting
+ R. gr T. This equation is solved by the quadratic formula, the solution, as shown in the Holton text notes given as part of the class lecture notes:
Homework #4 Key: Physical explanations 1.The way water drains down a sink, counterclockwise or clockwise, is independent of which hemisphere you are in. A draining sink is an example of vortex in cyclostrophic
Transmission loss (TL) can be predicted, to a very rough degree, solely on the basis of a few factors. These factors are range, and frequency.
Sonar Propagation By virtue of the fact that the speed that acoustic waves travel at depends on the properties of the medium (i.e. sea water), the propagation of sonar will be complicated. So complicated
LINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION:
LINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 1 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION: How far? Describing change in linear or angular position Distance (Scalar
Traveling on a Rotating Sphere
Traveling on a Rotating Sphere Table of Contents Page Click the titles below to jump through the lesson 2 Spin-offs of a Rotating Sphere 3 What Do You Know? 3 Heated Fluid Circulation 4 Where Do The Trade
Title: 4-Way-Stop Wait-Time Prediction Group members (1): David Held
Title: 4-Way-Stop Wait-Time Prediction Group members (1): David Held As part of my research in Sebastian Thrun's autonomous driving team, my goal is to predict the wait-time for a car at a 4-way intersection.
Lecture The Oceans
Lecture 22 -- The Oceans ATMOSPHERE CIRCULATION AND WINDS Coriolis effect Prevailing winds and vertical circulation Zones of pressure, evap. & ppt. Factors modifying global winds -- Differential heating
Anemometry. Anemometry. Wind Conventions and Characteristics. Anemometry. Wind Variability. Anemometry. Function of an anemometer:
Anemometry Anemometry Function of an anemometer: Measure some or all of the components of the wind vector In homogeneous terrain, vertical component is small express wind as -D horizontal vector For some
The Bruins I.C.E. School
The Bruins I.C.E. School Lesson 1: Area and Volume of a Cylinder Lesson 2: Using and Applying the Pythagorean Theorem Lesson 3: Investigating patterns of association in bivariate data Lesson 4: Investigating