If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the RightHandRule.


 Leon McDonald
 1 years ago
 Views:
Transcription
1 Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions. If it is possible to select one normal at each nonboundary point in such a way that the chosen normal varies continuously on the whole of, then the surface is said to be orientable, or twosided, and the selection of the normal gives an orientation to and thus makes an oriented surface. In such case, there are two possible orientations. Some surfaces are not orientable. For example, the Möbus band; it is onesided. Induced Orientation If is an oriented surface bounded by a curve C, then the orientation of induces an orientation for C, based on the RightHandRule. Flux Integrals Let F be a vector field and ˆn be the unit normal to the oriented surface, the flux integral over is F ˆn ds. 1
2 This integral gives the net flux through. The field strength (i.e. F ) can be measured as the amount of flux per unit area perpendicular to the local direction of F. When is the graph of a function f with continuous partials on a region R in the xy plane that is composed of vertically or horizontally simple regions, and its orientation is chosen to be directed upward (i.e. the k component of the unit normal is positive), then F ˆndS = [ F 1 f x F 2 f y + F 3 ]da R for F = F 1 i + F 2 j + F 3 k. Note Another way to remember this is F ˆndS = F NdA R where N = f x i + f y j + k is the upward normal to with the z component equal to 1. 2
3 Example Suppose is the part of the paraboloid z = 1 x 2 y 2 that lies above the xy plane and is oriented by the unit normal directed upward. Assume that the velocity of a fluid is v = x i + y j + 2z k. Determine the flow rate (volume/time) through. 2π Flux integrals can be defined for a surface composed of several oriented surfaces 1, 2, n, as F ˆndS = 1 F ˆndS + + n F ˆndS Example Let be the unit sphere x 2 + y 2 + z 2 = 1, oriented with the unit normal directed outward, and let F(x,y,z) = z k. Find F ˆndS. 4π/3 3
4 The Divergence Theorem Definition A solid region D is called a simple solid region if D is the solid region between the graphs of two functions f(x, y) and g(x, y) on a simple region R in the xy plane and if D has the corresponding properties with respect to the xz plane and the yz plane. Theorem Let D be a simple solid region whose boundary surface is oriented by the normal ˆn directed outward from D, and let F be a vector field whose component functions have continuous partial derivatives on D. Then F ˆndS = FdV. D pf: Let F = F 1 i + F 2 j + F 3 k, then F 1 F 1 i ˆndS = x dv F 2 j ˆndS = D D F 3 k ˆndS = D F 2 y dv F 3 z dv 4
5 Note The interpretation of flux through the boundary of D. F ˆndS is net outward Example Let D be the region bounded by the xy plane and the hemisphere x 2 + y 2 + z 2 = 4 with z 0, and let F(x,y,z) = 3x 4 i + 4xy 3 j + 4xz 3 k. Evaluate F ˆndS, where is the boundary of D. 0 Note If the domain of integration R is symmetry with respect to certain thing and the integrand is antisymmetric w.r.t. the same thing, then the integral is 0. For example, if f( x,y,z) = f(x,y,z), the function f is antisymmetric w.r.t. the yzplane. 5
6 The theorem can be applied to a not so simple solid region by considering the division of the region into a number of simple solid regions. Example Let F = q r 2 ˆr. Show that for any suface enclosing the origin, F ˆndS = 4πq. 6
7 Stokes Theorem Theorem Let be an oriented surface with normal ˆn (unit vector) and finite surface area. Assume that is bounded by a closed, piecewise smooth curve C whose orientation is induced by. Let F be a continuous vector field defined on, and assume that the component functions of F have continuous partial derivatives at each nonboundary point of. Then F d r = ( F) ˆndS C Example Let C be the intersection of the paraboloid z = x 2 +y 2 and the plane z = y, and give C a counterclockwise direction as viewed from the positive z axis. Evaluate 2xydx + x 2 dy + z 2 dz. C What about (2xy y)dx + x 2 dy + z 2 dz? C 0; π/4 7
8 Example Verify Stokes Theorem for F = 3y i xz j+ yz 2 k where is the surface of the paraboloid 2z = x 2 +y 2 bounded by z = 2, and C is its boundary. F=(z 2 +x) i (z+3) k 20π Problem Set 12 Proof of Stokes theorem ( F) N = ( y F 3 z F 2 )( f x )+( z F 1 x F 3 )( f y ) + ( x F 2 y F 1 ) = ( x F 2 +f x z F 2 +f y x F 3 ) ( y F 1 +f y z F 1 +f x y F 3 ) = x (F 2 (x,y,f(x,y)) + f y F 3 (x,y,f(x,y)) y (F 1 (x,y,f(x,y)) + f x F 3 (x,y,f(x,y)) Now apply Green s theorem on the xyplane as following. ( x (F 2 + f y F 3 ) ) y (F 1 + f x F 3 ) da R 8
9 = (F 1 + f x F 3 )dx + (F 2 + f y F 3 )dy R r(t)=(x(t),y(t),z(t)) where z(t)=f(x(t),y(t)) So that r (t)=(x (t),y (t),fxx (t)+fy y (t)) and F r (t)=f 1 x +F 2 y +(F 3 fxx +F 3 fyy ) = F 1 dx + F 2 dy + F 3 dz. Therefore, ( F) ˆndS = ( F) NdA = F d r. R 9
( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy
. Let F = ( yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F F = yz x B. curl F + ( 3xz) y + ( 9xy) = 0 + 0 + 0 = 0 z F = i j k x y z yz 3xz 9xy = 6xi 8yj+ 2zk C. div curl F F = 6x x + ( 8y)
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationSolutions  Homework sections 17.717.9
olutions  Homework sections 7.77.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points  and therefore the triangle between them  are on the plane x +
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More information52. The Del Operator: Divergence and Curl
52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy and xzplanes, etc. are For example, z = f(x, y), x =
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationChapter 17. Review. 1. Vector Fields (Section 17.1)
hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your
More informationMULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.
More information16.5: CURL AND DIVERGENCE
16.5: URL AN IVERGENE KIAM HEONG KWA 1. url Let F = P i + Qj + Rk be a vector field on a solid region R 3. If all firstorder partial derivatives of P, Q, and R exist, then the curl of F on is the vector
More informationF = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)
MATH 255 FINAL NAME: Instructions: You must include all the steps in your derivations/answers. Reduce answers as much as possible, but use exact arithmetic. Write neatly, please, and show all steps. Scientists
More informationFINAL EXAM SOLUTIONS Math 21a, Spring 03
INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic
More informationSolutions to Practice Final Exam
Math V22. Calculus IV, ection, pring 27 olutions to Practice Final Exam Problem Consider the integral x 2 2x dy dx + 2x dy dx x 2 x (a) ketch the region of integration. olution: ee Figure. y (2, ) y =
More information4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin
4.2. LINE INTEGRALS 1 4.2 Line Integrals MATH 294 FALL 1982 FINAL # 7 294FA82FQ7.tex 4.2.1 Consider the curve given parametrically by x = cos t t ; y = sin 2 2 ; z = t a) Determine the work done by the
More informationLine and surface integrals: Solutions
hapter 5 Line and surface integrals: olutions Example 5.1 Find the work done by the force F(x, y) x 2 i xyj in moving a particle along the curve which runs from (1, ) to (, 1) along the unit circle and
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More information4B. Line Integrals in the Plane
4. Line Integrals in the Plane 4A. Plane Vector Fields 4A1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationDivergence and Curl (3B)
Divergence and Curl (3B Divergence Curl Green's Theorem Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationVector Calculus Solutions to Sample Final Examination #1
Vector alculus s to Sample Final Examination #1 1. Let f(x, y) e xy sin(x + y). (a) In what direction, starting at (,π/), is f changing the fastest? (b) In what directions starting at (,π/) is f changing
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationFigure 1: u2 (x,y) D u 1 (x,y) yρ(x, y, z)dv. xρ(x, y, z)dv. M yz = E
Contiune on.7 Triple Integrals Figure 1: [ ] u2 (x,y) f(x, y, z)dv = f(x, y, z)dz da u 1 (x,y) Applications of Triple Integrals Let be a solid region with a density function ρ(x, y, z). Volume: V () =
More informationAP CALCULUS AB 2008 SCORING GUIDELINES
AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line
More informationProblem 1 (25 points)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam Three Solutions Problem 1 (25 points) Question 1 (5 points) Consider two circular rings of radius R, each perpendicular
More information(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
More informationChapter Eighteen. Stokes. Here also the socalled del operator = i + j + k x y z. i j k. x y z p q r
hapter Eighteen tokes 181 tokes's Theorem Let F:D R be a nice vector function If the curl of F is defined by F( x, = p( x, i + q( x, j + r( x, k, r q p r q p curlf = i + j + k y z z x x y Here also the
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationH.Calculating Normal Vectors
Appendix H H.Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use the OpenGL lighting facility, which is described in Chapter
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311  Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationMATH 1231 S2 2010: Calculus. Section 1: Functions of severable variables.
MATH 1231 S2 2010: Calculus For use in Dr Chris Tisdell s lectures Section 1: Functions of severable variables. Created and compiled by Chris Tisdell S1: Motivation S2: Function of two variables S3: Visualising
More informationAP Calculus BC 2004 FreeResponse Questions
AP Calculus BC 004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationvector calculus 2 Learning outcomes
29 ontents vector calculus 2 1. Line integrals involving vectors 2. Surface and volume integrals 3. Integral vector theorems Learning outcomes In this Workbook you will learn how to integrate functions
More informationChapter 3. Gauss s Law
3 3 30 Chapter 3 Gauss s Law 3.1 Electric Flux... 32 3.2 Gauss s Law (see also Gauss s Law Simulation in Section 3.10)... 34 Example 3.1: Infinitely Long Rod of Uniform Charge Density... 39 Example
More informationMATH 243312631. http://www.math.uh.edu/ ajajoo/math2433
MATH 243312631 Aarti Jajoo ajajoo@math.uh.edu Office : PGH 606 Lecture : MoWeFre 1011am in SR 116 Office hours : MW 11:3012:30pm and BY APPOINTMENT http://www.math.uh.edu/ ajajoo/math2433 A. Jajoo,
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information1 The basic equations of fluid dynamics
1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationAP Calculus AB 2004 FreeResponse Questions
AP Calculus AB 2004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationAP Calculus BC 2004 Scoring Guidelines
AP Calculus BC Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from
More informationQ28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P
Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P r + v r A. points in the same direction as v. B. points from point
More information27.3. Introduction. Prerequisites. Learning Outcomes
olume Integrals 27. Introduction In the previous two sections, surface integrals (or double integrals) were introduced i.e. functions were integrated with respect to one variable and then with respect
More informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #4, MATH 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #4, MATH 253 1. Prove that the following differential equations are satisfied by the given functions: (a) 2 u + 2 u 2 y + 2 u 2 z =0,whereu 2 =(x2 + y 2 + z 2 ) 1/2. (b)
More informationMath 53 Worksheet Solutions Minmax and Lagrange
Math 5 Worksheet Solutions Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial
More informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More information3. Double Integrals 3A. Double Integrals in Rectangular Coordinates
3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationM PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM
68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationReview B: Coordinate Systems
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B2 B.1.1 Infinitesimal Line Element... B4 B.1.2 Infinitesimal Area Element...
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationRecitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere
Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x
More informationChapter 4. Electrostatic Fields in Matter
Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the
More informationCurves and Surfaces. Goals. How do we draw surfaces? How do we specify a surface? How do we approximate a surface?
Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.110.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons
More informationMath 2443, Section 16.3
Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the
More informationPhysics 2220 Module 09 Homework
Physics 2220 Module 09 Homework 01. A potential difference of 0.050 V is developed across the 10cmlong wire of the figure as it moves though a magnetic field perpendicular to the page. What are the strength
More informationMath 21a Curl and Divergence Spring, 2009. 1 Define the operator (pronounced del ) by. = i
Math 21a url and ivergence Spring, 29 1 efine the operator (pronounced del by = i j y k z Notice that the gradient f (or also grad f is just applied to f (a We define the divergence of a vector field F,
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vectorvalued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationParametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) YanBin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationThis file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.
Jason Hill Math24 3 WeBWorK assignment number Exameview is due : 5/9/212 at 5:am MDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy other information.
More information299ReviewProblemSolutions.nb 1. Review Problems. Final Exam: Wednesday, 12/16/2009 1:30PM. Mathematica 6.0 Initializations
99ReviewProblemSolutions.nb Review Problems Final Exam: Wednesday, /6/009 :30PM Mathematica 6.0 Initializations R.) Put x@td = t  and y@td = t . Sketch on the axes below the curve traced out by 8x@tD,
More informationThe Fourth International DERIVETI92/89 Conference Liverpool, U.K., 1215 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVETI9/89 Conference Liverpool, U.K., 5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue NotreDame Ouest Montréal
More information3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important
More informationAP Calculus BC 2006 FreeResponse Questions
AP Calculus BC 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationCONSERVATION LAWS. See Figures 2 and 1.
CONSERVATION LAWS 1. Multivariable calculus 1.1. Divergence theorem (of Gauss). This states that the volume integral in of the divergence of the vectorvalued function F is equal to the total flux of F
More informationSolutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 2537, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 537, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More information4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complexvalued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complexvalued function of a real variable Consider a complex valued function f(t) of a real variable
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPHY 301: Mathematical Methods I Curvilinear Coordinate System (1012 Lectures)
PHY 301: Mathematical Methods I Curvilinear Coordinate System (1012 Lectures) Dr. Alok Kumar Department of Physical Sciences IISER, Bhopal Abstract The Curvilinear coordinates are the common name of
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More information