F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)


 Mervin Booth
 1 years ago
 Views:
Transcription
1 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 and engineers uphold very high standards of ethics: the work you submit in this exam must be yours. Be prepared to explain your answers in person. Also, please document your take home final: keep all of your calculations (work not included in your final exam submission until you get a final grade in your class).. (25 pts) Consider the vector field F = y + z, x + z, x + y. Determine whether the vector field is conservative and if so, find an associated scalar potential ψ(x, y, z). F =. Hence, conservative and thus F = ψ(x, y, z). We write x ψ = y + z () y ψ = x + z (2) z ψ = x + y (3) Integrate () with respect to x: ψ = (y + z)x + h(y, z). Differentiate this expression and match to (2): ψ y = x + y h = x + z. Hence y h(y, z) = z, which we integrate with respect to y: h(y, z) = yz + f(z). Hence ψ + (y + z)x + yz + f(z). We differentiate this expression with respect to z and match to (3): z ψ = x + y + f (z) = x + y, hence f (z) = which we integrate in z to get f(z) = c, a constant. Hence ψ = (y + z)x + yz + c.
2 2. (5 pts) Assume g is a scalar function and f a vector function. Use the identity [gf] = fg + g f to compute the divergence of F = r. Here r = xî + yĵ + zˆk, and r 3 r = x 2 + y 2 + z 2, the magnitude of r. r = 3. and r 3 = 3r 5 r. Hence F = rr 3 + r 3 r. Hence F = 3r 3 3r 5 r r = 3r 3 3r 5 r 2 =. 2
3 3. ( pts) Find a vector normal to the surface z = x 2 /4 y 2 /6. It does not have to be a unit vector. let g = z x 2 /4 y 2 /6 =. A level set. Now, Note that this is not a unit vector. N = g. g = ˆk + 2 xî( x2 /4 y 2 /6) /2 + 8 yĵ( x2 /4 y 2 /6) /2 This can be written as If g = z + x 2 /4 y 2 /6 =, then N = ˆk + y [xî + 4z 4ĵ]. N = ˆk y [xî + 4z 4ĵ]. 3
4 4. (5 pts) Find the circulation C F dr, where C is the perimeter of the triangle given by the plane 2x + y + z = 2, in the first octant. The field F = 3xzĵ. Answer: you can do it as a line integral or as a surface integral: Here we use Stokes s Theorem: C F dr = S Fˆndσ. S is the triangle. As a surface integral: We compute curlf = 3( xˆk + zˆk). n = g/ g, where g = 2x + y + z 2 =, or ˆn = 6 (2î + ĵ + ˆk). The differential surface: Hence the integral to be solved is S Fˆndσ = dσ = dxdy ˆn ˆk = 6dxdy. 2 2x As a line integral: F dr = 3xzdy. Hence C F dr = y 2 zdy + 2 3( 4x y + 2)dydx =. 3x(2 y)dy + 3xzdy =, since in the first integral z =, in the second one x =, and in the third z = and the limits of integration are. 4
5 5. ( pts) Find the surface area 2πah, of the side of a cylinder of radius a and height h using either the projection method or the Jacobiantransformation method Answer: parametrize, Then r(u, v) = a cos uî + a sin uĵ + vˆk. r r =< a sin u, a cos u, > u v =<,, >. Hence r u r =< a cos u, a sin u, >. v Finally, < a cos u, a sin u, > = a, u 2π, and v h. Then side ds = h 2π adudv = 2πah. 5
6 6. (25 pts) Let F = P (x, y, z)î + Q(x, y, z)ĵ + R(x, y, z)ˆk be a vector field. Derive the conditions on F that guarantee that C F dr is path independent. Here, r = xî + yĵ + zˆk, the path starts at P and ends at P, two points in 3D space. Write out explicitly the line integral in scalar form. Match the exact differential dψ = ψ dx integral. dx + ψ dy ψ dy + dz dz to the integrand of the line Find conditions on F that associate it to the different components of the exact differential. Write a vector identity that encapsulates the conditions on F found above. Write the answer to the integral in terms of ψ. Answer: C F dr = C P dx+qdy +Rdz. Now if F =, then P dx+qdy +Rdz = dψ, where dψ = x ψdx + y ψdy + z ψdz. Hence C F dr = P P dψ = ψ(p ) ψ(p ). 6
7 7. (3 pts) Let ρ(x, y, z) = z 3 be the density of a solid. Compute its total mass V ρdv, where V is bounded by z =, z = 4+sin(2x)+cos(2y), and π x π and π y π. Hint: exploit the Divergence Theorem. Answer: we use V FdV = S F ds. We let F = 4 z4ˆk, so that F = z 3. So we then compute the surface integral S F ds, where S is the surface of the box. The integral S 4 z4ˆk ds = flat bottom surface 4 z4ˆk ds+ sides of box 4 z4ˆk ds+ wavy top 4 z4ˆk ds. Since z = on the flat bottom surface, the first integral on the right hand side is zero. Since the dot product of F and the normal to the sides of the box is zero, no contribution from that integral. Hence Hence, S wavy top 4 z4ˆk ds = wavy top 4 z4ˆk ds. 4 z4ˆk ds = square 4 z4ˆk ˆn dxdy ˆn ˆk. on the wavy top, F = 4 [4 + sin(2x) + cos(2y)]4ˆk. Hence square π π 4 [4 + sin(2x) + cos(2y)]4 dxdy = π π 4 [4 + sin(2x) + cos(2y)]4 dxdy. The integrand contributes just a few nonzero terms. To find these, first expand a = (4 + cos(2y)), then (a + exp(i2x) exp( i2x) 2i ) 4 will have only the terms a 4 + 3a 2 + 3/8 nonsinusoidal in x. Then we expand (4 + exp(2y)/2 + exp( 2y)/2) 4 and (4 + exp(2y)/2 + exp( 2y)/2) 2 and retain only the nonsinusoidal terms in y. We thus obtain π π π π 4 [4 + sin(2x) + cos(2y)]4 dxdy = π π π π dxdy = 47π2 /4. 7
8 8. (3 pts) Consider a rectangular region D of the x y plane that excludes the origin. Find p such that the circulation on the perimeter of the region D is zero, for where r 2 = x 2 + y 2. F = y3 r p î xy2 r p ĵ, Answer: Compute the curl and set it to zero. We obtain p = 4. That is, if p = 4. curlf = ˆk[ y 2 r p ( r 2 px 2 r 2 py 2 + 4] = ˆk[ y 2 r p ( r 2 r 2 p + 4) =, 8
9 9. (4 pts) Let R be a region in a plane that has a unit normal ˆn = a, b, c and boundary C. Let F = bz, cx, ay. (a) Show that F = ˆn. (b) Show that the area of R is given by C F dr. (c) Consider a curve C given by r = 5 sin t, 3 cos t, 2 sin t, for t 2π. Prove that C lies in a plane by showing that r dr dt is constant for all t. (d) Use part (b) to find the area of the region enclosed by C in part (c). Hint: find the unit normal consistent with the orientation of C. Answer: This can be done as a surface or line integral thanks to Stokes theorem. In part (a) all you need to do is to compute F = ˆn. In part (b), F dr = ( F) ds = ˆn ds = ds = R. C R We compute v = 5 cos t, 3 sin t, 2 cos t, and then the cross product r v = 78 2,, which is a constant vector, for all t. To find a unit normal vector we compute r(π/2) r() = 5,, 2, 3,, and r(3π/2) r() = 5, 3, 2. Taking the cross product and normalizing n = 3 2,, 5. Hence F = 3, 5x, 2y. Thus, F v = 3(52 sin 2 t cos 2 t). Finally, 2π dtf v = 3 2 2π R R dt( ) = 3 3 π. 9
10 . Optional, extra credit. Worth up to 5 pts. Caculate the surface area of a hemisphere x 2 + y 2 + z 2 = 9. Answer: one can work this out using spherical or cylindrical polar coordinates. Recall that the surface of a sphere is 4πr 2, where r is its radius. Hence, the surface of this hemisphere is 8π. Using spherical coordinates, the surface S of a hemisphere is S = π/2 dφ 2π 9 sin φ = 8π π/2 dφ sin φ = 8π. Using cylindrical coordinates and the projection technique we find that the normal to the hemisphere is ˆr = r/ r. Hence ˆk ˆn = z/3, z >. So the surface area can be found by projecting on the x y plane which makes a shadow of a circle of radius 3. Thus S = R dxdy 2π 3 z/3 = 3 dθ rdr 3 = 6π 9 r 2 rdr 9 r 2. then a change of variable u = 9 r 2, so that du/2 = rdr, leads to the expected result after the integration 3π 9 u /2 du = 6πu /2 9 = 8π.. Optional, extra credit. Worth up to 5 pts. CHAPTER4 REVIEW, number 45. Answer: F = κ T, where T = exp( x 2 y 2 z 2 ). Let ρ 2 = x 2 + y 2 + z 2. Hence, F = 2κ exp( ρ 2 )r. We use the divergence theorem to find the total flux across the unit sphere, centered at the origin: F ds = FdV, S we use the volume integral. We need the divf = 2(3 2ρ 2 ) exp( ρ 2 ) unit sphere Fρ2 sin ψdρdφdψ = 2κ Integrating in θ and ψ, 6κπ V π 2π dψ sin ψ dθ dρρ 2 F. ρ 2 (3 2ρ 2 ) exp( ρ 2 )dρ = 6κπ ( 2ρ 3 exp( ρ 2 ) ρ= = 32πκ.
This 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 information( ) ( ) 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 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 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 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 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 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 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 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 informationIf Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the RightHandRule.
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.
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 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 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 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 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 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 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 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 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 informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
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 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 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 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 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 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 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 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 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 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 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 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 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 informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
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 informationModule 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems
Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Objectives In this lecture you will learn the following Define different coordinate systems like spherical polar and cylindrical coordinates
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 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 informationLINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents
LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE ROBERT LIPSHITZ Abstract We will discuss the notion of linear maps and introduce the total derivative of a function f : R n R m as a linear map We will
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 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 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 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 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 informationPhysics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings
1 of 11 9/7/2012 1:06 PM Logged in as Julie Alexander, Instructor Help Log Out Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings Course Home Assignments Roster Gradebook Item Library
More informationDivergence and Curl. . Here we discuss some details of the divergence and curl. and the magnetic field B ( r,t)
Divergence and url Overview and Motivation: In the upcoming two lectures we will be discussing Maxwell's equations. These equations involve both the divergence and curl of two vector fields the electric
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 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 informationElectromagnetism  Lecture 2. Electric Fields
Electromagnetism  Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More informationPROBLEM SET. Practice Problems for Exam #1. Math 2350, Fall 2004. Sept. 30, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam #1 Math 350, Fall 004 Sept. 30, 004 ANSWERS i Problem 1. The position vector of a particle is given by Rt) = t, t, t 3 ). Find the velocity and acceleration vectors
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 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 informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
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 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 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 informationFluid Dynamics and the NavierStokes Equation
Fluid Dynamics and the NavierStokes Equation CMSC498A: Spring 12 Semester By: Steven Dobek 5/17/2012 Introduction I began this project through a desire to simulate smoke and fire through the use of programming
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationRARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III
RARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III I. Basic Course Information A. Course Number and Title: MATH 251 Calculus III B. New or Modified Course: Modified Course C.
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationMATH 275: Calculus III. Lecture Notes by Angel V. Kumchev
MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. ThreeDimensional Coordinate Systems..................... 1 Lecture
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 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 informationExam 1 Practice Problems Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8 Spring 13 Exam 1 Practice Problems Solutions Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical
More informationConcepts in Calculus III
Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
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 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 informationRendering Area Sources D.A. Forsyth
Rendering Area Sources D.A. Forsyth Point source model is unphysical Because imagine source surrounded by big sphere, radius R small sphere, radius r each point on each sphere gets exactly the same brightness!
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 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 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 informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
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 information= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.
Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)
More informationSecondOrder Linear Differential Equations
SecondOrder Linear Differential Equations A secondorder linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1
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 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 2 Practice Problems Part 1 Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Exam Practice Problems Part 1 Solutions Problem 1 Electric Field and Charge Distributions from Electric Potential An electric potential V ( z
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 informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
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 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 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 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 information73 The BiotSavart Law and the Magnetic Vector Potential
11/14/4 section 7_3 The BiotSavart Law blank.doc 1/1 73 The BiotSavart Law and the Magnetic ector Potential Reading Assignment: pp. 818 Q: Given some field B, how can we determine the source J that
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 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 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 informationMath 432 HW 2.5 Solutions
Math 432 HW 2.5 Solutions Assigned: 110, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More informationElectromagnetism Laws and Equations
Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E and Dfields............................................. Electrostatic Force............................................2
More informationParametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston
Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm
More information