Math 20A lecture 22 Integral theorems in 3D

Size: px
Start display at page:

Download "Math 20A lecture 22 Integral theorems in 3D"

Transcription

1 Math 2A lecture 22 p. 1/12 Math 2A lecture 22 Integral theorems in 3D T.J. Barnet-Lamb Brandeis University

2 Math 2A lecture 22 p. 2/12 Announcements Homework eleven due Friday. Homework twelve is posted, due Tuesday after Thanksgiving. Office hours are 2 3.3pm today See the website for all sorts of course-related fun tbl/math2a/ It is your responsibility to log into LATTE and check that the grades are entered correctly. So far, HW 1 6 and HW 8 1 should be posted.

3 Previously on math 2a We looked at several kinds of integral in 3D. Vector line integrals measure how much the vector field is helping/hindering us as we try to move along a path Flux integrals measure how much vector stuff is going through a boundary which will be a surface in 3D. We looked at div (which measures how much vector stuff is being created or destroyed at a given point in a flow), and curl (which measures how much a little circle/sphere would want to rotate from the momentum the flow gives it). Math 2A lecture 22 p. 3/12

4 Math 2A lecture 22 p. 4/12 Previously on math 2a We saw two forms of Greens theorem (not counting the crazy one in the book!) One connected 2D flux integrals with (2D region) integrals of div. One connected 2D line integrals with (2D region) integrals of curl. We could see the first one quite clearly as a consequence of the law of conservation of vector stuff The other only became clear after quite a bit of algebra. But after we d found it, we could kinda-sorta see visually why it makes sense.

5 Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =.

6 Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =. Deja vu? This is the same example we did directly last time. It will be easier using Gauss s theorem (aka the divergence theorem).

7 Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =. Deja vu? This is the same example we did directly last time. It will be easier using Gauss s theorem (aka the divergence theorem). We work out F = 1.

8 Math 2A lecture 22 p. 6/12 Example: Gauss s theorem Thus the divergence theorem tells us we can work out the answer by integrating 1 over the inside of the region. (Let s call this 3D region R.) answer = = R = 2π 1dxdy dz = (1 r 2 )r dθ dr = r 2 1r dz dθ dr r r 3 dθ dr r r 3 dr = 2π[r 2 /2 r 4 /4] 1 = 2π(1/4) = π/2

9 Math 2A lecture 22 p. 7/12 Example: Gauss s theorem II Compute the flux of the vector field F(x,y,z) = z,y,x outward through the unit sphere. (Remember that the unit sphere can be parameterized by r(u,v) = sin u cos v, sin u sin v, cos u. ) The divergence is 1. Thus we get the answer by integrating 1 over the region (R, say) inside the unit sphere. We just get (4/3)π. We could also do the integral the long way. This integral is easiest in spherical polars, and I forgot to mention how to do integrals in spherical polars before! The only trick is that dxdy dz ρ 2 sin φdρdθ dφ

10 Math 2A lecture 22 p. 8/12 Example: Gauss s theorem II Doing the integral the long way: answer = = = R π π 1dxdy dz = π 1 3 [ρ3 ] 1 sin φdθ dφ = π 2π 3 sin φdφ = 2π 3 [ cos φ]π = 4π 3 1ρ 2 sin φdρdθ dφ 1 3 sinφdφ

11 Math 2A lecture 22 p. 9/12 Example: Stokes s theorem Compute the line integral F dr, where C is the circular C path p(t) = cos t, sin t, 1 + sin t + cos t and F is the vector field F(x,y,z) = y,x 2 x,z 2 z.

12 Math 2A lecture 22 p. 9/12 Example: Stokes s theorem Compute the line integral F dr, where C is the circular C path p(t) = cos t, sin t, 1 + sin t + cos t and F is the vector field F(x,y,z) = y,x 2 x,z 2 z. The curl is simply,, 2x. Thus we just need to integrate the flux of,, 2x through any surface with boundary C. Let s take the elliptical region in the plane z = 1 + x + y. This has parameterization r(u,v) = v cosu,v sin u, 1 + v cosu + v sin u for 2π and v 1.

13 Math 2A lecture 22 p. 1/12 Example: Stokes s theorem We work out (curl F)(r(u,v)) =,, 2v cosu r u = v sinu,v cosu, v sin u + v cosu r v = cosu, sin u, cos u + sinu r u r v =?,?, v sin 2 u v cos 2 u =?,?, v We want the flux upwards through the surface, so we want the other normal. Then the answer is (curl F)(r(u,v)) ( r u r v )dv du = 2v 2 cos udv du

14 Math 2A lecture 22 p. 11/12 Example: Stokes s theorem Working out the integral F(r(u,v)) ( r u r v )dv du = = 2 3 v3 1 cosudu = 2 3 cosudu = 2v 2 cosudv du

15 Math 2A lecture 22 p. 12/12 Exercise: Stokes theorem Compute the line integral F dr, where C is the circular C path p(t) = cost, sin t, 2 cos t and F is the vector field F(x,y,z) = y 2,x,z 2. Also set up the integral you would have to do if you did this directly as a line integral.

MATH2000 Flux integrals and Gauss divergence theorem (solutions)

MATH2000 Flux integrals and Gauss divergence theorem (solutions) DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,

More information

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then

More information

Math 23b Practice Final Summer 2011

Math 23b Practice Final Summer 2011 Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

More information

The Divergence Theorem

The Divergence Theorem Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points

More information

One side of each sheet is blank and may be used as scratch paper.

One side of each sheet is blank and may be used as scratch paper. Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever

More information

Figure 21:The polar and Cartesian coordinate systems.

Figure 21:The polar and Cartesian coordinate systems. Figure 21:The polar and Cartesian coordinate systems. Coordinate systems in R There are three standard coordinate systems which are used to describe points in -dimensional space. These coordinate systems

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 267 Answers to Stokes Practice Dr. Jones MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

More information

Figure 25:Differentials of surface.

Figure 25:Differentials of surface. 2.5. Change of variables and Jacobians In the previous example we saw that, once we have identified the type of coordinates which is best to use for solving a particular problem, the next step is to do

More information

MATH 255 Applied Honors Calculus III Winter Homework 11. Due: Monday, April 18, 2011

MATH 255 Applied Honors Calculus III Winter Homework 11. Due: Monday, April 18, 2011 MATH 255 Applied Honors Calculus III Winter 211 Homework 11 ue: Monday, April 18, 211 ection 17.7, pg. 1155: 5, 13, 19, 24. ection 17.8, pg. 1161: 3, 7, 13, 17 ection 17.9, pg. 1168: 3, 7, 19, 25. 17.7

More information

Read this cover page completely before you start.

Read this cover page completely before you start. I affirm that I have worked this exam independently, without texts, outside help, integral tables, calculator, solutions, or software. (Please sign legibly.) Read this cover page completely before you

More information

53. Flux Integrals. Here, R is the region over which the double integral is evaluated.

53. Flux Integrals. Here, R is the region over which the double integral is evaluated. 53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,

More information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

More information

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, Advanced Calculus for Engineers McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

More information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr. 1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

More information

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,

More information

Major Ideas in Calc 3 / Exam Review Topics

Major Ideas in Calc 3 / Exam Review Topics Major Ideas in Calc 3 / Exam Review Topics Here are some highlights of the things you should know to succeed in this class. I can not guarantee that this list is exhaustive!!!! Please be sure you are able

More information

ES.182A Topic 45 Notes Jeremy Orloff

ES.182A Topic 45 Notes Jeremy Orloff E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed

More information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final exam (practice 1) UCLA: Math 32B, Spring 2018 Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in

More information

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv 1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N

More information

Math 211, Fall 2014, Carleton College

Math 211, Fall 2014, Carleton College A. Let v (, 2, ) (1,, ) 1, 2, and w (,, 3) (1,, ) 1,, 3. Then n v w 6, 3, 2 is perpendicular to the plane, with length 7. Thus n/ n 6/7, 3/7, 2/7 is a unit vector perpendicular to the plane. [The negation

More information

Solutions to Sample Questions for Final Exam

Solutions to Sample Questions for Final Exam olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the

More information

Math 11 Fall 2018 Practice Final Exam

Math 11 Fall 2018 Practice Final Exam Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long

More information

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours) SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

More information

16.5 Surface Integrals of Vector Fields

16.5 Surface Integrals of Vector Fields 16.5 Surface Integrals of Vector Fields Lukas Geyer Montana State University M73, Fall 011 Lukas Geyer (MSU) 16.5 Surface Integrals of Vector Fields M73, Fall 011 1 / 19 Parametrized Surfaces Definition

More information

Chapter 3 - Vector Calculus

Chapter 3 - Vector Calculus Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f

More information

MAT 211 Final Exam. Fall Jennings.

MAT 211 Final Exam. Fall Jennings. MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv

More information

Assignment 11 Solutions

Assignment 11 Solutions . Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which

More information

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

HOMEWORK 8 SOLUTIONS

HOMEWORK 8 SOLUTIONS HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition

More information

APPM 2350 Final Exam points Monday December 17, 7:30am 10am, 2018

APPM 2350 Final Exam points Monday December 17, 7:30am 10am, 2018 APPM 2 Final Exam 28 points Monday December 7, 7:am am, 28 ON THE FONT OF YOU BLUEBOOK write: () your name, (2) your student ID number, () lecture section/time (4) your instructor s name, and () a grading

More information

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives. PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

More information

Lecture 10 - Moment of Inertia

Lecture 10 - Moment of Inertia Lecture 10 - oment of Inertia A Puzzle... Question For any object, there are typically many ways to calculate the moment of inertia I = r 2 dm, usually by doing the integration by considering different

More information

Solutions for the Practice Final - Math 23B, 2016

Solutions for the Practice Final - Math 23B, 2016 olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

More information

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Name: Student ID#: Section: Final Exam Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be

More information

MLC Practice Final Exam

MLC Practice Final Exam Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard

More information

Class 4 : Maxwell s Equations for Electrostatics

Class 4 : Maxwell s Equations for Electrostatics Class 4 : Maxwell s Equations for Electrostatics Concept of charge density Maxwell s 1 st and 2 nd Equations Physical interpretation of divergence and curl How do we check whether a given vector field

More information

CURRENT MATERIAL: Vector Calculus.

CURRENT MATERIAL: Vector Calculus. Math 275, section 002 (Ultman) Fall 2011 FINAL EXAM REVIEW The final exam will be held on Wednesday 14 December from 10:30am 12:30pm in our regular classroom. You will be allowed both sides of an 8.5 11

More information

MATH 52 FINAL EXAM DECEMBER 7, 2009

MATH 52 FINAL EXAM DECEMBER 7, 2009 MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB-

More information

( ) = x( u, v) i + y( u, v) j + z( u, v) k

( ) = x( u, v) i + y( u, v) j + z( u, v) k Math 8 ection 16.6 urface Integrals The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. uppose f is a function of three

More information

density = N A where the vector di erential aread A = ^n da, and ^n is the normaltothat patch of surface. Solid angle

density = N A where the vector di erential aread A = ^n da, and ^n is the normaltothat patch of surface. Solid angle Gauss Law Field lines and Flux Field lines are drawn so that E is tangent to the field line at every point. Field lines give us information about the direction of E, but also about its magnitude, since

More information

Page Points Score Total: 210. No more than 200 points may be earned on the exam.

Page Points Score Total: 210. No more than 200 points may be earned on the exam. Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200

More information

Math Review for Exam 3

Math Review for Exam 3 1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)

More information

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 CALCULUS AND MULTIVARIABLE CALCULUS MTHA4005Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Page Problem Score Max Score a 8 12b a b 10 14c 6 6 Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a

More information

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 234 final exam mainly consists of standard response questions where students

More information

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use

More information

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems Chapter 15. i j k Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

More information

Problem Points S C O R E

Problem Points S C O R E MATH 34F Final Exam March 19, 13 Name Student I # Your exam should consist of this cover sheet, followed by 7 problems. Check that you have a complete exam. Unless otherwise indicated, show all your work

More information

Divergence Theorem December 2013

Divergence Theorem December 2013 Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

Physics 9 Monday, March 19, 2012

Physics 9 Monday, March 19, 2012 Physics 9 Monday, March 19, 2012 learningcatalytics.com class session ID: 175557 Electricity: lots of new words & ideas, so ask lots of questions! Today: Gauss s law; electric potential (volts!); capacitance.

More information

Math 11 Fall 2016 Final Practice Problem Solutions

Math 11 Fall 2016 Final Practice Problem Solutions Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

Tutorial Exercises: Geometric Connections

Tutorial Exercises: Geometric Connections Tutorial Exercises: Geometric Connections 1. Geodesics in the Isotropic Mercator Projection When the surface of the globe is projected onto a flat map some aspects of the map are inevitably distorted.

More information

Ma 1c Practical - Solutions to Homework Set 7

Ma 1c Practical - Solutions to Homework Set 7 Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut

More information

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

the Cartesian coordinate system (which we normally use), in which we characterize points by two coordinates (x, y) and

the Cartesian coordinate system (which we normally use), in which we characterize points by two coordinates (x, y) and 2.5.2 Standard coordinate systems in R 2 and R Similarly as for functions of one variable, integrals of functions of two or three variables may become simpler when changing coordinates in an appropriate

More information

MATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name

MATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name MATH 223 Final Exam Solutions ecember 14, 25 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In

More information

MTH 234 Solutions to Exam 2 April 13, Multiple Choice. Circle the best answer. No work needed. No partial credit available.

MTH 234 Solutions to Exam 2 April 13, Multiple Choice. Circle the best answer. No work needed. No partial credit available. MTH 234 Solutions to Exam 2 April 3, 25 Multiple Choice. Circle the best answer. No work needed. No partial credit available.. (5 points) Parametrize of the part of the plane 3x+2y +z = that lies above

More information

MATH H53 : Final exam

MATH H53 : Final exam MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out

More information

MATH 52 FINAL EXAM SOLUTIONS

MATH 52 FINAL EXAM SOLUTIONS MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

More information

Coordinates 2D and 3D Gauss & Stokes Theorems

Coordinates 2D and 3D Gauss & Stokes Theorems Coordinates 2 and 3 Gauss & Stokes Theorems Yi-Zen Chu 1 2 imensions In 2 dimensions, we may use Cartesian coordinates r = (x, y) and the associated infinitesimal area We may also employ polar coordinates

More information

Math 114: Make-up Final Exam. Instructions:

Math 114: Make-up Final Exam. Instructions: Math 114: Make-up Final Exam Instructions: 1. Please sign your name and indicate the name of your instructor and your teaching assistant: A. Your Name: B. Your Instructor: C. Your Teaching Assistant: 2.

More information

Math Exam IV - Fall 2011

Math Exam IV - Fall 2011 Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and

More information

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the 1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015 Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

More information

First Derivative Test

First Derivative Test MA 2231 Lecture 22 - Concavity and Relative Extrema Wednesday, November 1, 2017 Objectives: Introduce the Second Derivative Test and its limitations. First Derivative Test When looking for relative extrema

More information

Math 31CH - Spring Final Exam

Math 31CH - Spring Final Exam Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate

More information

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

More information

Mathematics 205 Solutions for HWK 23. e x2 +y 2 dxdy

Mathematics 205 Solutions for HWK 23. e x2 +y 2 dxdy Mathematics 5 Solutions for HWK Problem 1. 6. p7. Let D be the unit disk: x + y 1. Evaluate the integral e x +y dxdy by making a change of variables to polar coordinates. D Solution. Step 1. The integrand,

More information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device

More information

v n ds where v = x z 2, 0,xz+1 and S is the surface that

v n ds where v = x z 2, 0,xz+1 and S is the surface that M D T P. erif the divergence theorem for d where is the surface of the sphere + + = a.. Calculate the surface integral encloses the solid region + +,. (a directl, (b b the divergence theorem. v n d where

More information

Tom Robbins WW Prob Lib1 Math , Fall 2001

Tom Robbins WW Prob Lib1 Math , Fall 2001 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment due 9/7/0 at 6:00 AM..( pt) A child walks due east on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 7 miles

More information

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11 1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +

More information

A1. Let r > 0 be constant. In this problem you will evaluate the following integral in two different ways: r r 2 x 2 dx

A1. Let r > 0 be constant. In this problem you will evaluate the following integral in two different ways: r r 2 x 2 dx Math 6 Summer 05 Homework 5 Solutions Drew Armstrong Book Problems: Chap.5 Eercises, 8, Chap 5. Eercises 6, 0, 56 Chap 5. Eercises,, 6 Chap 5.6 Eercises 8, 6 Chap 6. Eercises,, 30 Additional Problems:

More information

Practice Final Solutions

Practice Final Solutions Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )

More information

MA227 Surface Integrals

MA227 Surface Integrals MA7 urface Integrals Parametrically Defined urfaces We discussed earlier the concept of fx,y,zds where is given by z x,y.wehad fds fx,y,x,y1 x y 1 da R where R is the projection of onto the x,y - plane.

More information

Review problems for the final exam Calculus III Fall 2003

Review problems for the final exam Calculus III Fall 2003 Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)

More information

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π 1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:

More information

Math 234 Exam 3 Review Sheet

Math 234 Exam 3 Review Sheet Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the

More information

FINAL EXAM STUDY GUIDE

FINAL EXAM STUDY GUIDE FINAL EXAM STUDY GUIDE The Final Exam takes place on Wednesday, June 13, 2018, from 10:30 AM to 12:30 PM in 1100 Donald Bren Hall (not the usual lecture room!!!) NO books/notes/calculators/cheat sheets

More information

MATH 2400: Calculus III, Fall 2013 FINAL EXAM

MATH 2400: Calculus III, Fall 2013 FINAL EXAM MATH 2400: Calculus III, Fall 2013 FINAL EXAM December 16, 2013 YOUR NAME: Circle Your Section 001 E. Angel...................... (9am) 002 E. Angel..................... (10am) 003 A. Nita.......................

More information

D = 2(2) 3 2 = 4 9 = 5 < 0

D = 2(2) 3 2 = 4 9 = 5 < 0 1. (7 points) Let f(, ) = +3 + +. Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = 3 + + 1 f = f = 3 f = Both f = and f = onl at (

More information

MATH 251 Spring 2017 EXAM IV - VERSION A

MATH 251 Spring 2017 EXAM IV - VERSION A MATH 251 Spring 2017 EXAM IV - VERSION A LAST NAME: FIRST NAME: SETION NUMBER: UIN: DIRETIONS: 1. You may use a calculator on this exam. Programming formulas into the calculator is considered academic

More information

UNIVERSITY OF INDONESIA FACULTY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING

UNIVERSITY OF INDONESIA FACULTY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF INDONESIA FACULTY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING ENGINEERING MATHEMATICS (MCS-21007) 1. Course Name/Units : Engineering Mathematics/4 2. Department/Semester : Mechanical

More information

Math 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall.

Math 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall. Math 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall. Name: SID: GSI: Name of the student to your left: Name of the student to your right: Instructions: Write all answers

More information

Summary for Vector Calculus and Complex Calculus (Math 321) By Lei Li

Summary for Vector Calculus and Complex Calculus (Math 321) By Lei Li Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e

More information

is the curve of intersection of the plane y z 2 and the cylinder x oriented counterclockwise when viewed from above.

is the curve of intersection of the plane y z 2 and the cylinder x oriented counterclockwise when viewed from above. The questions below are representative or actual questions that have appeared on final eams in Math from pring 009 to present. The questions below are in no particular order. There are tpicall 10 questions

More information

Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution

Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution Jim Lambers MAT 8 Fall emester 6-7 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

More information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final exam (practice 1) UCLA: Math 32B, Spring 2018 Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions

More information

Math 11 Fall 2007 Practice Problem Solutions

Math 11 Fall 2007 Practice Problem Solutions Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

MA 351 Fall 2008 Exam #3 Review Solutions 1. (2) = λ = x 2y OR x = y = 0. = y = x 2y (2x + 2) = 2x2 + 2x 2y = 2y 2 = 2x 2 + 2x = y 2 = x 2 + x

MA 351 Fall 2008 Exam #3 Review Solutions 1. (2) = λ = x 2y OR x = y = 0. = y = x 2y (2x + 2) = 2x2 + 2x 2y = 2y 2 = 2x 2 + 2x = y 2 = x 2 + x MA 5 Fall 8 Eam # Review Solutions. Find the maimum of f, y y restricted to the curve + + y. Give both the coordinates of the point and the value of f. f, y y g, y + + y f < y, > g < +, y > solve y λ +

More information

xy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.

xy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1. Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density

More information

Review for the First Midterm Exam

Review for the First Midterm Exam Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers

More information

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In

More information

Triple Integrals. y x

Triple Integrals. y x Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object

More information

Math 221 Examination 2 Several Variable Calculus

Math 221 Examination 2 Several Variable Calculus Math Examination Spring Instructions These problems should be viewed as essa questions. Before making a calculation, ou should explain in words what our strateg is. Please write our solutions on our own

More information

Name: Instructor: Lecture time: TA: Section time:

Name: Instructor: Lecture time: TA: Section time: Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right

More information