16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

Size: px
Start display at page:

Download "16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21"

Transcription

1 16.2 Line Integrals Lukas Geyer Montana State University M273, Fall 211 Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

2 Scalar Line Integrals Definition f (x) ds = lim { s i } N f (P i ) s i i=1 Riemann sum illustration. Subdivision into arcs by red crosses, sample points P i as green stars, the s i are the length of the red line segments. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

3 Applications of Scalar Line Integrals Length length() = 1 ds Mass of a wire of linear density ρ(x) mass = ρ(x)ds Surface area of (one side of) a fence of height h(x) area = h(x)ds Electric Potential of a charged wire with linear charge density ρ(x) ρ(x) V (y) = k ds, where k is oulomb s constant. x y Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

4 alculation of Scalar Line Integrals Theorem Let c(t) be a parametrization of a curve for a t b. If f (x) and c (t) are continuous, then b f (x) ds = f (c(t)) c (t) dt. Differential version a ds = c (t) dt. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

5 Scalar Line Integral I alculate xy + z ds over the helix parametrized by c(t) = cos 3t, sin 3t, 4t for t π. ompute ds c (t) = 3 sin 3t, 3 cos 3t, 4 c (t) = 9 sin 2 3t + 9 cos 2 3t + 16 = 25 = 5 ds = c (t) dt = 5dt. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

6 Scalar Line Integral II alculate xy + z ds over the helix parametrized by c(t) = cos 3t, sin 3t, 4t for t π. ompute ds Write out integrand and evaluate xy + z ds = ds = c (t) dt = 5dt. = 5 = 5 π π (cos 3t sin 3t + 4t)5 dt cos 3t sin 3t dt + 2 ] π [ t2 [ 1 6 sin2 3t π ] π t dt = 1π 2 Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

7 Scalar Line Integral Applied I Find the mass of a wire in the shape of a circle of radius 2 centered at (3,4) with linear mass density ρ(x, y) = y 2. Parametrize c(t) = 3, cos t, sin t = cos t, sin t, where t 2π. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

8 Scalar Line Integral Applied II Find the mass of a wire in the shape of a circle of radius 2 centered at (3,4) with linear mass density ρ(x, y) = y 2. Parametrize ompute ds c(t) = cos t, sin t, t 2π. c (t) = 2 sin t, 2 cos t c (t) = 4 sin 2 t + 4 cos 2 t = 4 = 2 ds = c (t) dt = 2dt. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

9 Scalar Line Integral Applied III Find the mass of a wire in the shape of a circle of radius 2 centered at (3,4) with linear mass density ρ(x, y) = y 2. Parametrize, compute ds c(t) = cos t, sin t, t 2π, ds = 2dt. Write out integral and evaluate m = = 2π y 2 ds = 2π (4 + 2 sin t) 2 2 dt sin t + 8 sin 2 t dt = [32t 32 cos t + 4(t sin t cos t)] 2π = 64π + 8π = 72π Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

10 Vector Line Integrals Definition The line integral of a vector field F along an oriented curve is the scalar line integral of the tangential component of F. F ds = (F T)ds. Vector field F(x, y) =, 1 with an oriented curve (in green) and one unit tangent vector (in red). Question Is F ds positive, negative, or zero? Positive Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

11 Applications of Vector Line Integrals Work done by a force field F on a particle moving along W = F ds. Work done against a force field F, moving along W = F ds. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

12 omputing a Vector Line Integral Theorem If c(t) is a regular parametrization of an oriented curve for a t b, then b F ds = F(c(t)) c (t) dt. Differential version ds = c (t)dt c(t) is regular if it is smooth and c (t). a Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

13 Scalar vs. Vector Line Integrals Orientation matters for vector line integrals: If is the curve in negative orientation, then F ds = F ds Orientation does not matter for scalar line integrals: F ds = Fds F ds F ds Why? The tangential component F T always satisfies F T F. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

14 Vector Line Integral I Evaluate F ds, where F(x, y) =, 1, and is the part of the graph of y = 1 2 x 3 x from (2, 2) to ( 2, 2). Parametrize c(t) = t, 1 2 t3 t, 2 t 2. Problem The orientation is wrong. What to do about this? The easiest solution is to calculate with the wrong orientation and reverse the sign of the result. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

15 Vector Line Integral II Evaluate F ds, where F(x, y) =, 1, and is the part of the graph of y = 1 2 x 3 x from (2, 2) to ( 2, 2). Parametrize c(t) = t, 1 2 t3 t, 2 t 2. Write out integral and evaluate F ds = = F(c(t)) c (t) dt = 3 2 t2 + 1 dt = = = [ 1 2 t3 + t, 1 1, 3 2 t2 1 dt Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21 ] 2 2

16 Vector Line Integral III Evaluate F ds, where F(x, y) =, 1, and is the part of the graph of y = 1 2 x 3 x from (2, 2) to ( 2, 2). Write out integral and evaluate F ds = 4 Fix the orientation F ds = F ds = ( 4) = 4. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

17 Alternative Notation for Vector Line Integrals 2-D Version For F(x, y) = F 1 (x, y), F 2 (x, y) we write ds = dx, dy and F ds = F 1 dx + F 2 dy 3-D Version For F(x, y, z) = F 1 (x, y, z), F 2 (x, y, z), F 3 (x, y, z) we write ds = dx, dy, dz and F ds = F 1 dx + F 2 dy + F 3 dz Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

18 Alternative Notation II 3-D Parametrized Version For F(x, y, z) = F 1 (x, y, z), F 2 (x, y, z), F 3 (x, y, z) and a regular parametrization c(t) = x(t), y(t), z(t), a t b of, F ds = F 1 dx + F 2 dy + F 3 dz = b a F 1 (c(t))x (t) + F 2 (c(t))y (t) + F 3 (c(t))z (t) dt Differential version dx = x (t) dt, dy = y (t) dt, dz = z (t) dt Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

19 3-D Line Integral I alculate yz dx + xz dy + xy dz over the line segment from (1, 1, 1) to (3, 2, ). Parametrize c(t) = 1, 1, 1 + t( 3, 2, 1, 1, 1 ) = 1 + 2t, 1 + t, 1 t, t 1. Write out differentials dx = x (t) dt = 2 dt dy = y (t) dt = dt dz = z (t) dt = dt Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

20 3-D Line Integral II alculate yz dx + xz dy + xy dz over the line segment from (1, 1, 1) to (3, 2, ). Parametrize, write out differentials c(t) = 1 + 2t, 1 + t, 1 t, t 1, dx = 2 dt, dy = dt, dz = dt Write out integral and evaluate yz dx + xz dy + xy dz = 1 (1 + t)(1 t) 2 + (1 + 2t)(1 t) + (1 + 2t)(1 + t)( 1) dt Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

21 3-D Line Integral III alculate yz dx + xz dy + xy dz over the line segment from (1, 1, 1) to (3, 2, ). Write out integral and evaluate yz dx + xz dy + xy dz = = 1 1 (1 + t)(1 t) 2 + (1 + 2t)(1 t) + (1 + 2t)(1 + t)( 1) dt 2 2t t 2t 2 1 3t 2t 2 dt = = [ 2t t 2 2t 3] 1 = = t 6t 2 dt Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8 Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is

More information

16.3 Conservative Vector Fields

16.3 Conservative Vector Fields 16.3 Conservative Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.3 Conservative Vector Fields M273, Fall 2011 1 / 23 Fundamental Theorem for Conservative Vector

More information

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.

More information

Topic 5.1: Line Element and Scalar Line Integrals

Topic 5.1: Line Element and Scalar Line Integrals Math 275 Notes Topic 5.1: Line Element and Scalar Line Integrals Textbook Section: 16.2 More Details on Line Elements (vector dr, and scalar ds): http://www.math.oregonstate.edu/bridgebook/book/math/drvec

More information

Lecture 11: Arclength and Line Integrals

Lecture 11: Arclength and Line Integrals Lecture 11: Arclength and Line Integrals Rafikul Alam Department of Mathematics IIT Guwahati Parametric curves Definition: A continuous mapping γ : [a, b] R n is called a parametric curve or a parametrized

More information

16.1 Vector Fields. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.1 Vector Fields M273, Fall / 16

16.1 Vector Fields. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.1 Vector Fields M273, Fall / 16 16.1 Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.1 Vector Fields M273, Fall 2011 1 / 16 Vector Fields Definition An n-dimensional vector field is a function

More information

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

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

Exercises for Multivariable Differential Calculus XM521

Exercises for Multivariable Differential Calculus XM521 This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done

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

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

V11. Line Integrals in Space

V11. Line Integrals in Space V11. Line Integrals in Space 1. urves in space. In order to generalize to three-space our earlier work with line integrals in the plane, we begin by recalling the relevant facts about parametrized space

More information

Created by T. Madas LINE INTEGRALS. Created by T. Madas

Created by T. Madas LINE INTEGRALS. Created by T. Madas LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )

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

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3 M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly

More information

MATH 317 Fall 2016 Assignment 5

MATH 317 Fall 2016 Assignment 5 MATH 37 Fall 26 Assignment 5 6.3, 6.4. ( 6.3) etermine whether F(x, y) e x sin y îı + e x cos y ĵj is a conservative vector field. If it is, find a function f such that F f. enote F (P, Q). We have Q x

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

for 2 1/3 < t 3 1/3 parametrizes

for 2 1/3 < t 3 1/3 parametrizes Solution to Set 4, due Friday April 1) Section 5.1, Problem ) Explain why the path parametrized by ft) = t, t 1 ) is not smooth. Note this is true more specifically if the interval of t contains t = 1

More information

Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals

Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1

More information

0, such that. all. for all. 0, there exists. Name: Continuity. Limits and. calculus. the definitionn. satisfying. limit. However, is the limit of its

0, such that. all. for all. 0, there exists. Name: Continuity. Limits and. calculus. the definitionn. satisfying. limit. However, is the limit of its L Marizzaa A Bailey Multivariable andd Vector Calculus Name: Limits and Continuity Limits and Continuity We have previously defined limit in for single variable functions, but how do we generalize this

More information

NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:

NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing: AP Calculus AB PRACTICE MIDTERM EXAM Read each choice carefully and find the best answer. Your midterm exam will be made up of 8 of these questions. I reserve the right to change numbers and answers on

More information

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0) eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your

More information

Review Questions for Test 3 Hints and Answers

Review Questions for Test 3 Hints and Answers eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,

More information

Tangent and Normal Vector - (11.5)

Tangent and Normal Vector - (11.5) Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N

More information

Section 14.1 Vector Functions and Space Curves

Section 14.1 Vector Functions and Space Curves Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a

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

Calculus with Analytic Geometry 3 Fall 2018

Calculus with Analytic Geometry 3 Fall 2018 alculus with Analytic Geometry 3 Fall 218 Practice Exercises for the Final Exam I present here a number of exercises that could serve as practice for the final exam. Some are easy and straightforward;

More information

Math 106 Answers to Exam 3a Fall 2015

Math 106 Answers to Exam 3a Fall 2015 Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical

More information

Topic 2-2: Derivatives of Vector Functions. Textbook: Section 13.2, 13.4

Topic 2-2: Derivatives of Vector Functions. Textbook: Section 13.2, 13.4 Topic 2-2: Derivatives of Vector Functions Textbook: Section 13.2, 13.4 Warm-Up: Parametrization of Circles Each of the following vector functions describe the position of an object traveling around the

More information

Line and Surface Integrals. Stokes and Divergence Theorems

Line and Surface Integrals. Stokes and Divergence Theorems Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function

More information

Vector Calculus, Maths II

Vector Calculus, Maths II Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent

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

Final Exam Review Sheet : Comments and Selected Solutions

Final Exam Review Sheet : Comments and Selected Solutions MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,

More information

ENGI 4430 Line Integrals; Green s Theorem Page 8.01

ENGI 4430 Line Integrals; Green s Theorem Page 8.01 ENGI 4430 Line Integrals; Green s Theorem Page 8.01 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence

More information

Midterm Exam 2. Wednesday, May 7 Temple, Winter 2018

Midterm Exam 2. Wednesday, May 7 Temple, Winter 2018 Name: Student ID#: Section: Midterm Exam 2 Wednesday, May 7 Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be organized and use

More information

ENGI Parametric Vector Functions Page 5-01

ENGI Parametric Vector Functions Page 5-01 ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric

More information

The Calculus of Vec- tors

The Calculus of Vec- tors Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 3 1 The Calculus of Vec- Summary: tors 1. Calculus of Vectors: Limits and Derivatives 2. Parametric representation of Curves r(t) = [x(t), y(t),

More information

Math 207 Honors Calculus III Final Exam Solutions

Math 207 Honors Calculus III Final Exam Solutions Math 207 Honors Calculus III Final Exam Solutions PART I. Problem 1. A particle moves in the 3-dimensional space so that its velocity v(t) and acceleration a(t) satisfy v(0) = 3j and v(t) a(t) = t 3 for

More information

x 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3

x 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3 Version: 4/1/06. Note: These notes are mostly from my 5B course, with the addition of the part on components and projections. Look them over to make sure that we are on the same page as regards inner-products,

More information

Vector Calculus handout

Vector Calculus handout Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f

More information

ENGI 4430 Line Integrals; Green s Theorem Page 8.01

ENGI 4430 Line Integrals; Green s Theorem Page 8.01 ENGI 443 Line Integrals; Green s Theorem Page 8. 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence

More information

Math 20C Homework 2 Partial Solutions

Math 20C Homework 2 Partial Solutions Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

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

21-256: Partial differentiation

21-256: Partial differentiation 21-256: Partial differentiation Clive Newstead, Thursday 5th June 2014 This is a summary of the important results about partial derivatives and the chain rule that you should know. Partial derivatives

More information

MATH 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2

MATH 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2 MATH 3A: MIDTERM REVIEW JOE HUGHES 1. Curvature 1. Consider the curve r(t) = x(t), y(t), z(t), where x(t) = t Find the curvature κ(t). 0 cos(u) sin(u) du y(t) = Solution: The formula for curvature is t

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

Math 20E Midterm II(ver. a)

Math 20E Midterm II(ver. a) Name: olutions tudent ID No.: Discussion ection: Math 20E Midterm IIver. a) Fall 2018 Problem core 1 /24 2 /25 3 /26 4 /25 Total /100 1. 24 Points.) Consider the force field F 5y ı + 3y 2 j. Compute the

More information

Calculus of Vector-Valued Functions

Calculus of Vector-Valued Functions Chapter 3 Calculus of Vector-Valued Functions Useful Tip: If you are reading the electronic version of this publication formatted as a Mathematica Notebook, then it is possible to view 3-D plots generated

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

SOME PROBLEMS YOU SHOULD BE ABLE TO DO

SOME PROBLEMS YOU SHOULD BE ABLE TO DO OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson JUST THE MATHS UNIT NUMBER 13.1 INTEGRATION APPLICATIONS 1 (Second moments of an arc) by A.J.Hobson 13.1.1 Introduction 13.1. The second moment of an arc about the y-axis 13.1.3 The second moment of an

More information

f dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1

f dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1 hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals

More information

10.3 Parametric Equations. 1 Math 1432 Dr. Almus

10.3 Parametric Equations. 1 Math 1432 Dr. Almus Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line. Check your CASA account for Quiz

More information

Section 4.3 Vector Fields

Section 4.3 Vector Fields Section 4.3 Vector Fields DEFINITION: A vector field in R n is a map F : A R n R n that assigns to each point x in its domain A a vector F(x). If n = 2, F is called a vector field in the plane, and if

More information

Dr. Allen Back. Sep. 10, 2014

Dr. Allen Back. Sep. 10, 2014 Dr. Allen Back Sep. 10, 2014 The chain rule in multivariable calculus is in some ways very simple. But it can lead to extremely intricate sorts of relationships (try thermodynamics in physical chemistry...

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

Multivariable Calculus Midterm 2 Solutions John Ross

Multivariable Calculus Midterm 2 Solutions John Ross Multivariable Calculus Midterm Solutions John Ross Problem.: False. The double integral is not the same as the iterated integral. In particular, we have shown in a HW problem (section 5., number 9) that

More information

Math 180, Final Exam, Fall 2007 Problem 1 Solution

Math 180, Final Exam, Fall 2007 Problem 1 Solution Problem Solution. Differentiate with respect to x. Write your answers showing the use of the appropriate techniques. Do not simplify. (a) x 27 x 2/3 (b) (x 2 2x + 2)e x (c) ln(x 2 + 4) (a) Use the Power

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

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l. . If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r

More information

Lecture 13 - Wednesday April 29th

Lecture 13 - Wednesday April 29th Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131

More information

Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8. The chain rule with one independent variable Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

More information

Arc Length and Surface Area in Parametric Equations

Arc Length and Surface Area in Parametric Equations Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length

More information

Without fully opening the exam, check that you have pages 1 through 11.

Without fully opening the exam, check that you have pages 1 through 11. 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 11. Show all your work on the standard

More information

Lecture Wise Questions from 23 to 45 By Virtualians.pk. Q105. What is the impact of double integration in finding out the area and volume of Regions?

Lecture Wise Questions from 23 to 45 By Virtualians.pk. Q105. What is the impact of double integration in finding out the area and volume of Regions? Lecture Wise Questions from 23 to 45 By Virtualians.pk Q105. What is the impact of double integration in finding out the area and volume of Regions? Ans: It has very important contribution in finding the

More information

Final Review Worksheet

Final Review Worksheet Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;

More information

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C 15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.

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 2014 MTH 234 FINAL EXAM December 8, 2014 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 2 5 1 3 5 4 5 5 5 6 5 7 5 2 8 5 9 5 10

More information

Math 53: Worksheet 9 Solutions

Math 53: Worksheet 9 Solutions Math 5: Worksheet 9 Solutions November 1 1 Find the work done by the force F xy, x + y along (a) the curve y x from ( 1, 1) to (, 9) We first parametrize the given curve by r(t) t, t with 1 t Also note

More information

Practice Problems for the Final Exam

Practice Problems for the Final Exam Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of

More information

Problem Solving 1: Line Integrals and Surface Integrals

Problem Solving 1: Line Integrals and Surface Integrals A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is

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

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line

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

Arc Length and Riemannian Metric Geometry

Arc Length and Riemannian Metric Geometry Arc Length and Riemannian Metric Geometry References: 1 W F Reynolds, Hyperbolic geometry on a hyperboloid, Amer Math Monthly 100 (1993) 442 455 2 Wikipedia page Metric tensor The most pertinent parts

More information

Name: ID: Math 233 Exam 1. Page 1

Name: ID: Math 233 Exam 1. Page 1 Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are

More information

25. Chain Rule. Now, f is a function of t only. Expand by multiplication:

25. Chain Rule. Now, f is a function of t only. Expand by multiplication: 25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).

More information

Sec. 1.1: Basics of Vectors

Sec. 1.1: Basics of Vectors Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x

More information

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics 3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

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

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. MTH 34 Solutions to Exam April 9th, 8 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show

More information

Line Integrals and Gradient Fields

Line Integrals and Gradient Fields Line Integrals and Gradient Fields Based on notes by Ben Woodruff When you make your lesson plan, it should explain and contain examples of the following: 1. Describe how to integrate a function along

More information

Lecture 7 - Separable Equations

Lecture 7 - Separable Equations Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only

More information

16.2. Line Integrals

16.2. Line Integrals 16. Line Integrals Review of line integrals: Work integral Rules: Fdr F d r = Mdx Ndy Pdz FT r'( t) ds r t since d '(s) and hence d ds '( ) r T r r ds T = Fr '( t) dt since r r'( ) dr d dt t dt dt does

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

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS4613 SEMESTER: Autumn 2002/03 MODULE TITLE: Vector Analysis DURATION OF EXAMINATION:

More information

e x2 dxdy, e x2 da, e x2 x 3 dx = e

e x2 dxdy, e x2 da, e x2 x 3 dx = e STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete

More information

Parametric Curves. Calculus 2 Lia Vas

Parametric Curves. Calculus 2 Lia Vas Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider

More information

1. (a) The volume of a piece of cake, with radius r, height h and angle θ, is given by the formula: [Yes! It s a piece of cake.]

1. (a) The volume of a piece of cake, with radius r, height h and angle θ, is given by the formula: [Yes! It s a piece of cake.] 1. (a The volume of a piece of cake, with radius r, height h and angle θ, is given by the formula: / 3 V (r,h,θ = 1 r θh. Calculate V r, V h and V θ. [Yes! It s a piece of cake.] V r = 1 r θh = rθh V h

More information

Section 17.4 Green s Theorem

Section 17.4 Green s Theorem Section 17.4 Green s Theorem alculating Line Integrals using ouble Integrals In the previous section, we saw an easy way to determine line integrals in the special case when a vector field F is conservative.

More information

Math 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.

Math 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π. Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =

More information

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question

More information

f. D that is, F dr = c c = [2"' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2"' dt = 2

f. D that is, F dr = c c = [2' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2' dt = 2 SECTION 16.4 GREEN'S THEOREM 1089 X with center the origin and radius a, where a is chosen to be small enough that C' lies inside C. (See Figure 11.) Let be the region bounded by C and C'. Then its positively

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

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

Partial Derivatives. w = f(x, y, z).

Partial Derivatives. w = f(x, y, z). Partial Derivatives 1 Functions of Several Variables So far we have focused our attention of functions of one variable. These functions model situations in which a variable depends on another independent

More information

Math 210, Final Exam, Practice Fall 2009 Problem 1 Solution AB AC AB. cosθ = AB BC AB (0)(1)+( 4)( 2)+(3)(2)

Math 210, Final Exam, Practice Fall 2009 Problem 1 Solution AB AC AB. cosθ = AB BC AB (0)(1)+( 4)( 2)+(3)(2) Math 2, Final Exam, Practice Fall 29 Problem Solution. A triangle has vertices at the points A (,,), B (, 3,4), and C (2,,3) (a) Find the cosine of the angle between the vectors AB and AC. (b) Find an

More information

Vector-Valued Functions

Vector-Valued Functions Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1

More information

Stokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem

Stokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem tokes Theorem MATH 311, alculus III J. Robert Buchanan Department of Mathematics ummer 2011 Background (1 of 2) Recall: Green s Theorem, M(x, y) dx + N(x, y) dy = R ( N x M ) da y where is a piecewise

More information

ARNOLD PIZER rochester problib from CVS Summer 2003

ARNOLD PIZER rochester problib from CVS Summer 2003 ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.

More information