- Conservative Vector Fields
- In this section, we continue the study of conservative vector fields. We examine the Fundamental Theorem for Line Integrals, which is a useful generalization of the Fundamental Theorem of Calculus to line integrals of conservative vector fields. We also discover show how to test whether a given vector field is conservative, and determine how to build a potential function for a vector field known to be conservative.

- Divergence and Curl
- Divergence and curl are two important operations on a vector field. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering.

- Green's Theorem
- Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region D in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve C and a double integral over the region enclosed by C.

- Introduction to Vector Field Chapter
- Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. A deep understanding of physics or engineering is impossible without an understanding of vector fields. Furthermore, vector fields have mathematical properties that are worthy of study in their own right. In particular, vector fields can be used to develop several higher-dimensional versions of the Fundamental Theorem of Calculus.

- Line Integrals
- Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.Line integrals have many applications to engineering and physics. And, they are closely connected to the properties of vector fields, as we shall see.
- Line Integrals (Exercises)

- Stokes' Theorem
- In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes’ theorem relates a vector surface integral over surface S in space to a line integral around the boundary of S.

- Surface Integrals
- If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.

- The Divergence Theorem
- We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study.

- Vector Calculus (Exercises)
- These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.

- Vector Fields
- Vector fields are an important tool for describing many physical concepts, such as gravitation and electromagnetism, which affect the behavior of objects over a large region of a plane or of space. They are also useful for dealing with large-scale behavior such as atmospheric storms or deep-sea ocean currents. In this section, we examine the basic definitions and graphs of vector fields so we can study them in more detail in the rest of this chapter.
- Vector Fields (Exercises)

## MAT 2500 Syllabus

12.1 Three Dimensional Coordinate Systems

12.2 Vectors

12.3 The Dot Product

12.4 Cross Products

10.1 Curves Defined by Parametric Equations - Review

12.5 Equations of Lines and Planes

12.6 Quadratic Surfaces (Optional)

**Chapter 13: Vector Functions**

13.1 Vector Functions and Space Curves

13.2 Derivatives and Integrals of Vector Functions

13.3 Arc Length and Curvature

13.4 Motion in Space: Velocity and Acceleration

**Chapter 14: Partial Derivatives**

14.1 Functions of Several Variables

14.2 Limits and Continuity

14.3 Partial Derivatives

14.4 Tangent Planes and Linear Approximations

14.5 The Chain Rule

14.6 Directional Derivatives and the Gradient

14.7 Maximum and Minimum Values

14.8 Lagrange Multipliers (Optional)

**Chapter 15: Multiple Integrals**

15.1 Double Integrals over Rectangles

15.2 Double Integrals over General Regions

10.3 Polar Coordinates - Review

15.3 Double Integrals in Polar Coordinates

15.4 Applications of Double Integrals

15.5 Surface Area (Optional)

15.6 Triple Integrals

15.7 Triple Integrals in Cylindrical Coordinates

15.8 Triple Integrals in Spherical Coordinates (Optional)

**Chapter 16: Vector Calculus**

16.1 Vector Fields

16.2 Line Integrals

16.3 Fundamental Theorem for Line Integrals

16.4 Green's Theorem

16.5 Curl and Divergence

16.6 Parametric Surfaces & Their Areas (Optional)

16.7 Surface Integrals (Optional)

16.8 Stokes' Theorem (Optional)

16.9 The Divergence Theorem (Optional)

This material is covered over a 14 week (56 class hours) semester.

Faculty have the option of utilizing WebAssign, a web-based supplement to our textbook. This portal provides graded and practice homework problems, online quizzes, video instruction and an eBook. WebAssign registration requires an access code, which is included with the text purchased from the Villanova Bookstore.

Note: If you choose to rent/purcase a book that comes without a WebAssign code, you will be required to purchase the registration code separately if your instructor uses WebAssign.

Upon the start of each semester, students will received a second code, called a class enrollment code, from their instructor. This code enrolls them into the specific course set up by their instructor for that semester.

**Computer Algebra System (CAS)**

Instructors will be using Maple or a comparable Computer Algebra System in the course.

## Solution Manual for Calculus 9th Edition Stewart

Diagnostic Tests.

A Preview of Calculus.

1. FUNCTIONS AND LIMITS.

Four Ways to Represent a Function. Mathematical Models: A Catalog of Essential Functions. New Functions from Old Functions. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. The Precise Definition of a Limit. Continuity. Review. Principles of Problem Solving.

2. DERIVATIVES.

Derivatives and Rates of Change. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Differentiation Formulas. Applied Project: Building a Better Roller Coaster. Derivatives of Trigonometric Functions. The Chain Rule. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Discovery Project: Families of Implicit Curves. Rates of Change in the Natural and Social Sciences. Related Rates. Linear Approximations and Differentials. Discovery Project: Polynomial Approximations. Review. Problems Plus.

3. APPLICATIONS OF DIFFERENTIATION.

Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. The Mean Value Theorem. What Derivatives Tell Us About the Shape of a Graph. Limits at Infinity Horizontal Asymptotes. Summary of Curve Sketching. Graphing with Calculus and Technology. Optimization Problems. Applied Project: The Shape of a Can. Applied Project: Planes and Birds: Minimizing Energy. Newton’s Method. Antiderivatives. Review. Problems Plus.

4. INTEGRALS.

The Area and Distance Problems. The Definite Integral. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Indefinite Integrals and the Net Change Theorem. Writing Project: Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Review. Problems Plus.

5. APPLICATIONS OF INTEGRATION.

Areas Between Curves. Applied Project: The Gini Index. Volumes. Volumes by Cylindrical Shells. Work. Average Value of a Function. Applied Project: Calculus and Baseball. Review. Problems Plus.

6. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS.

Inverse Functions. Instructors may cover either Sections 6.2–6.4 or Sections 6.2*–6.4*. Exponential Functions and Their Derivatives. Logarithmic Functions. Derivatives of Logarithmic Functions. The Natural Logarithmic Function. The Natural Exponential Function. General Logarithmic and Exponential Functions. Exponential Growth and Decay. Applied Project: Controlling Red Blood Cell Loss During Surgery. Inverse Trigonometric Functions. Applied Project: Where to Sit at the Movies. Hyperbolic Functions. Indeterminate Forms and l’Hospital’s Rule. Writing Project: The Origins of l’Hospital’s Rule. Review. Problems Plus.

7. TECHNIQUES OF INTEGRATION.

Integration by Parts. Trigonometric Integrals. Trigonometric Substitution. Integration of Rational Functions by Partial Fractions. Strategy for Integration. Integration Using Tables and Technology. Discovery Project: Patterns in Integrals. Approximate Integration. Improper Integrals. Review. Problems Plus.

8. FURTHER APPLICATIONS OF INTEGRATION.

Arc Length. Discovery Project: Arc Length Contest. Area of a Surface of Revolution. Discovery Project: Rotating on a Slant. Applications to Physics and Engineering. Discovery Project: Complementary Coffee Cups. Applications to Economics and Biology. Probability. Review. Problems Plus.

9. DIFFERENTIAL EQUATIONS.

Modeling with Differential Equations. Direction Fields and Euler’s Method. Separable Equations. Applied Project: How Fast Does a Tank Drain? Models for Population Growth. Linear Equations. Applied Project: Which Is Faster, Going Up or Coming Down? Predator-Prey Systems. Review. Problems Plus.

10. PARAMETRIC EQUATIONS AND POLAR COORDINATES.

Curves Defined by Parametric Equations. Discovery Project: Running Circles Around Circles. Calculus with Parametric Curves. Discovery Project: Bézier Curves. Polar Coordinates. Discovery Project: Families of Polar Curves. Calculus in Polar Coordinates. Conic Sections. Conic Sections in Polar Coordinates. Review. Problems Plus.

11. SEQUENCES, SERIES, AND POWER SERIES.

Sequences. Discovery Project: Logistic Sequences. Series. The Integral Test and Estimates of Sums. The Comparison Tests. Alternating Series and Absolute Convergence. The Ratio and Root Tests. Strategy for Testing Series. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series. Discovery Project: An Elusive Limit. Writing Project: How Newton Discovered the Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Review. Problems Plus.

12. VECTORS AND THE GEOMETRY OF SPACE.

Three-Dimensional Coordinate Systems. Vectors. Discovery Project: The Shape of a Hanging Chain. The Dot Product. The Cross Product. Discovery Project: The Geometry of a Tetrahedron. Equations of Lines and Planes. Discovery Project: Putting 3D in Perspective. Cylinders and Quadric Surfaces. Review. Problems Plus.

13. VECTOR FUNCTIONS.

Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space: Velocity and Acceleration. Applied Project: Kepler’s Laws. Review. Problems Plus.

14. PARTIAL DERIVATIVES.

Functions of Several Variables. Limits and Continuity. Partial Derivatives. Discovery Project: Deriving the Cobb-Douglas Production Function. Tangent Planes and Linear Approximations. Applied Project: The Speedo LZR Racer. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Discovery Project: Quadratic Approximations and Critical Points. Lagrange Multipliers. Applied Project: Rocket Science. Applied Project: Hydro-Turbine Optimization. Review. Problems Plus.

15. MULTIPLE INTEGRALS.

Double Integrals over Rectangles. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical Coordinates. Discovery Project: The Intersection of Three Cylinders. Triple Integrals in Spherical Coordinates. Applied Project: Roller Derby. Change of Variables in Multiple Integrals. Review. Problems Plus.

16. VECTOR CALCULUS.

Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green’s Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes’ Theorem. The Divergence Theorem. Summary. Review. Problems Plus.

APPENDIXES.

Numbers, Inequalities, and Absolute Values. Coordinate Geometry and Lines. Graphs of Second-Degree Equations. Trigonometry. Sigma Notation. Proofs of Theorems. Answers to Odd-Numbered Exercises.

INDEX.

## Solution Manual (Downloadable Files) for Calculus, 8th Edition, James Stewart, ISBN-10: 1285740629, ISBN-13: 9781285740621

Table of Contents

Preface.

To the Student.

Diagnostic Tests.

A Preview of Calculus.

1. FUNCTIONS AND LIMITS.

Four Ways to Represent a Function. Mathematical Models: A Catalog of Essential Functions. New Functions from Old Functions. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. The Precise Definition of a Limit. Continuity. Review. Principles of Problem Solving.

2. DERIVATIVES.

Derivatives and Rates of Change. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Differentiation Formulas. Applied Project: Building a Better Roller Coaster. Derivatives of Trigonometric Functions. The Chain Rule. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Laboratory Project: Families of Implicit Curves. Rates of Change in the Natural and Social Sciences. Related Rates. Linear Approximations and Differentials. Laboratory Project: Taylor Polynomials. Review. Problems Plus.

3. APPLICATION OF DIFFERENTIATION.

Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. The Mean Value Theorem. How Derivatives Affect the Shape of a Graph. Limits at Infinity Horizontal Asymptotes. Summary of Curve Sketching. Graphing with Calculus and Calculators. Optimization Problems. Applied Project: The Shape of a Can. Applied Project: Planes and Birds: Minimizing Energy. Newton’s Method. Antiderivatives. Review. Problems Plus.

4. INTEGRALS.

Areas and Distances. The Definite Integral. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Indefinite Integrals and the Net Change Theorem. Writing Project: Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Review. Problems Plus.

5. APPLICATIONS OF INTEGRATION.

Areas Between Curves. Applied Project: The Gini Index. Volumes. Volumes by Cylindrical Shells. Work. Average Value of a Function. Applied Project: Calculus and Baseball. Review. Problems Plus.

6. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS.

Inverse Functions. Instructors may cover either Sections 6.2-6.4 or Sections 6.2*-6.4* Exponential Functions and Their Derivatives. Logarithmic Functions. Derivatives of Logarithmic Functions. The Natural Logarithmic Function The Natural Exponential Function. General Logarithmic and Exponential Functions. Exponential Growth and Decay. Applied Project: Controlling Red Blood Cell Loss During Surgery. Inverse Trigonometric Functions. Applied Project: Where to Sit at the Movies. Hyperbolic Functions. Indeterminate Forms and l’Hospital’s Rule. Writing Project: The Origins of l’ Hospital’s Rule Review. Problems Plus.

7. TECHNIQUES OF INTEGRATION.

Integration by Parts. Trigonometric Integrals. Trigonometric Substitution. Integration of Rational Functions by Partial Fractions. Strategy for Integration. Integration Using Tables and Computer Algebra Systems. Discovery Project: Patterns in Integrals. Approximate Integration. Improper Integrals. Review. Problems Plus.

8. FURTHER APPLICATIONS OF INTEGRATION.

Arc Length. Discovery Project: Arc Length Contest. Area of a Surface of Revolution. Discovery Project: Rotating on a Slant. Applications to Physics and Engineering. Discovery Project: Complementary Coffee Cups. Applications to Economics and Biology. Probability. Review. Problems Plus.

9. DIFFERENTIAL EQUATIONS.

Modeling with Differential Equations. Direction Fields and Euler’s Method. Separable Equations. Applied Project: How Fast Does a Tank Drain? Applied Project: Which is Faster, Going Up or Coming Down? Models for Population Growth. Linear Equations. Predator-Prey Systems. Review. Problems Plus.

10. PARAMETRIC EQUATIONS AND POLAR COORDINATES.

Curves Defined by Parametric Equations. Laboratory Project: Running Circles Around Circles. Calculus with Parametric Curves. Laboratory Project: Bézier Curves. Polar Coordinates. Laboratory Project: Families of Polar Curves. Areas and Lengths in Polar Coordinates. Conic Sections. Conic Sections in Polar Coordinates. Review. Problems Plus.

11. INFINITE SEQUENCES AND SERIES.

Sequences. Laboratory Project: Logistic Sequences. Series. The Integral Test and Estimates of Sums. The Comparison Tests. Alternating Series. Absolute Convergence and the Ratio and Root Tests. Strategy for Testing Series. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series. Laboratory Project: An Elusive Limit. Writing Project: How Newton Discovered the Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Review. Problems Plus.

12. VECTORS AND THE GEOMETRY OF SPACE.

Three-Dimensional Coordinate Systems. Vectors. The Dot Product. The Cross Product. Discovery Project: The Geometry of a Tetrahedron. Equations of Lines and Planes. Laboratory Project: Putting 3D in Perspective Cylinders and Quadric Surfaces. Review. Problems Plus.

13. VECTOR FUNCTIONS.

Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space: Velocity and Acceleration. Applied Project: Kepler’s Laws. Review. Problems Plus.

14. PARTIAL DERIVATIVES.

Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Linear Approximation. Applied Project: The Speedo LZR Racer Suit. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Applied Project: Designing a Dumpster. Discovery Project: Quadratic Approximations and Critical Points. Lagrange Multipliers. Applied Project: Rocket Science. Applied Project: Hydro-Turbine Optimization. Review. Problems Plus.

15. MULTIPLE INTEGRALS.

Double Integrals over Rectangles. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical Coordinates. Discovery Project: The Intersection of Three Cylinders. Triple Integrals in Spherical Coordinates. Applied Project: Roller Derby. Change of Variables in Multiple Integrals. Review. Problems Plus.

16. VECTOR CALCULUS.

Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green’s Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes’ Theorem. Writing Project: Three Men and Two Theorems. The Divergence Theorem. Summary. Review. Problems Plus.

17. SECOND-ORDER DIFFERENTIAL EQUATIONS.

Second-Order Linear Equations. Nonhomogeneous Linear Equations. Applications of Second-Order Differential Equations. Series Solutions. Review.

APPENDIXES.

A Numbers, Inequalities, and Absolute Values. B Coordinate Geometry and Lines. C Graphs of Second-Degree Equations. D Trigonometry. E Sigma Notation. F Proofs of Theorems. G Complex Numbers. H Answers to Odd-Numbered Exercises.

INDEX.

## Calculus

Calculus, Third Edition emphasizes the techniques and theorems of calculus, including many applied examples and exercises in both drill and applied-type problems. This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, proof of l'Hôpital's rule, and approximation using Taylor polynomials. Other topics include the rectangular coordinate system in space, higher-order partial derivatives, line integrals in space, and vibratory motion. This publication is valuable to students taking calculus.

Calculus, Third Edition emphasizes the techniques and theorems of calculus, including many applied examples and exercises in both drill and applied-type problems. This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, proof of l'Hôpital's rule, and approximation using Taylor polynomials. Other topics include the rectangular coordinate system in space, higher-order partial derivatives, line integrals in space, and vibratory motion. This publication is valuable to students taking calculus.

## Calculus: Early Transcendentals

The figure shows a curve $ C $ and a contour map of a function $ f $ whose gradient is continuous. Find $ int_C
abla f cdot d extbf

### Problem 2

A table of values of a function $ f $ with continuous gradient is given. Find $ int_C
abla f cdot d extbf

$ x = t^2 + 1 $ $ y = t^3 + t $ $ 0 leqslant t leqslant 1 $

### Problem 3

Determine whether or not $ extbf

$ extbf*+ (x^2 + 2xy) , extbf $*

* *

*Problem 4*

Determine whether or not $ extbf

$ extbf*+ 2xy , extbf $*

* *

*Problem 5*

Determine whether or not $ extbf

$ extbf*+ (1 + xy)e^ , extbf *

* *

*Problem 6*

$ extbf*+ (e^x + e^y) , extbf $*

* *

*Problem 7*

$ extbf*+ (e^x + x cos y) , extbf $*

*
*

### Problem 8

$ extbf*+ (x^2 - 2xy^<-3>) , extbf $, $ y > 0 $*

* *

*Problem 9*

$ extbf*+ (2y sin x - x sin y) , extbf $*

* *

*Problem 10*

$ extbf*+ (ln x + x/y) , extbf $*

* *

*Problem 11*

The figure shows the vector field $ extbf

(a) Explain why $ int_C extbf

(b) What is this common value?

### Problem 12

(a) Find a function $ f $ such that $ extbf

$ extbf*+ 2x^2y , extbf $, *

$ C $ is the arc of the hyperbola $ y = 1/x $ from $ (1, 1) $ to $ (4, frac<1><4>) $

* *

*Problem 13*

(a) Find a function $ f $ such that $ extbf

$ extbf*+ x^3y^2 , extbf $, *

$ C $: $ extbf

* *

*Problem 14*

(a) Find a function $ f $ such that $ extbf

$ extbf*+ x^2e^ , extbf *

$ C $: $ extbf

*+ 2 sin t , extbf*$, $ 0 leqslant t leqslant pi/2 $

* *

*Problem 15*

*Problem 15*

$ extbf*+ xz , extbf + (xy + 2z) extbf *

$ C $ is the line segment from $ (1, 0, -2) $ to $ (4, 6, 3) $

* *

*Problem 16*

$ extbf*+ 2xyz , extbf + (xy^2 + 2x^2z) , extbf *

$ C $: $ x = sqrt

* *

*Problem 17*

$ extbf*+ e^ , extbf *

$ C $: $ extbf

*+ (t^2 - 1) , extbf*+ (t^2 - 2t) , extbf $,

$ 0 leqslant t leqslant 2 $

$ 0 leqslant t leqslant 2 $

* *

*Problem 18*

*Problem 18*

$ extbf*+ (x cos y + cos z) , extbf - ysin z , extbf *

$ C $: $ extbf

*+ t , extbf + 2t , extbf*$, $ 0 leqslant t leqslant pi/2 $

* *

*Problem 19*

*Problem 19*

Show that the line integral is independent of path and evaluate the integral.

$ int_C 2xe^ <-y>, dx + (2y - x^2e^<-y>) , dy $, $ C $ is any path from $ (1, 0) $ to $ (2, 1) $.

### Problem 20

Show that the line integral is independent of path and evaluate the integral.

$ int_C sin y , dx + (x cos y - sin y) , dy $, $ C $ is any path from $ (2, 0) $ to $ (1, pi) $.

### Problem 21

Supposed you're asked to determine the curve that requires the least work for a force field $ extbf

### Problem 22

Suppose an experiment determines that the amount of work required for a force field $ extbf

### Problem 23

Find the work done by the force field $ extbf

$ extbf*+ y^3 , extbf $ $ P(1, 0) $, $ Q(2, 2) $*

* *

*Problem 24*

Find the work done by the force field $ extbf

$ extbf*+ x , extbf $ $ P(1, 1) $, $ Q(4, 3) $*

* *

*Problem 25*

Is the vector field shown in the figure conservative? Explain.

### Problem 26

Is the vector field shown in the figure conservative? Explain.

### Problem 27

If $ extbf*+ (1 + x cos y) , extbf $, use a plot to guess whether $ extbf *

* *

*Problem 28*

Let $ extbf

(a) $ displaystyle int_

### Problem 29

Show that if the vector field $ extbf *+ Q , extbf + R , extbf *

$ dfrac

* *

*Problem 30*

Use Exercise 29 to show that the line integral $ int_C y , dx + x , dy + xyz , dz $ is not independent of path.

### Problem 31

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.

### Problem 32

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.

### Problem 33

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.

### Problem 34

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.

### Problem 35

(a) Show that $ partial P/ partial y = partial Q/ partial x $.

(b) Show that $ displaystyle int_C extbf

### Problem 36

(a) Suppose that $ extbf *+ y , extbf + z , extbf *

* *

*(b) An example of an inverse square field is the gravitational field $ extbf = -(mMG) extbf*

* *

*(c) Another example of an inverse square field is the electric force field $ extbf = varepsilon qQ extbf*

* *

*16: Vector Fields, Line Integrals, and Vector Theorems - Mathematics*

MATH 317 Section 202

Calculus IV: Vector Calculus

2014W Term 2

- Assignment 10 on webwork is due on Tuesday, April 7th at 9 a.m.

Text: *Multivariable Calculus* 7th Edition by James Stewart.

Main MATH 317 Website maintained by Prof. Jim Bryan.

- Vector valued functions of one variable (Chapter 13):

Parameterized curves, velocity, acceleration, arc length.

Includes curvature, normal and binormal vectors, tangential and normal components of acceleration. - Vector valued functions of several variables (Chapter 16):

vector fields, line integrals, conservative fields, fundamental theorem of line integrals,

Green's theorem, gradient, curl, divergence,

parameterized surfaces, suface area, surface integrals,

Stoke's theorem, divergence theorem.

Instructor: Mark Mac Lean

Email: maclean (domain: math.ubc.ca)

Office: MATH 113

Phone: 604-827-3038

Hours: By appointment. I am also happy to answer questions by email.

Your final mark in this course will be determined by the following breakdown:

- Final grade computation. It is given by which ever is greater,

Homework/etc.*20% + Midterms*30% + FinalExam*50%

OR

FinalExam - 10.

The second option is your safety net : even if you perform very badly on the midterms, you can still get a good grade in the class by doing well on the final.

There will be regular assignments, both written and on WebWork. Mathematics is a subject one learns by DOING problems, so please take these quizzes and assignments seriously.

- Due Wednesday, January 14th
**at the start of class**:**13.1**#30, 40, 42, 48,**13.2**# 28, 34, 54 - Due Monday, January 26th
**at the start of class**:**13.3**#16, 18, 22, 48, 50, 66 - Due Friday, January 30th
**at the start of class**:**13.4**#16, 40, 42, 44, 46 - Due Wednesday, February 25th at the start of class:
**16.1**#26, 29 -- 32,**16.2**#4, 8, 18, 20,**16.3**#4, 8, 12, 25, 26, 35.

## Calculus - Analytic Geometry III

**Spring 2015**

Hi, here is some information about my course Calculus III (CRN: 22678, MAC 2313-007, 4 credits). We meet Mondays, Wednesdays, Thursdays and Fridays, 1:00 - 1:50 p.m. in BU 208.

The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the "method of exhaustion". Limits arise not only when finding areas of a region, but also when computing the slope of a tangent line to a curve, the velocity of a car, or the sum of an infinite series. In each case, one quantity is computed as the limit of other, easily calculated quantities. Sir Isaac Newton invented his version of calculus in order to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast coffee prices rise, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas.

## APEX Calculus

We have studied functions of two and three variables, where the input of such functions is a point (either a point in the plane or in space) and the output is a number.

We could also create functions where the input is a point (again, either in the plane or in space), but the output is a *vector*. For instance, we could create the following function: (vec F(x,y) = langle x+y, x-y
angle ext<,>) where (vec F(2,3) = langle 5,-1
angle ext<.>) We are to think of (vec F) assigning the vector (langle 5,-1
angle) to the point ((2,3) ext<>) in some sense, the vector (langle 5,-1
angle) lies at the point ((2,3) ext<.>)

Such functions are extremely useful in any context where magnitude and direction are important. For instance, we could create a function (vec F) that represents the electromagnetic force exerted at a point by a electromagnetic field, or the velocity of air as it moves across an airfoil.

Because these functions are so important, we need to formally define them.

###### Definition 16.2.2 . Vector Field.

A *vector field in the plane* is a function (vec F(x,y)) whose domain is a subset of (mathbb

A *vector field in space* is a function (vec F(x,y,z)) whose domain is a subset of (mathbb

This definition may seem odd at first, as a special type of function is called a “field.” However, as the function determines a “field of vectors”, we can say the field is *defined by* the function, and thus the field *is* a function.

Visualizing vector fields helps cement this connection. When graphing a vector field in the plane, the general idea is to draw the vector (vec F(x,y)) at the point ((x,y) ext<.>) For instance, using (vec F(x,y) = langle x+y,x-y angle) as before, at ((1,1)) we would draw (langle 2,0 angle ext<.>)

In Figure 16.2.3.(a), one can see that the vector (langle 2,0
angle) is drawn *starting from* the point ((1,1) ext<.>) A total of 8 vectors are drawn, with the (x)- and (y)-values of (-1,0,1 ext<.>) In many ways, the resulting graph is a mess it is hard to tell what this field “looks like.”

In Figure 16.2.3.(b), the same field is redrawn with each vector (vec F(x,y)) drawn *centered on* the point ((x,y) ext<.>) This makes for a better looking image, though the long vectors can cause confusion: when one vector intersects another, the image looks cluttered.

A common way to address this problem is limit the length of each arrow, and represent long vectors with thick arrows, as done in Figure 16.2.4.(a). Usually we do not use a graph of a vector field to determine exactly the magnitude of a particular vector. Rather, we are more concerned with the relative magnitudes of vectors: which are bigger than others? Thus limiting the length of the vectors is not problematic.

Drawing arrows with variable thickness is best done with technology search the documentation of your favorite graphing program for terms like “vector fields” or “slope fields” to learn how. Technology obviously allows us to plot many vectors in a vector field nicely in Figure 16.2.4.(b), we see the same vector field drawn with many vectors, and finally get a clear picture of how this vector field behaves. (If this vector field represented the velocity of air moving across a flat surface, we could see that the air tends to move either to the upper-right or lower-left, and moves very slowly near the origin.)

We can similarly plot vector fields in space, as shown in Figure 16.2.5, though it is not often done. The plots get very busy very quickly, as there are lots of arrows drawn in a small amount of space. In Figure 16.2.5 the field (vec F = langle -y,x,z angle) is graphed. If one could view the graph from above, one could see the arrows point in a circle about the (z)-axis. One should also note how the arrows far from the origin are larger than those close to the origin.

It is good practice to try to visualize certain vector fields in one's head. For instance, consider a point mass at the origin and the vector field that represents the gravitational force exerted by the mass at any point in the room. The field would consist of arrows pointing toward the origin, increasing in size as they near the origin (as the gravitational pull is strongest near the point mass).

### Subsection 16.2.1 Vector Field Notation and Del Operator

Definition 16.2.2 defines a vector field (vec F) using the notation

That is, the components of (vec F) are each functions of (x) and (y) (and also (z) in space). As done in other contexts, we will drop the “of (x ext<,>) (y) and (z)” portions of the notation and refer to vector fields in the plane and in space as

respectively, as this shorthand is quite convenient.

Another item of notation will become useful: the “del operator.” Recall in Section 14.3 how we used the symbol ( abla) (pronounced “del”) to represent the gradient of a function of two variables. That is, if (z = f(x,y) ext<,>) then “del (f)” (= abla f = langle f_x, f_y angle ext<.>)

We now define ( abla) to be the “del operator.” It is a vector whose components are partial derivative operations.

With this definition of ( abla ext<,>) we can better understand the gradient ( abla f ext<.>) As (f) returns a scalar, the properties of scalar and vector multiplication gives

Now apply the del operator ( abla) to vector fields. Let (vec F = langle x+sin y,y^2+z,x^2 angle ext<.>) We can use vector operations and find the dot product of ( abla) and (vec F ext<:>)

We can also compute their cross products:

We do not yet know why we would want to compute the above. However, as we next learn about properties of vector fields, we will see how these dot and cross products with the del operator are quite useful.

### Subsection 16.2.2 Divergence and Curl

Two properties of vector fields will prove themselves to be very important: divergence and curl. Each is a special “derivative” of a vector field that is, each measures an instantaneous rate of change of a vector field.

If the vector field represents the velocity of a fluid or gas, then the *divergence* of the field is a measure of the “compressibility” of the fluid. If the divergence is negative at a point, it means that the fluid is compressing: more fluid is going into the point than is going out. If the divergence is positive, it means the fluid is expanding: more fluid is going out at that point than going in. A divergence of zero means the same amount of fluid is going in as is going out. If the divergence is zero at all points, we say the field is *incompressible*.

It turns out that the proper measure of divergence is simply ( abla cdot vec F ext<,>) as stated in the following definition.

###### Definition 16.2.9 . Divergence of a Vector Field.

The of a vector field (vec F) is

In the plane, with (vec F = langle M,N angle ext<,>) (divv vec F = M_x+N_y ext<.>)

In space, with (vec F = langle M,N,P angle ext<,>) (divv vec F = M_x+N_y+P_z ext<.>)

*Curl* is a measure of the spinning action of the field. Let (vec F) represent the flow of water over a flat surface. If a small round cork were held in place at a point in the water, would the water cause the cork to spin? No spin corresponds to zero curl counterclockwise spin corresponds to positive curl and clockwise spin corresponds to negative curl.

In space, things are a bit more complicated. Again let (vec F) represent the flow of water, and imagine suspending a tennis ball in one location in this flow. The water may cause the ball to spin along an axis. If so, the curl of the vector field is a *vector* (not a *scalar*, as before), parallel to the axis of rotation, following a right hand rule: when the thumb of one's right hand points in the direction of the curl, the ball will spin in the direction of the curling fingers of the hand.

In space, it turns out the proper measure of curl is ( abla imes vec F ext<,>) as stated in the following definition. To find the curl of a planar vector field (vec F = langle M,N angle ext<,>) embed it into space as (vec F = langle M, N, 0 angle) and apply the cross product definition. Since (M) and (N) are functions of just (x) and (y) (and not (z)), all partial derivatives with respect to (z) become 0 and the result is simply (langle 0,0,N_x-M_y angle ext<.>) The third component is the measure of curl of a planar vector field.

###### Definition 16.2.11 . Curl of a Vector Field.

Let (vec F = langle M,N
angle) be a vector field in the plane. The *curl* of (vec F) is (curl vec F = N_x - M_y ext<.>)

Let (vec F = langle M,N,P
angle) be a vector field in space. The *curl* of (vec F) is (curl vec F =
abla imes vec F = langle P_y-N_z,M_z-P_x,N_x - M_y
angle ext<.>)

We adopt the convention of referring to curl as ( abla imes vec F ext<,>) regardless of whether (vec F) is a vector field in two or three dimensions.

We now practice computing these quantities.

###### Example 16.2.13 . Computing divergence and curl of planar vector fields.

For each of the planar vector fields given below, view its graph and try to visually determine if its divergence and curl are 0. Then compute the divergence and curl.

(vec F = langle cos y, sin x angle) (see Figure 16.2.15.(b))

The arrow sizes are constant along any horizontal line, so if one were to draw a small box anywhere on the graph, it would seem that the same amount of fluid would enter the box as exit. Therefore it seems the divergence is zero it is, as

At any point on the (x)-axis, arrows above it move to the right and arrows below it move to the left, indicating that a cork placed on the axis would spin clockwise. A cork placed anywhere above the (x)-axis would have water above it moving to the right faster than the water below it, also creating a clockwise spin. A clockwise spin also appears to be created at points below the (x)-axis. Thus it seems the curl should be negative (and not zero). Indeed, it is:

It appears that all vectors that lie on a circle of radius (r ext<,>) centered at the origin, have the same length (and indeed this is true). That implies that the divergence should be zero: draw any box on the graph, and any fluid coming in will lie along a circle that takes the same amount of fluid out. Indeed, the divergence is zero, as

Clearly this field moves objects in a circle, but would it induce a cork to spin? It appears that yes, it would: place a cork anywhere in the flow, and the point of the cork closest to the origin would feel less flow than the point on the cork farthest from the origin, which would induce a counterclockwise flow. Indeed, the curl is positive:

Since the curl is constant, we conclude the induced spin is the same no matter where one is in this field.

At the origin, there are many arrows pointing out but no arrows pointing in. We conclude that at the origin, the divergence must be positive (and not zero). If one were to draw a box anywhere in the field, the edges farther from the origin would have larger arrows passing through them than the edges close to the origin, indicating that more is going from a point than going in. This indicates a positive (and not zero) divergence. This is correct:

One may find this curl to be harder to determine visually than previous examples. One might note that any arrow that induces a clockwise spin on a cork will have an equally sized arrow inducing a counterclockwise spin on the other side, indicating no spin and no curl. This is correct, as

One might find this divergence hard to determine visually as large arrows appear in close proximity to small arrows, each pointing in different directions. Instead of trying to rationalize a guess, we compute the divergence:

Perhaps surprisingly, the divergence is 0. Will all the loops of different directions in the field, one is apt to reason the curl is variable. Indeed, it is:

Depending on the values of (x) and (y ext<,>) the curl may be positive, negative, or zero.

###### Example 16.2.16 . Computing divergence and curl of vector fields in space.

Compute the divergence and curl of each of the following vector fields.

(displaystyle vec F = langle x^2+y+z, -x-z, x+y angle)

(displaystyle vec F = langle e^

We compute the divergence and curl of each field following the definitions.

For this particular field, no matter the location in space, a spin is induced with axis parallel to (langle 2,0,-2 angle ext<.>)

###### Example 16.2.18 . Creating a field representing gravitational force.

The force of gravity between two objects is inversely proportional to the square of the distance between the objects. Locate a point mass at the origin. Create a vector field (vec F) that represents the gravitational pull of the point mass at any point ((x,y,z) ext<.>) Find the divergence and curl of this field.

The point mass pulls toward the origin, so at ((x,y,z) ext<,>) the force will pull in the direction of (langle -x, -y, -z angle ext<.>) To get the proper magnitude, it will be useful to find the unit vector in this direction. Dividing by its magnitude, we have

The magnitude of the force is inversely proportional to the square of the distance between the two points. Letting (k) be the constant of proportionality, we have the magnitude as (dsfrac

We leave it to the reader to confirm that (divv vec F = 0) and (curl vec F = vec 0 ext<.>)

The analogous planar vector field is given in Figure 16.2.19. Note how all arrows point to the origin, and the magnitude gets very small when “far” from the origin.

A function (f(x,y)) naturally induces a vector field, (vec F =
abla f = langle f_x,f_y
angle ext<.>) Given what we learned of the gradient in Section 14.3, we know that the vectors of (vec F) point in the direction of greatest increase of (f ext<.>) Because of this, (f) is said to be the *potential function* of (vec F ext<.>) Vector fields that are the gradient of potential functions will play an important role in the next section.

###### Example 16.2.20 . A vector field that is the gradient of a potential function.

Let (f(x,y) = 3-x^2-2y^2) and let (vec F = abla f ext<.>) Graph (vec F ext<,>) and find the divergence and curl of (vec F ext<.>)

Given (f ext<,>) we find (vec F = abla f = langle -2x,-4y angle ext<.>) A graph of (vec F) is given in Figure 16.2.21.(a). In Figure 16.2.21.(b), the vector field is given along with a graph of the surface itself one can see how each vector is pointing in the direction of “steepest uphill”, which, in this case, is not simply just “toward the origin.”

We leave it to the reader to confirm that (divv vec F = -6) and (curl vec F = 0 ext<.>)

There are some important concepts visited in this section that will be revisited in subsequent sections and again at the very end of this chapter. One is: given a vector field (vec F ext<,>) both (divvvec F) and (curlvec F) are measures of rates of change of (vec F ext<.>) The divergence measures how much the field spreads (diverges) at a point, and the curl measures how much the field twists (curls) at a point. Another important concept is this: given (z=f(x,y) ext<,>) the gradient ( abla f) is also a measure of a rate of change of (f ext<.>) We will see how the integrals of these rates of change produce meaningful results.

This section introduces the concept of a vector field. The next section “applies calculus” to vector fields. A common application is this: let (vec F) be a vector field representing a force (hence it is called a “force field,” though this name has a decidedly comic-book feel) and let a particle move along a curve (C) under the influence of this force. What work is performed by the field on this particle? The solution lies in correctly applying the concepts of line integrals in the context of vector fields.

### Exercises 16.2.3 Exercises

###### Terms and Concepts

Give two quantities that can be represented by a vector field in the plane or in space.

In your own words, describe what it means for a vector field to have a negative divergence at a point.

In your own words, describe what it means for a vector field to have a negative curl at a point.

The divergence of a vector field (vec F) at a particular point is 0. Does this mean that (vec F) is incompressible? Why/why not?

###### Problems

In the following exercises, sketch the given vector field over the rectangle with opposite corners ((-2,-2)) and ((2,2) ext<,>) sketching one vector for every point with integer coordinates (i.e., at ((0,0) ext<,>) ((1,2) ext<,>) etc.).

In the following exercises, find the divergence and curl of the given vector field.

(vec F = langle cos (xy), sin (xy) angle)

(dsvec F = la x^2+z^2,x^2+y^2,y^2+z^2 a)

(vec F = abla f ext<,>) where (f(x,y) = frac12x^2+frac13y^3 ext<.>)

(vec F = abla f ext<,>) where (f(x,y) = x^2y ext<.>)

(vec F = abla f ext<,>) where (f(x,y,z) = x^2y+sin z ext<.>)

(vec F =
abla f ext<,>) where (ds f(x,y,z) = frac1

## Course Details

**Instructor:** Michael Woodbury (x4-4988, 247 Mathematics, [email protected])

**Office Hours:** T 12:30pm-1:30pm, W 1:00pm-2:00pm, Mathematics 427, or by appointment.

**Teaching Assistant:** Dili Wang, [email protected], Office Hours: Thursday 10am-11am, 5pm-6pm in the Math help room (406 Mathematics)

**Text:** James Stewart *Calculus: Early Transcendentals*, sixth edition, Brooks/Cole, 2008.

**Course description:** We will cover chapters 15 (Multiple Integrals) and 16 (Vector Calculus). The main topics are:

- multiple integrals (using rectangular coordinates)
- integrals using polar, cylindrical, and spherical coordinates
- vector fields
- line integrals
- gradients, curl, and divergence.
- Green's Theorem, Stoke's Theorem, Divergence Theorem

We will also cover topics from basic complex analysis. The required material will be presented in class. Additional reference materials will also be provided.

Here are course notes written by Prof. Herve Jacquet. This (together with the lectures) is the main reference for the section on complex numbers.

Complex Numbers

Complex Functions and the Cauchy Riemann Equations

Contour Integrals and Cauchy's Theorem

Here are some notes prepared by someone who taught this class a few years ago. Note that the treatment is somewhat different from the notes above. I only provide it as a resource for those who want to see things from another perspective. It has some nice exercises too for those who want extra practice. However, you are not required to know this material except to the extent that it is represented by the notes above or by my lectures.

Complex Variables: Lecture 1

Complex Variables: Lecture 2

If you feel like you would benefit from additional reading material, here is an online book on complex analysis:

Complex Analysis by George Cain

It is pretty readable and has a number of good exercises for practice.

**Prerequisites:** All material covered in Calculus I-III (Stewart chapters 1-14, except those involving differential equations) will be assumed. Knowing the theory of integration in one variable (very) well will help you generalize to our multivariable setting.