Multivariable Calculus Workbook Developed By: Jerry Morris, Sonoma State University

Multivariable Calculus Workbook Developed by: Jerry Morris, Sonoma State University

A Companion to Multivariable Calculus McCallum, Hughes-Hallett, et. al. c Wiley, 2007

Note to Students: (Please Read) This workbook contains examples and exercises that will be referred to regularly during class. Please purchase or print out the rest of the workbook before our next class and bring it to class with you every day.

1. To Purchase the Workbook. Go to the Sonoma State University campus bookstore, where the workbook is available for purchase. The copying charge will probably be between $10.00 and $20.00.

2. To Print Out the Workbook. Go to the Canvas page for our course and click on the link “Math 261 Workbook”, which will open the ﬁle containing the workbook as a .pdf ﬁle. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. If you do choose to try to print it, please leave yourself enough time to purchase the workbook before our next class in case your printing attempt is unsuccessful. 2 Sonoma State University Table of Contents

Course Introduction and Expectations ...... 3

Preliminary Review Problems ...... 4

Chapter 12 – Functions of Several Variables Section 12.1-12.3 – Functions, Graphs, and Contours ...... 5 Section 12.4 – Linear Functions (Planes) ...... 11 Section 12.5 – Functions of Three Variables ...... 14

Chapter 13 – Vectors Section 13.1 & 13.2 – Vectors ...... 18 Section 13.3 & 13.4 – Dot and Cross Products ...... 21

Chapter 14 – Derivatives of Multivariable Functions Section 14.1 & 14.2 – Partial Derivatives ...... 29 Section 14.3 – Local Linearity and the Diﬀerential ...... 34 Section 14.4 – Gradients and Directional Derivatives in the Plane ...... 38 Section 14.5 – Gradients and Directional Derivatives in Space ...... 44 Section 14.6 – The Chain Rule ...... 48 Section 14.7 – Second-Order Partial Derivatives ...... 51

Chapter 16 – Integration Section 16.1 – The Deﬁnite Integral of a Function of Two Variables ...... 54 Section 16.2 – Iterated Integrals ...... 60 Section 16.3 – Triple Integrals ...... 65 Section 16.4 – Integrals in Polar Coordinates ...... 72 Section 16.5 – Integration in Cylindrical and Spherical Coordinates ...... 78

Chapter 17 – Parameterized Curves Section 17.1 – Parameterized Curves ...... 86 Section 17.2 – Motion, Velocity, and Accleration ...... 94 Section 17.3 – Vector Fields ...... 98

Chapter 18 – Line Integrals Sections 18.1 & 18.2 – Line Integrals ...... 101 Section 18.3 – Gradient Vector Fields and Path Independent Fields ...... 109 Section 18.4 – Green’s Theorem ...... 119

Chapter 19 – Flux Integrals Section 19.1 & 19.2 – Flux Integrals ...... 123

Chapter 20 – Calculus of Vector Fields Section 20.1 – The Divergence of a Vector Field ...... 139 Section 20.2 – The Divergence Theorem ...... 145 Section 20.3 – The Curl of a Vector Field ...... 151 Section 20.4 – Stokes’ Theorem ...... 158 Multivariable Calculus Workbook, Wiley, 2007 3 Multivariable Calculus – An Introduction

Welcome to Math 261, the world of calculus in 3 dimensions! In this course, we take many of the concepts from first-semester calculus, such as derivatives and integrals, and extend them to higher dimensions. Since the focus and expectations in this course are somewhat different than those of previous calculus courses you’ve taken, I want to take a few minutes to specifically point some of these out. 1. Unlike calculus 2 and parts of calculus 1, this course is mainly concept-driven rather than procedure-driven. In other words, understanding the practical and geometric significance of a calculation is often just as important as knowing how to do the calculation. For example, knowing how to calculate a derivative is nice, but understanding that it represents a rate of change is more useful in the real world. Therefore, there will be a lot problems in this class where you will need to choose for yourself an appropriate calculation method based on your understanding of the concept behind the method. This ability should develop naturally over the course of the semester provided that you take seriously the “give an explanation” portion of homework problems. In most cases, simply memorizing how to do a calculation but having no understanding of its significance will not get you very far. So don’t be afraid to ask me those “why” types of questions early and often both inside and outside of class; the more curious you are about the deeper meaning behind what we do, the easier things will be for you! 2. Unlike some other calculus textbooks, this textbook is intended to be read by students both in the worked out examples and in the explanations between the examples. When you are stuck on a homework problem and are looking for assistance in the textbook, you won’t always find a worked-out example exactly like it; rather, you may have to read the section and try to gain a deeper understanding of the big picture before you can attempt the problem. While I understand that this can be frustrating and time consuming, the goal is for you to gain a deeper understanding of the concepts in the course, and this understanding cannot be obtained by just copying the steps of similar worked-out examples. Keep in mind that making a sincere effort to read the book is important to your success in the course. Also keep in mind that you are not expected to understand everything in the textbook on your own! So don’t be shy about asking for help when you are stuck, this is a normal part of the course that I am happy to help you with. 3. Due to the amount of material and the time that it takes to cover it, more is expected of you outside of class in multivariable calculus than in previous calculus courses. Since a typical problem in this course tends to take a long time, there will not be time to do as many different types of examples together in class as we could in a previous calculus course. Therefore, it will be important for you to carefully study examples in the textbook, to keep up with the list of practice problems I ask you to work through, and to be diligent about asking for help when you need it. This outside-of-class involvement in the course will be critical for you to be successful in the course. 4. One thing you will immediately notice about the exercises in our textbook is that a lot of them call for detailed explanations. While some students tend to write off such questions as being “not that important, since the calculations are what really matter in a math class”, I couldn’t disagree more! Mathematical calculations are of no use if their significance and connection to the real world cannot be explained. Therefore, I will expect you to show work when you do calculations and to give complete explanations when you are asked to explain or describe something. In particular, here are some specific grading criteria for homework, quiz, and exam items:

• On any question that contains the key words “explain,” “describe,” “why,” or other words that call for an explanation or interpretation, answer the question in complete sentence form. If your answer to the question involves quantities with units (like meters or seconds, for instance), make sure the units are included in your answer. • Show work on all questions that call for a calculation, regardless of whether or not the instructions in the problem ask you to show your work. Also make it clear what your final answer is; circling your answer is one way of doing this. Remember that your intermediate comments and calculations are just as important for grading purposes as your final answer. • On homework problems that you turn in for grading purposes, make sure that your work is neat and easy to follow. Work should not be crammed into small spaces, and items should not be scribbled out on the final draft that you turn in. Many problems require a lot of thought and trial and error, and some will take more space than you first expected. You may find that you need to do the problems first on scratch paper and then turn in a neater, rewritten final version. My policy is to warn you once about neatness and organization and to dock points if it continues. (continued on next page −→) 4 Sonoma State University In conclusion, I want to remind everyone that I am always happy to help you with any problems you might be having with the course, and I hope you will feel comfortable approaching me when you need this help. I look forward to working with all of you and hope that we will all have a great semester!

Note. The following problems deal with review topics that should already be familiar to you from previous calculus and algebra courses. Being able to do problems similar to those below eﬃciently will be important to your success in Math 261.

1. Find the derivatives of each of the following with respect to x. You may assume that a is a constant: 2 2 1 1 (a) y = sin(xeax ) (b) y = sin(aexa ) (c) y = (d) y = x + 2a2 a + 2x2 2. Calculate the following integrals:

Z e 1 Z 1 + e2x (a) 2 dx (d) x dx 1 x(1 + ln x) e Z Z ex (b) xex dx (e) dx 1 + e2x √ Z sin( x) Z (c) √ dx (f) (1 + ax2)2 dx x

3. Find the exact value of the area bounded by the curves y = ln x, y = 2, the x-axis, and the y-axis. 4. The kinetic energy, E, in Joules, of a moving object with ﬁxed mass is given by E = f(v), where v is the speed of the object in meters per second. (a) Give a practical interpretation of f(5) in the context of this problem. Include all appropriate units. (b) Give a practical interpretation of f 0(5) in the context of this problem. Include all appropriate units. x 0.0 0.2 0.4 0.6 0.8 1.0 5. f(x) 2.3 2.9 3.4 3.7 4.0 4.2

Z 1 (a) Approximate f(x)dx. 0 (b) Approximate f 0(0.4) and f 0(0.8).

6. A helicopter ﬂies along a north-south straight line. Throughout the ﬂight, its velocity is recorded. Velocity 500 to the north is recorded as positive and to the south is recorded as negative. The velocity graph is given to the 300 right.

(a) Does the helicopter reverse its direction during the feet per minute 100 recorded ﬂight? If so, at what approximate time(s)?

(b) Does the helicopter change its velocity during the recorded flight? If so, approximately when is it flying slowest and when is it flying fastest? 10 20 30 40 50 60 time (in minutes)

(c) Does the helicopter change its acceleration during the recorded flight? If so, approximately when is it accelerating the most? (d) Does the helicopter ever decelerate during the recorded flight? If so, approximately when is it decelerating the most? (e) Approximately how far does the helicopter fly during the first 20 minutes of its flight? Multivariable Calculus Workbook, Wiley, 2007 5 Sections 12.1 – 12.3: Functions, Graphs, and Contours

Example 1. Plot the points P = (0, 0, 2),Q = (3, 4, 0),R = (0, 4, 5), and S = (3, 4, 5) on the coordinate axes to the right. z

y x

Example 2. Consider the functions f(x, y) = x2 + y2 and g(x, y) = 3. Evaluate f at several values of x and y, and sketch a graph of g.

Example 3. Windchill temperature is a tem- w \ T 35 30 25 20 15 10 5 0 perature which tells you how cold it feels as a result of the combination of wind and tempera- 5 31 25 19 13 7 1 -5 -11 ture. Let C = f(w, T ), where C is the windchill 10 27 21 15 9 3 -4 -10 -16 temperature (in degrees Fahrenheit) that is asso- 15 25 19 13 6 0 -7 -13 -19 ciated with a wind speed of w miles per hour and 20 24 17 11 4 -2 -9 -15 -22 a temperature of T degrees Fahrenheit. A table of 25 23 16 9 3 -4 -11 -17 -24 values for the function f is given to the right:

(a) Evaluate and interpret f(20, 5).

(b) How fast does the wind need to blow for it to feel like −10◦F when the air temperature is really 5◦F? 6 Sonoma State University Methods for Analyzing Functions of Two Variables

Deﬁnitions. Let z = f(x, y) be a function of two variables, x and y, and let c be a constant. 1. The curves z = f(c, y) and z = f(x, c) are called cross-sections of f(x, y). Geometrically, we get cross-sections by intersecting the graph of f with the planes x = c and y = c.

2. The curve c = f(x, y) is called a contour, or level curve of f(x, y). Geometrically, we get contours by intersecting the graph of f with the plane z = c.

Example. Let z = f(x, y) = y2 − x2. Find equations for the cross-sections z = f(1/2, y), z = f(x, −1/2), and for the contour at z = 0.3. The pictures below illustrate these three curves.

x -1 -0.5 0 0.5

0.5 z 0

-0.5

-1 -1 -0.5 0 y 0.5 1

x -1 -0.5 0 0.5 1 1

0.5 z 0

-0.5

-1 -1 -0.5 0 0.5 y 1

x -1 -0.5 0 0.5 1 1

0.5

z 0

-0.5

-1-1 -0.5 0

y 0.5 1 Multivariable Calculus Workbook, Wiley, 2007 7 Example. To the right is the graph of z = f(x, y) = x2 + y2 together with the planes z = 1, z = 5, and z = 9. 10 (a) Find equations for the contours associated with the planes in the diagram to the right. 8

0 -4 -4 -2 -2 0 0

2 2

4 4

(b) To the right is a contour plot of z = x2 + y2. Note that the circles in the plot are not equally spaced. What information 4 does this give you about the function z = x2 + y2? 3 2 1

y 0 -1 1 5 -2 9 13 -3

-3 -2 -1 0 1 2 3 4 x 8 Sonoma State University Exercises

1. Let T = f(d, t), where T is the temperature, in degrees Fahrenheit, of a room t minutes after a heater is turned on, at a distance of d meters from the heater.

(a) Is T an increasing or a decreasing function of t? Ex- plain. Sketch possible cross sections of f for d = 1, 2, and 3 meters.

(b) Is T an increasing or a decreasing function of d? Ex- plain. Sketch possible cross sections of f for t = 0, 5, and 10 minutes.

2. Make a contour plot for the function z = 2y − x with at least 3 diﬀerent contours. Label each contour with the corresponding value of z. Multivariable Calculus Workbook, Wiley, 2007 9

3. Given below are the formulas for several functions of two variables, followed by their contour plots and graphs. Match each formula to its corresponding graph and to its corresponding contour plot.

p 4 (a) f(x, y) = x2 + y2 (b) f(x, y) = x2 + y2 (c) f(x, y) = x2 + y2 p (d) f(x, y) = sin( 3x2 + 3y2) (e) f(x, y) = sin2 x sin2 y (f) f(x, y) = cos(xy/3)

(I) (II) 3 3

2 2

1 1

y 0 y 0

-1 -1

-2 -2

-3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x x

(III) (IV) 3 3

2 2

1 1

y 0 y 0

-1 -1

-2 -2

-3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x x

(V) (VI) 3 3

2 2

1 1

y 0

- 1 -1

-2 -2

-3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x x 10 Sonoma State University

(A) (B)

z 1

y x

3 3

y y x x

(E) (F) 9 1

y x

y x Multivariable Calculus Workbook, Wiley, 2007 11 Section 12.4 – Linear Functions (Planes)

Deﬁnition. A function z = f(x, y) is called linear if it has a constant slope in each direction. The graph of a linear function is a plane in 3-dimensional space. z

y x

If a plane (i.e., a linear function z = f(x, y)) has slope m in the x-direction, slope n in the y-direction, and passes through the point (x0, y0, z0), then its equation is

z =

Example 1. An airline sells discount fares and regular Regular fares, f fares. Let d be the number of discount fares the airline 0 100 200 300 sells, let f be the number of regular (full) fares the airline sells, and let R be the resulting revenue, in thousands of 0 0 24 48 72 dollars. The table to the right gives R as a function of d Discount 100 8 32 56 80 and f. Find a formula for R. fares, d 200 16 40 64 88 300 24 48 72 96 12 Sonoma State University Example 2. Find the equation of the plane that goes through the points (4, 0, 0), (0, 3, 0), and (0, 0, 2).

Example 3. Which of the tables of values below could represent linear functions? Explain.

x \ y 0 5 10 1 2 4 6 5 4 8 12 9 8 16 32

x \ y 0 3 6 0 1 -4 -9 2 4 -1 -6 4 7 2 -3 6 10 5 0

x \ y 0 1 2 3 0 1 1 1 1 1 2 2 2 2 2 1 1 1 1 3 2 2 2 2 Multivariable Calculus Workbook, Wiley, 2007 13 Example 4. Which of the following contour plots could represent linear functions? For those that could be linear, ﬁnd a possible formula for the function. y 4 3 10 2 8 1 6 x −1 4 −2 −3 2 0 −4 −4 −3 −2 −1 1 2 3 4

y 4 3 0 2 1 2 x −1 4 6 −2 −3 −4 −4 −3 −2 −1 1 2 3 4

y 4 3 2 6 3 3 6 1 x −1 −2 −3 −4 −4 −3 −2 −1 1 2 3 4 14 Sonoma State University Section 12.5 – Functions of Three Variables

Preliminary Example A. Find an equation for a sphere of radius r centered at the origin.

Preliminary Example B. Sketch graphs of the following three surfaces in three-dimensional space: x2 + y2 = 4, x2 + z2 = 9, and y = x2.

Example 1. The temperature on a 6 meter by 6 meter metal plate at the point (x, y) is given by T (x, y) = x2 + y2 degrees Celsius. Discuss the signiﬁcance of the objects shown below in the context of this problem.

T 3 2 7 4 1 1

y 0 -1 -2

y -2 -1 0 1 2 3 x x Multivariable Calculus Workbook, Wiley, 2007 15 Example 2. A block of ice is located at the point (0, 0, 0). The temperature at the point (x, y, z) is given by 1 2 2 2 T (x, y, z) = 4 (x + y + z ) degrees Celsius, where x, y and z are measured in meters. (a) How many dimensions would be required to represent the graph of T ?

(b) Pictured to the right are the surfaces T (x, y, z) = 1,T (x, y, z) = 5, and T (x, y, z) = 9. Find formulas for these surfaces and explain their signiﬁcance in the context of this problem.

Deﬁnition. Let w = f(x, y, z) be a function of three variables. A level surface of f is a surface of the form f(x, y, z) = c, where c is constant. The function f can be represented by the family of level surfaces obtained by letting c vary. Exercises

1. A long, thin rope containing insect repellant is placed along the z-axis. The density of insects at a point (x, y, z) is given by p(x, y, z) = 5(x2 + y2) insects per cubic foot, where x, y and z are measured in feet. (a) What is the insect density at the point (1, 1, 2)? Does the density of insects increase or decrease as you move away from the z-axis? 16 Sonoma State University

(b) Describe the set of all points at which the insect density is 20 insects per cubic foot.

2. A company makes three types of cookies: shortbread, mint, and fudge swirl. The cost for producing s pounds of shortbread, m pounds of mint, and f pounds of fudge swirl is given by C = s + 2m + 3f dollars. (a) Sketch the portion of the level surface C = 12 that lies in the ﬁrst octant. (Hint: Label your 3 axes with s, m, and f, and start by ﬁnding the s, m, and f intercepts.)

(b) Explain the signiﬁcance of the level surface you drew in part (a) in the context of this problem. Multivariable Calculus Workbook, Wiley, 2007 17

3. Describe the level surfaces of each of the following functions.

2 2 (a) f(x, y, z) = e−(x +y ) (b) g(x, y, z) = z − y (c) V (x, y, z) = ln(x2 + y2 + z2) 18 Sonoma State University Sections 13.1 & 13.2 – Vectors

Deﬁnition. A vector is a quantity that has both a magnitude and a direction. Graphically, vectors are represented by arrows.

Preliminary Example A. Consider the vectors given to the right. You may assume that successive dotted lines are separated by one unit. w v p q

Preliminary Example B. Let ~a = 2~i+3~j and ~b = 4~i+~j. Sketch the vectors ~a, ~b, and ~a + ~b. You may assume that successive dotted lines are separated by one unit.

Information About Vectors ~ 1. For vectors a1~i + a2~j and b1~i + b2~j + b3k, we have q ~ ~ ~ ~ 2 2 Length of a1i + a2j = ka1i + a2jk = a1 + a2 q ~ ~ ~ ~ ~ ~ 2 2 2 Length of b1i + b2j + b3k = kb1i + b2j + b3kk = b1 + b2 + b3

2. Two vectors ~u and ~v are parallel if ~u = λ~v for some scalar λ. ~p 3.A unit vector is a vector of length 1. Given a nonzero vector ~p, the vector is a unit vector that k~pk points in the same direction as ~p. Multivariable Calculus Workbook, Wiley, 2007 19 Exercises 1. In the diagram to the right, sketch each of the following z vectors:

(a) ~u + ~v v (b) ~u + ~v + ~w (c) ~u − ~v

u w y x

2. Give some examples of scalar quantities, and give some examples of vector quantities.

3. A car is driving counterclockwise around a racetrack, accelerating on the straight stretches and slowing down for the curves so it doesn’t ﬂy oﬀ the track. Use qualitative reasoning to draw in possible velocity vectors for the car at the points labeled on the track to the right. 20 Sonoma State University

4. Three ropes are ﬁrmly attached to an indestructible holiday fruitcake in the shape of a donut. A group of chemists pulls one of the ropes with a force of 300 pounds due east, while a group of mathematicians pulls a second rope southeast with a force of 400 pounds. With what force (give components) would a group of physicists have to pull the third rope so that the fruitcake remains stationary? Multivariable Calculus Workbook, Wiley, 2007 21 Sections 13.3 & 13.4 – Dot and Cross Products

~ ~ Algebraic Deﬁnition of the Dot Product. Let ~v = v1~i+v2~j +v3k, and let ~w = w1~i+w2~j +w3k. Then the dot product, ~v · ~w, of ~v and ~w is deﬁned as follows:

~v · ~w = v1w1 + v2w2 + v3w3.

Example 1. Calculate ~v · ~w if ~v =~i + 2~j + 3~k and ~w = 3~i − 2~j + ~k.

Example 2. Calculate ~u ·~v for each of the following:

u v v u u

~u = ~i + 2~j ~u = 3~i +~j ~u = 3~i +~j ~v = 3~i +~j ~v = −~i + 3~j ~v = −2~i − 2~j ~u ·~v = ~u ·~v = ~u ·~v =

~ ~ Algebraic Deﬁnition of the Cross Product. Let ~v = v1~i+v2~j+v3k, and let ~w = w1~i+w2~j+w3k. Then the cross product, ~v × ~w, of ~v and ~w is deﬁned as follows: ~ ~v × ~w = (v2w3 − v3w2)~i + (v3w1 − v1w3)~j + (v1w2 − v2w1)k.

Calculation of the Cross Product: 22 Sonoma State University

Example 3. Let ~u =~i + 2~j and ~v = −~i + 2~j + ~k. Find ~u × ~v and ~v × ~u and draw them in on the provided diagram. z H2,-1,4L

Ó v Ó u y x H-2,1,-4L

Section 13.3 & 13.4 Information

Theorem (Geometric Deﬁnition of the Dot Product)

~v · ~w = k~vkk~wk cos θ,

where θ is the angle between ~v and ~w and 0 ≤ θ ≤ π.

Theorem (Geometric Deﬁnition of the Cross Product. Let ~v and ~w be vectors in 3- dimensional space. Then ~v × ~w has the following three properties: 1. ~v × ~w is perpendicular to both ~v and ~w. 2. ~v × ~w points in the direction determined by v the right-hand rule. w 3. The length of ~v × ~w equals the area of the parallelogram determined by ~v and ~w. v

Notes on Dot and Cross Products: 1. The cross product of two vectors is a , but the dot product of two vectors is a . 2. To ﬁnd a vector perpendicular to both ~v and ~w, we calculate . 3. The vectors ~v and ~w are orthogonal if and only if . Multivariable Calculus Workbook, Wiley, 2007 23 Normal Vectors and the Equation of a Plane

Deﬁnition. A normal vector of a plane is any vector ~n that is perpendicular to the plane.

P0 = (x0 , y0 , z0 )

P = (x, y, z)

Various Ways of Writing the Equation for a Plane

~ 1. If ~n = a~i + b~j + ck is a normal vector to a plane and (x0, y0, z0) is a point on the plane, the equation for the plane is given by

2. If m is the slope of a plane in the x-direction, n is the slope of the plane in the y-direction, and (x0, y0, z0) is a point on the plane, the equation for the plane is given by 24 Sonoma State University Examples and Exercises

1. Let ~v = 3~i + 3~j + 4~k and ~w = 3~i + 4~k. (a) Find the measure of the angle between ~v and ~w.

(b) Find a unit vector that is perpendicular to both ~v and ~w.

(c) Find the value of t for which the vector t~i + 2t~j + 3~k is perpendicular to ~v. Multivariable Calculus Workbook, Wiley, 2007 25

2. Which of the following are equations of planes? For those that are, write down a normal vector for the plane.

(a)2 x + 3y − 5z = 4 (c) y = 2x − 1 (b)2 x2 + 3(y − 2)2 − 4(z − 1)2 = 0 (d) f(x, y) = 4 − 2y

3. Which of the following planes are parallel to one another? Which are perpendicular to one another?

(a)4 x + 6y − 2z = 4 (c)2 x + 3y + z = 4 (b) f(x, y) = 2x + 3y (d)4 x − 5y + 7z = 2 26 Sonoma State University

4. Find the equation of the plane that contains the points (1, 2, −1), (2, 3, 0), and (3, −1, 2).

5. Find the equation of the plane parallel to z = 1 − x + 6y that contains the point (1, 1, 1). Multivariable Calculus Workbook, Wiley, 2007 27

6. Find the equation of the plane that contains the point (1, 0, −1) and the line x − 2y = 4 in the xy-plane.

7. Let P1 be the plane x − 3y + z = 1.

(a) Does the point (1, 1, 3) lie on the plane P1?

(b) Find two distinct points that lie on P1. 28 Sonoma State University

(d) If you did part (c) above correctly, ~n ·~v should equal 0. Check that this is true; then, explain geometrically why it makes sense that ~n ·~v = 0.

Deﬁnition. If a constant force F~ is applied to an object, resulting in some displacement d,~ then the work done, W, on the object by the force is given by

W = F~ · d.~

8. A crate is hauled 6 meters up a ramp under a constant force of 300 Newtons applied at an angle of 25◦ to the ramp. Use the above deﬁnition to calculate the work done. F

25o D Multivariable Calculus Workbook, Wiley, 2007 29 Sections 14.1 & 14.2 – Partial Derivatives

Example 1. The temperature, T, in degrees Celsius, at a point (x, y) on an 8 meter by 8 meter metal plate is given by the function T = f(x, y). The positions described by x and y are both measured in meters. Given below is a table and a graph that both represent the temperature function f. Our goal is to discuss the problem of ﬁnding the rate at which the temperature of the plate changes, at a particular point, as you move away from the point in either the x-direction or the y-direction.

x, meters T -4 -3 -2 -1 0 1 2 3 4 -4 28 35 40 43 44 43 40 35 28 -3 35 42 47 50 51 50 47 42 35 -2 40 47 52 55 56 55 52 47 40 -1 43 50 55 58 59 58 55 50 43 y, 0 44 51 56 59 60 59 56 51 44 meters 1 43 50 55 58 59 58 55 50 43 2 40 47 52 55 56 55 52 47 40 3 35 42 47 50 51 50 47 42 35 y 4 28 35 40 43 44 43 40 35 28 x

Partial Derivatives of f With Respect to x and y For all points at which the limits exist, we deﬁne the partial derivatives at the point (a, b) by

f(a + h, b) − f(a, b) fx(a, b) = Rate of change of f with respect to x at (a, b) = lim , h→0 h f(a, b + h) − f(a, b) fy(a, b) = Rate of change of f with respect to y at (a, b) = lim . h→0 h

If we let a and b vary, we have the partial derivative functions fx(x, y) and fy(x, y).

Alternative Notation for Partial Derivatives. Let z = f(x, y). 30 Sonoma State University Examples and Exercises.

1. Let T = f(x, t), where T is the temperature, in degrees Celsius, of a room t (minutes)

t minutes after a heater is turned on, at a distance of x meters from the 60 30 heater. A contour plot for f is given to the right. 50 40 25 (a) Estimate fx(10, 25). Then, give a practical interpretation of this quantity. 30 20 20 15 10 10

5 10 15 20 25 30 x (meters)

(b) Estimate ft(10, 25). Then, give a practical interpretation of this quantity.

2. Let P = f(A, r, N), where P represents the monthly payment for a mortgage, A represents the initial amount borrowed, in dollars, r represents the annual interest rate, in percent, and N represents the number of years before the loan is paid oﬀ. (a) Give a practical interpretation of the statement f(92000, 14, 30) = 1090.8.

(b) Give a practical interpretation of the statement fr(92000, 14, 30) = 72.82. Multivariable Calculus Workbook, Wiley, 2007 31

∂P (c) Is ∂N positive or negative? Justify your answer.

3. Given to the right is a table of values for the function x \ y 0 2 5 z = f(x, y). Assuming that f is diﬀerentiable, estimate 1 1 2 4 f (3, 2) and f (3, 2). x y 3 -1 1 2 6 -3 0 0

4. Given to the right is the graph of the function f(x, y) = z 1 −x/2 2 ye on the domain 0 ≤ x ≤ 4 and 0 ≤ y ≤ 4. (a) Use the graph to the right to rank the following quantities in order from smallest to largest: fx(3, 2), fx(1, 2), fy(3, 2), fy(1, 2), 0 y

(b) Use algebra to calculate the numerical values of the four derivatives from part (a) and conﬁrm that your answer to part (a) is correct. 32 Sonoma State University 5. For each of the following functions, calculate the partial derivatives with respect to each independent variable.

mv2 (a) f(x, y, z) = z ln(y cos x) (d) f(m, r, v) = r (b) z = x2y + y2x a √ (e) z = e sin(a + b) (c) f(x, y) = xe xy Multivariable Calculus Workbook, Wiley, 2007 33

6. When riding your bike on a still day during an Iowa winter, the windchill temperature is a measure of how cold you feel as a result of the induced breeze caused by your travel. If W represents windchill temperature (in ◦F) that you experience, then W = f(T, v), where T is the actual air temperature (in ◦F) and v is your speed, in meters per second. Match each of the practical interpretations below and to the left with a mathematical statement that most accurately describes it below and to the right. For the remaining mathematical statement, give a practical interpretation.

(i) “The faster you ride, the colder you’ll feel.” (a) fT (T, v) > 0 (ii) “The warmer the day, the warmer you’ll feel.” (b) f(0, v) ≤ 0

The equation of the tangent plane to a surface z = f(x, y) at the point (a, b, f(a, b)) is given by the equation

z = f(a, b) + m(x − a) + n(y − b),

where

m = n =

Tangent Plane Approximation to f(x, y) for (x, y) Near the Point (a, b) Provided f is diﬀerentiable at (a, b), we can approximate f(x, y) as follows:

f(x, y) ≈ f(a, b) + fx(a, b)(x − a) + fy(a, b)(y − b)

Note: The closer (x, y) is to (a, b), the more accurate the approximation will be.

2 2 Example 1. Let f(x, y) = (y + 1)ex +y . (a) Find the tangent plane to f at the point (0, 0).

(b) Use the tangent plane to approximate f(0.1, 0.1) and compare your answer with the exact value of f(0.1, 0.1). Multivariable Calculus Workbook, Wiley, 2007 35 Example 2. In a room, the temperature is given by T = f(x, t) degrees Celsius, where x is the distance from a heater (in meters) and t is the elapsed time (in minutes) since the heat has been turned on. A person standing 3 meters from the heater 5 minutes after it has been turned on observes the following: (1) The temperature is increasing by 1.2◦C per minute, and (2) Walking away from the heater, the temperature decreases by 2◦C per meter as time is held constant. Estimate how much cooler or warmer it would be 2.5 meters from the heater after 6 minutes.

The Diﬀerential of a Function z = f(x, y): The diﬀerential, df (or dz) at a point (a, b) is given by df = fx(a, b) dx + fy(a, b) dy. Conceptually, df represents the approximate change in the output values of f if x is changed by dx and y is changed by dy.

Example 3. Given to the right is the graph of a surface z z = f(x, y) and its tangent plane at (a, b) (the plane that f T contains the point S). Find the z-coordinates of R, S, and S ∆f T and use them to ﬁnd formulas for df and ∆f. df (a,b, f(a,b)) R

y = dx ∆x P x (a,b,0) ∆y = dy

Changes in a function z = f(x, y): Let z = f(x, y) be a function. Conceptually, we have:

∆f = the exact change in f between (a, b) and (a + ∆x, b + ∆y). df = the approximate change in f between (a, b) and (a + ∆x, b + ∆y).

Computationally, we have

∆f =

df = 36 Sonoma State University Exercises. 1. (a) Find the diﬀerential of g(u, v) = u2 + uv.

(b) Use your answer to part (a) to estimate the change in g as you move from (1, 2) to (1.2, 2.1).

2. A right circular cylinder has a radius of 50 cm and a height of 100 cm. Use diﬀerentials to estimate the change in volume of the cylinder if its height and radius are both increased by 1 cm. Multivariable Calculus Workbook, Wiley, 2007 37

3. The relative humidity, measured in percent, is given by the function H = f(T,D) percent, where T is the air temperature and D is the dew-point temperature, both in degrees Fahrenheit. Note that the contours below give constant dew-point temperatures for various combinations of temperature and relative humidity; in other words, the contours are labeled with values of D, not values of H. H 80

70 60

60 40 50 20 40 0 30 -20 20

T 10 20 30 40 50 60 70 80

(a) Find a linear approximation of H, which is most accurate for air that is nearly 80◦F with a dew-point of 60◦F.

(b) Use your linear approximation to estimate the relative humidity in Cedar Falls, Iowa, where the temperature is 85◦F and the dew-point temperature is 65◦F. 38 Sonoma State University Section 14.4 – Gradients and Directional Derivatives in the Plane

Definition. The gradient vector of a differentiable function f at a point (a, b), which is denoted either by “grad f(a, b)” or “∇f(a, b)”, is defined by

∇f(a, b) = fx(a, b)~i + fy(a, b)~j.

Preliminary Example. The temperature at a point (x, y) on a metal plate is given by 1 2 1 2 T (x, y) = 1 − 4 x − 4 y degrees Celsius, where x and y are both measured in centimeters. Level curves of T Graph of T y 2 z

0 0.25 P 1 0.5 0.75 Q

0 x

-1 x Q P y

-2 -2 -1 0 1 2

(a) Calculate ∇T and use your formula to ﬁll in the following table. Then, sketch your vectors on the diagram. √ √ (a, b) (−1, 0) (−2, 0) (1, 1) (0, 2) (0, − 2) ∇T (a, b)

(b) Approximately how fast does the temperature change at the point P = (1, 1) as you move toward the point Q = (1.66, 0.5)? (See diagram above.) Include units with your answer. Multivariable Calculus Workbook, Wiley, 2007 39

Theorem. Let ~u = u1~i + u2~j be a unit vector. If f is a diﬀerentiable function, then the directional derivative, f~u, can be calculated as follows:

f~u(a, b) = Rate of change of f at (a, b) in the direction of ~u =

1 2 1 2 Example 1. Let T (x, y) = 1 − 4 x − 4 y be the temperature function from Preliminary Example. Find the derivative of T at P = (1, 1) as we move in the direction of Q = (1.7, 0.5). Include units, and compare with your answer to part (b) of the Preliminary Example. 40 Sonoma State University Example 2. The ﬂuid pressure, in atmospheres, at the bottom of a body of water of varying depths is given by x2y P (x, y) = 1 + , where x and y are measured in meters. 100 (a) Find ∇P (1, 2) and k∇P (1, 2)k and interpret them in the context of this problem.

(b) Find P~u(1, 2), where ~u is a unit vector in the direction of 3~i −~j. Also give an interpretation of your answer.

Notes on Gradients and Directional Derivatives

For all of the following, let f(x, y) be a diﬀerentiable function.

1. The vector ∇f(a, b) points in the of f at (a, b).

2. The scalar k∇f(a, b)k gives the of f as you move away from (a, b). Multivariable Calculus Workbook, Wiley, 2007 41 Examples and Exercises y 1. A heat source lies at the origin of the (x, y)-plane, and the temperature 2 2 2 at any point (x, y) is given by T (x, y) = 100/(1 + x + 2y ) degrees 1.5 Celsius (see contour diagram to the right), where x and y are measured 15 1 in meters. 30 45 0.5 (a) Calculate ∇T (1, 1) and give a practical interpretation of your answer. 0 x -0.5 -1 -1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2

(b) Calculate k∇T (1, 1)k and give a practical interpretation of your answer.

(c) Let ~u = √2 ~i + √1 ~j. Calculate T (1, 1), and give a practical interpretation of your answer. 5 5 ~u 42 Sonoma State University

2. You are standing at the point (1, 1, 3) on the hill whose equation is given by z = 5y − x2 − y2. (a) If you choose to climb in the direction of steepest ascent, what is your initial rate of ascent relative to the horizontal distance?

(b) If you decide to go straight northwest, will you be ascending or descending? At what rate?

3. Given to the right is a contour diagram for a diﬀerentiable function h. 5 Use the diagram to answer the questions that follow.

(a) Estimate ∇h(2, 1). Clearly demonstrate what you are doing to 4 arrive at an answer. 4 3 8 y 12 2 16 1 20

1 2 3 4 5 x Multivariable Calculus Workbook, Wiley, 2007 43

(b) Estimate the directional derivative of h at (2, 1) in the direction 5 of the point (4, 0). 4 4 3 8 y 12 2 16 1 20

1 2 3 4 5 x

4. Consider the graphs of f and g shown to the Figure 1. z = f(x, y) Figure 2. z = g(x, y) right. You may assume that each square in z the xy-plane has side length 1. z

(a) Use Figure 1 to determine whether H-2, -2, 3L f~u(1, 1) is positive, negative, or zero for each of the following values of ~u: (i) √1 (~i +~j) (ii) √1 (~i −~j) H2, 2, 3L 2 2 (iii) √1 (−~i +~j) (iv) √1 (−~i −~j) y y 2 2 x x

(b) Let ~u = √1 (~i −~j) and ~v = √1 (−~i +~j). Use Figure 2 to decide which is larger, g (1, 1) or g (1, 1). 2 2 ~u ~v Explain. 44 Sonoma State University Sections 14.5 – Gradients and Directional Derivatives in Space

Example 1. A thin column of ice lies along the z-axis, and the tem- Figure 1. T (x, y, z) = 20 2 2 perature at any point in space is given by T (x, y, z) = 5(x + y ) degrees z Celsius, where x, y, and z are measured in meters.

(a) Is the surface shown in Figure 1 the graph of the function T ? If not, P then what is it? Q y x

√ √ (b) Suppose that P = (0, 2, 2) and Q = ( 2, 2√, 0)√ are points on the level surface T (x, y, z) = 20 (see Figure 1). Calculate and draw in ∇T (0, 2, 2) and ∇T ( 2, 2, 0). Then, give a practical interpretation of both of these quantities.

(c) How fast does the temperature change as you move away from P = (0, 2, 2) in the direction of R = (0, 3, 3)? Include units with your answer. Multivariable Calculus Workbook, Wiley, 2007 45 Example 2. Find the equation of the tangent plane to the surface 2ez = cos z + x3yz + xy3 at the point (1, 1, 0). 46 Sonoma State University Summary of Gradients

z = f(x, y) w = g(x, y, z)

• The graph of f lives in 3-dimensional space. • The graph of g lives in space. The domain of f lives in 2-dimensional space. The domain of g lives in space.

Facts about ∇f(x, y) Facts about ∇g(x, y, z)

• Gradient lives in 2-dimensional space. • Gradient lives in space. • Gradient points in the direction that causes the • Gradient points in the direction that causes the fastest increase in the value of z. fastest increase in the value of . • k∇f(x, y)k gives the magnitude of the fastest • k∇g(x, y, z)k gives the magnitude of the fastest increase in z. increase in .

•∇ f(x, y) is orthogonal to the level curves of f. •∇ g(x, y, z) is normal to the level of g. • If ~u = u1~i + u2~j is a unit vector, then ~ • If ~u = u1~i + u2~j + u3k is a unit vector, then f~u(a, b) = ∇f(a, b) ·~u. g~u(a, b, c) = .

Examples and Exercises 1. The temperature in a spherical portion of space is given by T (x, y, z) = 2x2yz degrees Fahrenheit, where x, y, and z are measured in meters. How fast does the temperature change as you move away from the point (1, 1, 2) in the direction of the origin? Multivariable Calculus Workbook, Wiley, 2007 47

2 1 2 2 2. Let f(x, y) = xy − 2 x y . 2 1 2 2 f(x, y) = xy − 2 x y (a) Is the vector ∇f(x, y) normal to the graph of f? Explain. z

(b) Find two vectors, one that is normal to the graph of f at (1, 0) and one that is normal to the graph of f at (1, 1). Two such vectors x are illustrated in the ﬁgure to the right.

3. Find the equation of the tangent plane to the surface 4x2 + y2 + z2 = 24 at the point (2, 2, 2). 48 Sonoma State University Section 14.6 – The Chain Rule

Preliminary Example. Let T = f(x, y) represent the temperature at the point (x, y) in the plane, and let x = g(t) and y = h(t) represent the x and y-coordinates of the particle, in meters, after t seconds. Assuming that dT f, g, and h are all diﬀerentiable, derive a formula for . dt

Examples and Exercises

dz y 2 1. Find dt if z = xe , x = 2t, and y = 1 − t . Multivariable Calculus Workbook, Wiley, 2007 49

∂z ∂z 2 2 2 3 3 2 2. Find ∂u and ∂v if z = ln(xy), x = (u + v ) , and y = (u + v ) .

3. Suppose that z = f(x, y) is a diﬀerentiable function of x and y, where x = g(r, s, t) and y = h(r, s, t) are both diﬀerentiable functions of r, s, and t. Draw a dependence diagram and then write out the corresponding formula(s) for the Chain Rule. 50 Sonoma State University

4. Suppose that w = f(x, y, z) is a diﬀerentiable function of x, y, and z, where x = g(t), y = h(t), and z = k(t) are all diﬀerentiable functions of t. Draw a dependence diagram and write out the corresponding formula(s) for the Chain Rule.

5. The pressure, P, (in kilopascals), volume, V, (in cm3), and temperature, T, (in Kelvin) of n moles of an ideal gas are related by PV = nRT, where R = 8.314 × 103 is a constant of proportionality related to an ideal gas. A sample container having volume 100 cm3 contains 0.002 moles of an ideal gas at room temperature, which is 300 K. If this sample is compressed at a rate of 2 cm3/min and heated at a rate of 1 K/min, how fast is the pressure of the gas changing? Multivariable Calculus Workbook, Wiley, 2007 51 Section 14.7 – Second-Order Partial Derivatives

Second-Order Partial Derivatives of z = f(x, y)

∂ ∂z ∂2z = = f ∂x ∂x ∂x2 xx ∂ ∂z ∂2z = = f ∂y ∂x ∂y∂x xy ∂ ∂z ∂2z = = f ∂x ∂y ∂x∂y yx ∂ ∂z ∂2z = = f ∂y ∂y ∂y2 yy

Example 1. Let f(x, y) = x2y + 3y2ex. Compute all second-order partial derivatives of f.

Example 2. Given to the right is a contour diagram for z = f(x, y). y Decide whether each of the following are positive, negative, or zero: fx(3, 3), fy(3, 3), fxx(3, 3), and fxy(3, 3). 5 6 7 8 9

(3, 3)

x 52 Sonoma State University Examples and Exercises 1. Given to the right is a contour diagram for z = f(x, y). Decide whether y each of the following are positive, negative, or zero: 2 (a) fx(P ) 3

(b) fy(P ) 4 (c) f (P ) 5 xx x (d) fyy(P ) P 6 (e) fxy(P ) Multivariable Calculus Workbook, Wiley, 2007 53

2. Let f(x, y) be the function whose values are given in the table to the x \ y 2 4 6 8 right. Estimate each of the following, making it clear from your work how you are obtaining your answers. 2 2 4 6 8 4 3 6 9 12 (a) f~v(4, 4), where ~v = 2~i + 4~j. 6 4 8 12 16 8 5 10 15 20 (b) fxy(4, 4) 54 Sonoma State University Section 16.1 – The Deﬁnite Integral of a Function of Two Variables

Preliminary Example. The resort community of Dinisville lies along the shore of Lake Fubini. Let ρ(x, y) be the population density, in people per square mile, at the point (x, y), where x and y are measured in miles (see diagram below). A table of values for the function ρ is given below.

y \ x 0 5 10 10 0 10 15 20 5 9 13 20 y 5 10 8 11 14 Lake 0 Fubini 0 5 10 x

Goal: Approximate the population of Dinisville for the rectangle 0 ≤ x, y ≤ 10. Multivariable Calculus Workbook, Wiley, 2007 55 Exercise. Consider a 4 meter by 3 meter metal plate of varying y density given by the function f(x, y) grams per square meter at 3 the point (x, y), where x and y are measured in meters. A contour diagram for the function f is given to the right. Using ∆x = 2 100 and ∆y = 1, estimate and interpret R f(x, y) dA, where R is the R 2 rectangle given by 0 ≤ x ≤ 4 and 0 ≤ y ≤ 3. 200 300 1 400 500 x 2 4 56 Sonoma State University

A Graphical Interpretation of Double Integrals

Let z = f(x, y) be a surface, and let R denote a rectangular region in the xy-plane. 1. If z ≥ 0 for all points (x, y) in R, then the integral z R f(x, y) dA represents R f(x,y) =

c d y a ∆y ∆x

b x

2. In general, if z is both positive and negative on R, R z then R f(x, y) dA represents

y x Multivariable Calculus Workbook, Wiley, 2007 57 The Average Value of a Function T

Example. The temperature, T, in degrees Celsius, at a point (x, y) on an 8 meter by 8 meter metal plate is given by the function T = f(x, y). The positions described by x and y are both measured in meters. A graph of T is given to the right. Goal: Find an expression for the average temperature of the plate. y x 58 Sonoma State University Exercises.

1. Let R denote the rectangle [2, 10]×[2, 6] in the xy-plane, and suppose that a thin metal plate having density y (cm) ρ(x, y) = x+y grams per cm2 occupies this rectangular 6 region (see diagram to the right). Using ∆x = ∆y = 2, estimate the mass of the metal plate. 4 2 x (cm) 2 4 6 8 10

2. Estimate the average density of the metal plate from Exercise 1. Multivariable Calculus Workbook, Wiley, 2007 59

3. The table below gives heights of a mound, where x and y y \ x -1 0 1 are measured in meters away from the peak of the mound. Find an overestimate and an underestimate of the volume -1 2 3 2 of the mound. 0 3 4 3 1 2 3 2

4. Let D be the region of the xy-plane that is inside the right hand half of the unit circle centered at the origin. Without calculating them, decide whether the following integrals are positive, negative, or zero. Explain your answers.

R 2 (a) D xy dA

R 2 (b) D x y dA 60 Sonoma State University Section 16.2 – Iterated Integrals

y d

x a b

Theorem. If R is the rectangle a ≤ x ≤ b and c ≤ y ≤ d, and f is a continuous function over R, then the integral of f over R exists and can be written as Z f(x, y) dA = . R Multivariable Calculus Workbook, Wiley, 2007 61 Examples and Exercises

1. Find the volume under the surface z = x2 + y2 and above the rectangle 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

2. Consider the triangular region R shown to the right. y (2,4) R (a) Calculate R 6xy dA by integrating in the “dy dx” order. R (b) Calculate R 6xy dA by integrating in the “dx dy” order.

x 62 Sonoma State University

3. A thin triangular metal plate is bounded by the lines y = x, x = 1, and y = 0. The charge density of the plate 2 is given by ρ(x, y) = ex Coulombs per cm2. (a) Find the total charge on the plate, in Coulombs. (b) What is the average charge density on the plate? Multivariable Calculus Workbook, Wiley, 2007 63

4. Set up, but do not evaluate, an integral giving the volume between the plane z = 4 and the bowl z = x2 + y2. 64 Sonoma State University

8 2 Z Z 4 5. By reversing the order of integration, ﬁnd the exact value of ex dx dy. (Hint: You’ll need to sketch √ 0 3 y the region of integration before you can successfully reverse the order.)

6. Set up, but do not evaluate, an iterated integral that gives the volume of the solid region that lies below the paraboloid z = x2 + y2 and above the region in the xy-plane bounded by the curves y = 8x2 and y2 = x. Multivariable Calculus Workbook, Wiley, 2007 65 Section 16.3 – Triple Integrals

Preliminary Example. Let ρ(x, y, z) give the den- z sity, in grams per cm3, of the solid prism shown on the right, where x, y, and z are measured in centimeters. x=a z=s

Goal: Find an expression for the mass of the prism. x=b

y z=r

y=c y=d x 66 Sonoma State University Examples and Exercises

1. A solid object is given by the region E in 3-space bounded by the plane x + y + z = 1, and above the square 1 1 region given by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2 . (a) Find the mass of the region E if the density is given by ρ(x, y, z) = 48y. (b) Set up, but do not evaluate, an integral that gives the volume of the region. Multivariable Calculus Workbook, Wiley, 2007 67 z 2. Let E be the solid region in the ﬁrst octant bounded by the planes z = 1−x and the plane y = 1. Suppose that x, y, and z are measured in centimeters 1 and that the density of E is given by ρ(x, y, z) = 6x grams per cm3. (a) Find the mass of the solid region E.

(b) Without calculating the integral, explain what the triple integral 1 1 R dV represents in the context of this problem. E y x 68 Sonoma State University

3. Set up, but do not evaluate, an iterated integral that gives the volume of solid region enclosed by the paraboloids z = x2 + y2 and z = 8 − x2 − y2. Multivariable Calculus Workbook, Wiley, 2007 69

4. Given to the right is the region of integration for the integral z 4 1 2−2y 2 Z Z Z z=2-2y √ f(x, y, z) dz dy dx. x 0 2 0 Rewrite this integral as an equivalent iterated integral in the “dx dy dz” order AND as an equivalent iterated integral in the “dy dx dz” order.

y 1 x !!!!x=2y 70 Sonoma State University

5. Set up an iterated integral that is equivalent to the triple R z integral E z dV, where E is the region bounded by the planes x = 0, y = 0, z = 0, y + z = 1, and x + z = 1.

x Multivariable Calculus Workbook, Wiley, 2007 71

R 6. Set up an iterated integral that is equivalent to the triple integral E x dV, where E is the region bounded by the paraboloid y = 3x2 + 3z2 and the plane y = 3. 72 Sonoma State University Section 16.4 – Integrals in Polar Coordinates

x y r θ

Cartesian Coordinates Polar Coordinates

Example 1. Each of the following represent either a polar point or curve. Plot and label each of these objects on the polar grid to the right. (a) The point P (2, π).

(b) The point Q(3, 7π/6).

Conversion Equations P(x, y) = P(r, θ )

(0, 0) Multivariable Calculus Workbook, Wiley, 2007 73 Example 2. Convert the following points or equations into polar form if they’re given in Cartesian form or into Cartesian form if they’re given in polar form.

(a) r = 2 (c) The polar point (2, π/6)

2 2 (b) x + y = 2x + 1 (d) The Cartesian point (3, −3)

Cartesian Summing Polar Summing y y d

c x a b x Region = {(x, y): a ≤ x ≤ b, c ≤ y ≤ d} Region = dA = dx dy dA = 74 Sonoma State University Examples and Exercises

1. Below, several regions R are shown. Decide whether to use polar coordinates or Cartesian (rectangular) R coordinates and write R f(x, y) dA as an iterated integral, where f is an arbitrary continuous function on R.

y y 3

x x −3 −1 1 3 −3

y 2 y

(3,3)

x −2 2 x

−1 (−3,−3) Multivariable Calculus Workbook, Wiley, 2007 75

√ R −x2−y2 2 2. Evaluate D e dA, where D is the region bounded by the semicircle y = 9 − x and the x-axis. Hint: Change to polar coordinates!

3. The population density of a circular city of diameter 10 varies linearly from 100 people per square mile at the center of the city to 0 people per square mile on the outer boundary. Set up an iterated integral that represents the total population of the city. 76 Sonoma State University

√ √ Z 2 Z 4−y2 4. Convert xy dx dy to polar coordinates and evaluate it. (Hint: First, sketch the region of 0 y integration.) Multivariable Calculus Workbook, Wiley, 2007 77

5. Use polar coordinates to ﬁnd the volume of the solid region that lies inside the sphere x2 + y2 + z2 = 81, outside the z cylinder x2 + y2 = 9, and above the xy-plane.

y x 78 Sonoma State University Section 16.5 – Integration in Cylindrical and Spherical Coordinates Cylindrical Coordinates.

Q(r,θ,z)

z θ r

Spherical Coordinates.

Q(ρ, θ, φ) φ ρ

Example 1. Describe and provide a rough sketch of each of the following surfaces. π 2 (a) r = 2 (b) θ = 4 (c) z = r Multivariable Calculus Workbook, Wiley, 2007 79 Example 2. Describe and provide a rough sketch of each of the following surfaces. π π (a) ρ = 3 (b) φ = 4 (c) φ = 2

Example 3. Convert each of the following equations from rectangular form into both cylindrical and spherical coordinates. Which of the three equations for each surface is simplest?

(a) x2 + y2 + z2 = 4

(b) x2 + y2 − 2z = 1 80 Sonoma State University R Evaluation of E f(x, y, z) dV As an Iterated Integral

1. In rectangular coordinates, use dV = 2. In cylindrical coordinates, use dV =

3. In spherical coordinates, use dV =

Examples and Exercises

1. Find the volume of the solid region bounded between the sphere z x2 + y2 + z2 = 9 and above the cone z = px2 + y2 (see ﬁgure to the right).

y x Multivariable Calculus Workbook, Wiley, 2007 81

2. Find the volume of the region W bounded between the paraboloids z = x2 + y2 and z = 36 − 3x2 − 3y2 (see ﬁgure to the right). Hint: z Find the curve of intersection for these two surfaces.

x y 82 Sonoma State University

3. Set up an integral giving the mass of the solid region in the ﬁrst octant between the planes y = 0 and y = x and between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4. The density of the region at a point P is proportional to the square of the distance of P from the origin. Multivariable Calculus Workbook, Wiley, 2007 83

√ Z 1 Z 1 Z 1−x2 1 4. Evaluate the integral dy dx dz by converting to a more appropriate coordinate √ p 2 2 0 −1 − 1−x2 x + y system and then evaluating. 84 Sonoma State University

3 5. The density of a spherical solid of radius 2, centered at the origin, is given√ by D(ρ) = 4ρ grams per cm . Calculate the mass of the portion of the sphere lying above the plane z = 3. Multivariable Calculus Workbook, Wiley, 2007 85

6. The 8 octants in 3-space break the sphere x2 + y2 + z2 = 1 into 8 separate pieces. Using spherical coordinates, determine the limits of integration to integrate the function g(ρ, θ, φ) over the region inside the sphere lying in (a) the ﬁrst octant (x > 0, y > 0, z > 0) (b) the third octant (x < 0, y < 0, z > 0) (c) the sixth octant (x < 0, y > 0, z < 0) (d) the eighth octant (x > 0, y < 0, z < 0) 86 Sonoma State University Section 17.1 – Parameterized Curves

Preliminary Example. Sketch the space curve given by ~r(t) = x(t)~i + y(t)~j for 0 ≤ t ≤ 2π, where x(t) = −1 + 3 cos t and y(t) = 1 + 3 sin t. t x(t) y(t) ~r(t) 0 π/2 π 3π/2 2π

Example 1. As in the preliminary example above, sketch and describe each parametric curve below. (a) x = 4 cos t, y = −4 sin t, 0 ≤ t ≤ 2π (b) x = 2 cos t, y = 1, z = 2 sin t, 0 ≤ t ≤ 2π Multivariable Calculus Workbook, Wiley, 2007 87 Example 2. Sketch and describe the parametic curve given by x = 1 + 2t, y = 2 + 3t. 9 8 7 6 5 4 3 2 1

1 2 3 4 5

Example 3. Find parametric equations for the portion of the parabola y = x2 that starts at (0, 0) and ends at (2, 4).

Example 4. Find parametric equations for the portion of the parabola x = y2 that goes from (1, −1) to (4, 2). 88 Sonoma State University

In general, we will use the following two equivalent ways of representing a curve C:

1. Parametric Form: x = x(t), y = y(t), z = z(t), where a ≤ t ≤ b.

2. Vector Form: ~r(t) = x(t)~i + y(t)~j + z(t)~k, where a ≤ t ≤ b.

(x(t2 ), y(t 2), z(t2 )) (x(t ), y(t ), z(t )) 1 1 1 (x(b), y(b), z(b)) (x(t3 ), y(t 3), z(t 3))

(x(t5 ), y(t5 ), z(t5 )) (x(a), y(a), z(a))

(x(t4 ), y(t 4 ), z(t 4 ))

(0,0,0)

Parameterization of Special Curves

Circles in the xy-plane. The parametric equations for a circle of radius r in the xy-plane, centered at (a, b), and traversed once are given by

Counterclockwise: x = a + r cos t, y = b + r sin t, where 0 ≤ t ≤ 2π Clockwise: x = a + r cos t, y = b − r sin t, where 0 ≤ t ≤ 2π

Note: These formulas can be easily extended to give equations for circles in 3-dimensional space.

~ Lines. Let ~v = a~i + b~j + c k be a vector parallel to the line L, and let P = (x0, y0, z0) be a point on the line L (see diagram below). L V R P

1. Parametric equations for all of L: 2. Parametric equations for line segment PR :

Functions in the xy-plane. Given a curve y = f(x) in the xy-plane, i.e., x is the independent variable and y is the dependent variable, parametric equations for the portion of the curve running from (a, f(a)) to (b, f(b)) are given by

x = t, y = f(t), where a ≤ t ≤ b

Note: These formulas can be easily extended to give equations for functions of y or curves in 3-dimensional space. Multivariable Calculus Workbook, Wiley, 2007 89 Examples and Exercises

1. Match the parametric equation on the left with the graph of the circle it represents on the right. Also, draw an arrow on each circle to indicate its orientation. (A) ~r(t) = 4 cos t~i + 4 sin t~j + 2~k, 0 ≤ t ≤ 2π z (B) ~r(t) = 2 cos t~i − 2 sin t~j + 4~k, 0 ≤ t ≤ 2π

(C) ~r(t) = cos t~i + 3~j + sin t~k, 0 ≤ t ≤ 2π (D) ~r(t) = (2 + cos t)~i + (2 + sin t)~j, 0 ≤ t ≤ 2π (E) ~r(t) = 3~i + cos t~j + sin t~k, 0 ≤ t ≤ 2π x y

2. Find parametric equations for each of the following lines and curves. (a) The line segment starting at (0, 0, 0) and ending at (1, 2, 3).

(b) The line segment starting at (1, 2, 3) and ending at (2, −5, 8).

(d) The circle of radius 5, centered at (0, 0), in the xy-plane, oriented counterclockwise as seen from above. 90 Sonoma State University

(e) The circle of radius 5 in the plane z = 1, centered at the point (0, 0, 1), oriented counterclockwise as seen from above.

(f) The semicircle from (0, 0, −2) to (0, 0, 2) in the yz-plane with y ≥ 0.

(g) The circle of radius 3 in the plane x = 2, centered at the point (2, 0, 0).

(h) The curve y = x2 + 1 from (−1, 2) to (3, 10). Multivariable Calculus Workbook, Wiley, 2007 91

(i) The curve x = y2 + 1 from (2, −1) to (5, 2).

(j) The semicircle from (1, 0, 0) to (−1, 0, 0) in the xy-plane, with y ≤ 0.

(k) The hyperbola xy = 4 from the point (4, 1) to the point (2, 2).

~ (l) The line that goes through the point P0 = (2, −1, 2) and is parallel to the vector ~v = −~i +~j + k. 92 Sonoma State University

3. Two space ships are traveling along straight paths given by the vector equations ~ ~ ~r1(t) = (−1 + t)~i + (4 − t)~j + (−1 + 2t)k and ~r2(t) = (−7 + 2t)~i + (−6 + 2t)~j + (−1 + t)k, respectively, where t represents time. (a) Do the space ships ever collide? If so, when and where?

(b) Do the paths of the two space ships ever cross? If so, where? Multivariable Calculus Workbook, Wiley, 2007 93

4. Match each parametric equation shown to the (A) x = cos(3t), y = t, z = sin(3t) right with the appropriate graph below. Note that the scales may be diﬀerent from one graph to an- (B) x = cos t, y = sin t, z = sin(5t) other, and the scales on the distinct axes of one (C) x = cos t, y = sin t, z = t2 − 500 graph may also not be the same.

z z z

x y x y x y 94 Sonoma State University Section 17.2 – Motion, Velocity, and Acceleration

Deﬁnition. Let ~r(t) represent the position vector of an object at time t. Then the derivative, ~r 0(t), is given by

d~r ~r(t + h) − ~r(t) ~r 0(t) = = lim . dt h→0 h The velocity vector of the object at time t, denoted ~v(t), is deﬁned to be the derivative of ~r(t). In other words,

~v(t) = ~r 0(t).

Theorem. Let ~r(t) = x(t)~i + y(t)~j + z(t)~k be the position vector of an object at time t. Then

~r 0(t) = x0(t)~i + y0(t)~j + z0(t)~k.

Other Deﬁnitions. The acceleration vector, ~a(t), of an object with position vector ~r(t) is deﬁned by

~a(t) = .

The speed of the object is deﬁned by . Examples and Exercises

1. A water balloon launcher is aimed so that a propelled balloon impacts a target on the side of a building that lies 4 meters above level ground (see diagram to the right). The position of the balloon while it is in the air t seconds after launch is given by Launch x = 17.3t, y = −4.9t2 + 10t, Site where x gives the horizontal displacement and y gives the vertical displacement, both in meters, of the balloon from the launch site. You may assume that the balloon hits the building on the way down rather than on the way up.

(a) Find the initial velocity and initial speed of the water balloon. Multivariable Calculus Workbook, Wiley, 2007 95

(b) Find the horizontal distance between launch site and the building.

(d) What is the maximum height reached by the balloon during its ﬂight? 96 Sonoma State University

2. The path of a child on a merry-go-round is given by the parametric equations x = 4 sin(t/2), y = 4 cos(t/2), z = 0, where t is measured in seconds and x, y, and z are measured in feet. (a) To the right, sketch a graph of the path of the child.

(b) Calculate the position, velocity, and acceleration of the child at t = 0 seconds. Draw the velocity and acceleration vectors in appropriate places on your sketch from part (a).

(d) True or False: A moving object that has constant speed has zero acceleration. Justify your answer. Multivariable Calculus Workbook, Wiley, 2007 97

3. Does the curve given by ~r(t) = (t + 3)~i + t2~j + (t − 6)~k intersect the graph of the plane x + y − z = 13? If so, at what point(s)? 98 Sonoma State University Section 17.3 – Vector Fields Point of Notation: Throughout this section, and for the remainder of the course, the vector ~r will represent x~i + y~j (in 2-space) and x~i + y~j + z~k (in 3-space).

Definition. A vector field in 2-space is a function F~ (x, y) whose value at a point (x, y) is a two-dimensional vector. Similarly, a vector field in 3-space is a function F~ (x, y, z) whose values are 3-dimensional vectors.

Example. On the grids shown to the right, sketch a portion of the following two vector ﬁelds: F~1(x, y) = −y~i + x~j and F~2(~r) = ~r.

Examples and Exercises

1. Shown below are two vector ﬁelds. One is given by F~1(x, y) = −y~i + x~j and the other is given by −y~i + x~j F~2(x, y) = . Determine which is which and explain your reasoning. px2 + y2

2 2

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Multivariable Calculus Workbook, Wiley, 2007 99

2. Match the vector ﬁelds given below with their plots. (A) F~ (x, y) = 2y~i + 2x~j (C) F~ (x, y) = (sin y)~i + (sin x)~j (B) F~ (x, y) = (x + y)~i + (x − y)~j (D) F~ (x, y) = x~i + y2~j

2 6 4 1 2

0 0 -2 -1 -4

-6 -2 -6 -4 -2 0 2 4 6 -2 -1 0 1 2 2 2 1 1 0 0 -1 -1 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2

3. The gradient of a scalar function generates a vector field. For example, the paraboloid f(x, y) = x2 + y2 generates the gradient vector field ∇f(x, y) = 2x~i + 2y~j. Below, you are given level curve diagrams of three scalar functions. Match each level curve diagram with its corresponding gradient vector field.

3 3 3

2.5 2.5 2.5

2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

3 3 3

2.5 2.5 2.5

2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 100 Sonoma State University

4. Match the vector ﬁelds given below with their plots. (A) F~ (x, y, z) =~i + (z/2)~j + 2~k (D) F~ (x, y, z) =~i + 2~j + (z/2)~k (B) F~ (x, y, z) = −y~i + x~j (E) F~ (x, y, z) =~i + 2~j + ~k (C) F~ (~r) = ~r z z

y x x y

z z

y x y x

x y Multivariable Calculus Workbook, Wiley, 2007 101 Sections 18.1 & 18.2 – Line Integrals Example 1. Each of the following situations illustrates an object being subjected to a constant force, F,~ and experiencing a displacement, ~r. Given the formula Work = F~ ·~r, calculate the work done in each case.

||F~ || = 10 N, ||~r|| = 2 m ||F~ || = 10 N, ||~r|| = 2 m ||F~ || = 10 N, ||~r|| = 2 m r r r

o o 120 60 F F F

Work = Work = Work =

Example 2. A particle moves along the path C, which is given by ~r(t) = (t+1)~i+(4−t2)~j. Find the work done by the force ﬁeld F~ (x, y) = 4 2y~i + 2x~j between t = 0 and t = 2. 3

1 2 3 4 102 Sonoma State University

Example 3. Let F~ (x, y) = (x + 2)~j and let C be the circle of radius 1 centered at the origin, oriented counterclockwise (see Figure 2 to the right). 1 R ~ Find a parameterization of C and then calculate C F · d~r. 0

-1

-1 0 1

Figure 2. F~ (x, y) = (x+2)~j

R ~ ~ Example 4. Calculate C F · d~r, where F is the vector ﬁeld from Example 3 and C is the line segment starting at (0, 0) and ending at (2, −2). (Hint: Start by ﬁnding a parameterization for this line segment and writing it in “~r(t)” form.) Multivariable Calculus Workbook, Wiley, 2007 103

Definition. The line integral of a vector field F~ along an oriented curve C is defined by

n−1 Z X F~ · d~r = lim F~ (~ri) · ∆~ri. k∆~r k→0 C i i=0

If an oriented curve C is given parametrically as ~r(t) = x(t)~i + y(t)~j + z(t)~k for a ≤ t ≤ b, then this integral can be calculated as follows: Z Z b F~ · d~r = F~ (~r(t)) ·~r 0(t) dt. C a

Notes. 1. Roughly speaking, a line integral tells us whether a particular path goes with or against the flow of a vector ~ R ~ field overall. If a curve C moves more “with the flow” of F, then C F · d~r is . If C moves ~ R ~ more “against the flow” of F, then C F · d~r is . ~ R ~ 2. If F represents a force exerted on a particle as it moves along C, then C F · d~r represents .

3. The notation “F~ (~r(t))” means “F~ (x(t), y(t))” (in 2 dimensions) or “F~ (x(t), y(t), z(t))” (in 3 dimensions).

Deﬁnition. A curve is said to be oriented if we have chosen a direction of travel on it. 104 Sonoma State University Examples and Exercises

1. Given to the right is a vector ﬁeld F~ (x, y), together with oriented curves C1,C2,C3,C4,C5, and C6. C5 (a) Is R F~ · d~r positive, negative, or zero? Explain. C6

C6 C2

C1 C3

R ~ (b) Let C be the closed curve C1 + C2 + C3 + C4. Is C F · d~r positive, negative, or zero? Explain.

(c) Arrange the following quantities in order from smallest to largest: Z Z Z Z Z Z F~ · d~r, F~ · d~r, F~ · d~r, F~ · d~r, F~ · d~r, F~ · d~r C1 C2 C3 C4 C5 C6 Multivariable Calculus Workbook, Wiley, 2007 105

2. Let F~ (x, y) = −y~i + x~j, and let C1,C2, and C3 be the paths described below (see also diagram to the right): 1

C1 : ~r(t) = t~i +~j, −1 ≤ t ≤ 1

C2 : ~r(t) = t~i + t~j, −1 ≤ t ≤ 1 0.5

C3 : ~r(t) = cos t~i + sin t~j, 0 ≤ t ≤ 2π 0 (a) For each of the integrals R F~ · d~r, R F~ · d~r, and C1 C2 R F~ · d~r, decide whether the integral appears to be -0.5 C3 positive, negative, or zero WITHOUT DOING ANY CALCULATIONS. Also explain your answers. -1 -1 -0.5 0 0.5 1

Figure 1. F~ (x, y) = −y~i + x~j, 106 Sonoma State University

2. (Continued) Let F~ (x, y) = −y~i + x~j, and let C1,C2, and C3 be the paths described below (see also diagram to the 1 right): ~ ~ C1 : ~r(t) = t i + j, −1 ≤ t ≤ 1 0.5

C2 : ~r(t) = t~i + t~j, −1 ≤ t ≤ 1 ~ ~ C3 : ~r(t) = cos t i + sin t j, 0 ≤ t ≤ 2π 0

(b) Calculate the integrals R F~ · d~r, R F~ · d~r, and C1 C2 -0.5 R F~ · d~r, and compare to your answers from part C3 (a). -1 -1 -0.5 0 0.5 1

Figure 1. F~ (x, y) = −y~i + x~j, Multivariable Calculus Workbook, Wiley, 2007 107

3. Let C be an oriented curve parameterized by ~r(t), where a ≤ t ≤ b. Suppose that the following two items are also true: (i) F~ is always tangent to C in the direction of orientation, and (ii) kF~ k is constant, say kF~ k = m, everywhere along C (see diagram below). (a) What is the angle between F~ and d~r? C

F (b) Use the geometric deﬁnition of the dot product to ex- dr plain why F~ · d~r must equal kF~ kkd~rk.

4. Find the work done by the force ﬁeld F~ (x, y, z) = y~i − x~j + z2~k on a particle that moves along the line segment from (1, 2, 0) to (1, 4, 3).

−y x 5. Consider the vector ﬁeld given by F~ (x, y) = ~i + ~j, and px2 + y2 px2 + y2 1 let C denote the closed curve consisting of the line segment from (0, 0) to (1, 0), followed by the quarter circle of radius 1, centered at the origin, from (1, 0) to (0, 1), followed by the line segment from (0, 1) to (0, 0) (see Figure R ~ 3 to the right). Calculate C F · d~r. Hint: Use the result of Exercise 3 for the circular portion of your path.

0 0 1

Figure 3. Multivariable Calculus Workbook, Wiley, 2007 109 Section 18.3 – Gradient Fields and Path Independent Fields Preliminary Example. As in Example 2 on page 101, let F~ (x, y) = F~ (x, y) = 2y~i + 2x~j 2y~i + 2x~j, and let C be the curve ~r(t) = (t + 1)~i + (4 − t2)~j starting at (1, 4) and ending at (3, 0). 4

(a) Try to guess a scalar function f(x, y) such that F~ = ∇f. 3

1 2 3 4

R ~ (b) Calculate f(3, 0) − f(1, 4) and compare the result to the value of C F · d~r obtained in Example 2 on page 101.

Definition. Let F~ be a vector field. If it is possible to find a function f such that F~ = ∇f, then we call F~ a vector field or a vector field, and we call f a for F.~

The Fundamental Theorem of Line Integrals. Suppose C is a piecewise smooth oriented path with starting point P and endpoint Q, and let F~ be a gradient vector ﬁeld with F~ = ∇f. If f is a function whose gradient is continuous on the path C, then

Notes. ~ R ~ 1. In a conservative vector field F, the value of C F · d~r depends only on the starting and ending point of the path C. In other words, conservative vector fields are . 2. If C is a closed curve in a conservative vector field F,~ then Z F~ · d~r = . C

3. The above theorem only works if we can ﬁnd a potential function f; that is, only if F~ is conservative. 110 Sonoma State University Derivation of the Fundamental Theorem

Let F~ be a conservative vector ﬁeld, and let f be a potential F~ (x, y) ~ function for F. Let C be a smooth, oriented curve starting at a 1.5 point P and ending at Q.

0.5

-0.5

-1

-1 -0.5 0 0.5 1 1.5

Example 1. Let F~ = 2x~i + 2y~j. Use the Fundamental Theorem of Line 5 Integrals to calculate the line integral of F~ along each of the three described paths. 4 3 C (a) C1: the line segment from (0, 4) to (3, 1). 1 2 (b) C2: the parabola y = (x − 2) from (0, 4) to (3, 1). 2 (c) C3: the line segment from (1, −1) to (4, −1). 1 C2 0 -1 C3 0 1 2 3 4 Multivariable Calculus Workbook, Wiley, 2007 111 How Do We Know When a Vector Field Is Conservative?

~ Definition of Curl. Let F~ = F1~i + F2 ~j + F3 k. Then we define a new vector field, curl F,~ as follows: ∂F ∂F ∂F ∂F ∂F ∂F curl F~ = ∇×F~ = 3 − 2 ~i + 1 − 3 ~j + 2 − 1 ~k ∂y ∂z ∂z ∂x ∂x ∂y

~ Theorem (Test for Conservative Vector Fields). Let F~ = F1~i + F2 ~j + F3 k be a vector ﬁeld such that F1,F2, and F3 have continuous partial derivatives everywhere. Then F~ is conservative, i.e., it is possible to ﬁnd a potential function for F,~ if and only if curl F~ = ~0 everywhere.

2 ~ ~ ~ Theorem (Test for Conservative Vector Fields in R ). Let F (x, y) = F1 i+F2 j be a vector ﬁeld such that F1 and F2 have continuous partial derivatives everywhere. Then F~ is conservative if and only if 112 Sonoma State University

Example 2. Show that F~ (x, y) = (3x2 + 3y)~i + (3x + 2y)~j is a conservative vector field, and find a potential R ~ function. Then, calculate C F · d~r, where C is the curve y = sin x starting at (0, 0) and ending at (2π, 0). Multivariable Calculus Workbook, Wiley, 2007 113 Examples and Exercises 1. Let f(x, y) = x2 + y2, and let C be the portion of the parabola y = x2 that starts at (0, 0) and ends at (1, 1). R Calculate C ∇f · d~r in two different ways: (a) without using the Fundamental Theorem of Line Integrals, and (b) using the Fundamental Theorem of Line Integrals. 114 Sonoma State University

R ~ 2. For each of the following, use the Fundamental Theorem of Line Integrals to calculate C F · d~r exactly. (a) F~ = (2x + y)~i + (x + 2y)~j, and C is the quarter circle of radius 2, centered at the origin, starting at (2, 0) and ending at (0, 2). Multivariable Calculus Workbook, Wiley, 2007 115

(b) F~ = (ex + y3)~i + (3xy2 + 2y)~j, and C is the parabola x = y2 starting at (0, 0) and ending at (4, 2). 116 Sonoma State University z 3. Let F~ (x, y, z) = 2xyz~i + (x2z + ez)~j + (x2y + yez + 1)~k, and let C1 C1 be the path from (0, 0, 0) to (1, 1, 1) shown in the ﬁgure to the 1 right.

(a) Calculate R F~ · d~r. C1

y 1

1 x

R (b) Calculate F~ · d~r, where C2 is the line segment from (0, 0, 0) to (1, 1, 1). C2 Multivariable Calculus Workbook, Wiley, 2007 117

~ ~ ~ R ~ 2 4. Let F (x, y) = yi + 2j. Calculate C F · d~r, where C is the portion of the parabola y = x that starts at (0, 0) and ends at (3, 9).

5. Let F~ and G~ be the vector ﬁelds F~ (x, y) = −y~i + x~j G~ (x, y) = x~i + y~j whose formulas and graphs are 2 2 shown to the right.

(a) Is F~ conservative? Justify 1 1 your answer graphically, with-

out doing any calculations. 0 0

-1 -1

-2 -2 -2 -1 0 1 2 -2 -1 0 1 2 118 Sonoma State University

(b) Is G~ conservative? Justify your answer.

R ~ (c) Find C G · d~r, where C is the closed curve shown below and to the right.

−1 1

−1 Multivariable Calculus Workbook, Wiley, 2007 119 Section 18.4 – Green’s Theorem Green’s Theorem. Suppose C is a piecewise smooth simple closed curve C (meaning that the curve doesn’t cross itself) that is the boundary of an open region R in the plane, oriented so that the region is on the left as we move around the curve (see diagram to the right). Suppose F~ = F1~i + F2 ~j is a smooth vector ﬁeld on an open region containing R and C. Then R Z Z ∂F ∂F F~ · d~r = 2 − 1 dA. C R ∂x ∂y

Notes on Green’s Theorem.

1. The purpose of Green’s Theorem is to replace a tedious line integral with a hopefully easier double integral.

2. Green’s Theorem applies only to around curves.

3. For Green’s Theorem to work, the vector ﬁeld F~ must be “nice” everywhere in R. In particular, F1 and F2 must have continuous derivatives on R. 120 Sonoma State University Examples and Exercises

~ 2 y~ 2 ~ R ~ 1. Let F = −3x e i + sin(y ) j. Use Green’s Theorem to evaluate C F · d~r, where C is the boundary of the square whose vertices are given by (1, 1), (1, −1), (−1, 1), and (−1, −1), oriented counterclockwise. Multivariable Calculus Workbook, Wiley, 2007 121

2. Use Green’s Theorem to evaluate the line integral C R ~ ~ 2~ ~ C F · d~r, where F = −y i + 2xy j, and C is the curve shown to the right consisting of a semicircular arc and a line segment.

(−2,0) (2,0)

−y x 3. Let F~ = F ~i + F ~j = ~i + ~j. 1 2 x2 + y2 x2 + y2

2 2 R ~ (a) Let C denote the unit circle x + y = 1 oriented counterclockwise. Calculate C F · d~r without using Green’s Theorem. That is, parameterize C and evaluate this line integral directly. 122 Sonoma State University (b) Calculate the double integral Z ∂F ∂F 2 − 1 dA, D ∂x ∂y where D is the region inside the unit circle C.

(c) Note that your answers to (a) and (b) are not equal. Why does this not contradict Green’s Theorem? Multivariable Calculus Workbook, Wiley, 2007 123 Sections 19.1 & 19.2 – Flux and Flux Integrals

Preliminary Example 1. Imagine a screen immersed in a river that is ﬂowing in the positive y- direction; that is, left to right (see Figures 1 and 2 below). The velocity of the water at each point is represented by the vectors in these diagrams.

Figure 1. Screen sitting straight up and down, im- Figure 2. Same screen as in Figure 1, but mersed in river flow. tilted at an angle relative to river flow. Discussion Questions: • Through which screen will one observe the greatest rate of fluid flux (flow) of water, measured in volume of water per unit time? • What specific physical quantities influence the amount of flow through the screen?

The Orientation of a Surface. At each point on a smooth, orientable surface, there are two unit normal vectors. Choosing an orientation means picking one of these unit normal vectors at each point in a continuous way. Once this has been done, we use the notation ~n to denote a unit normal vector in the direction of orientation.

Preliminary Example 2. Each figure below shows a portion of the plane z = 0, it gives you an orientation for the plane, and it gives you a vector field. Decide whether the “flux” of F~ through the surface is positive, negative, or zero.

F~ = ~j + ~k F~ = ~j + ~k F~ = ~j upward orientation downward orientation upward orientation

F~ = −~j − ~k F~ = −~j − ~k F~ = ~j upward orientation downward orientation downward orientation 124 Sonoma State University Example 3. For each of the following surfaces, illustrate the indicated orientation by drawing in some unit normal vectors, ~n.

Example 4. Suppose that F~ = 3~j + 3~k represents the flow velocity of a fluid, each component in m/sec, and let S be a 2 meter by 2 meter square screen parallel to the xy-plane, oriented upward. Find the flux of F~ through S, in cubic meters per second.

S Multivariable Calculus Workbook, Wiley, 2007 125 The Flux of a Constant Vector Field Through a Flat Surface

Fact. Let F~ be a constant vector field, and let S be a flat surface. Then the flux of F~ through S is given by F Flux Of F~ Through S = A where A~ is a vector that is normal to S in the direction of orientation and whose magnitude equals the area of S. We call A~ the area vector of S. S

Flux Integrals

Goal: Devise a method for calculating the flux of a vector field F~ through a surface S that works even if F~ is not constant and/or S is not flat. F z

∆A

x y

Definition. The flux integral of F~ through the oriented surface S is defined by

Z X F~ · dA~ = lim F~ · ∆A.~ S kA~k→0 126 Sonoma State University R ~ ~ Formulas for the Flux Integral S F · dA

(1) A General Formula: dA~ = ~ndA

~ (2) Flux of F through z = f(x, y): Let S be a portion of the z surface given by the formula z = f(x, y). z=f(x,y) v u ~ (∗) dA~ = (−fx~i − fy ~j + k) dx dy (oriented upward) ∆y

R ~ ~ ~ (∗) To use this formula to calculate S F · dA, the vector ﬁeld F must ∆x be written entirely in terms of the two variables of integration: x and y x Multivariable Calculus Workbook, Wiley, 2007 127 R ~ ~ Formulas for the Flux Integral S F · dA

(3) Flux of F~ through a Cylinder. Let S be a portion of a z right circular cylinder of radius R centered along the z-axis. R∆θ (∗) dA~ = (cos θ~i + sin θ~j) R dθ dz (oriented away from z-axis) ∆θ R ~ ~ ~ ∆z (∗) To use this formula to calculate S F · dA, the vector ﬁeld F must be written entirely in terms of the variables of integration: θ and z. R n

x 128 Sonoma State University R ~ ~ Formulas for the Flux Integral S F · dA ~ (4) Flux of F through a Sphere. Let S be a portion of a z n sphere of radius R centered at the origin. R (∗) dA~ = (sin φ cos θ~i + sin φ sin θ~j + cos φ~k) R2 sin φ dφ dθ ∆φ R sin (oriented away from origin) ∆φ φ∆θ

R ~ ~ ~ ∆θ (∗) To use this formula to calculate S F · dA, the vector ﬁeld F must be written entirely in terms of the variables of integration: φ and θ. Multivariable Calculus Workbook, Wiley, 2007 129 Examples and Exercises

1. Let F~ (x, y, z) = 2~i + 3~j. Let S be the surface of the cube pictured z to the right, oriented outward. Find the ﬂux of F~ through each 10 of the following four surfaces: (a) S1, the right side of the cube (y = 10), (b) S2, the left side of the cube (y = 0), (c) S3, the front of the cube (x = 10), and (d) S4, the top of the cube (z = 10).

y 10 10 x 130 Sonoma State University

~ 2. Let F~ (x, y, z) = x~i + y~j + (z + x)k, and let S1,S2,S3, and S4 be four circular disks, each having radius 2, as described below:

• S1 has center (5, 0, 0), is perpendicular to the x-axis, and is oriented in the positive x direction.

• S2 has center (8, 0, 0), is perpendicular to the x-axis, and is oriented in the negative x direction.

• S3 has center (0, 6, 0), is perpendicular to the y-axis, and is oriented in the negative y direction.

• S4 has center (0, 0, 7), is perpendicular to the z-axis, and is oriented in the positive z direction. Determine whether each of the following integrals is positive, negative, or zero: R F~ · dA,~ R F~ · dA,~ R F~ · dA,~ and R F~ · dA.~ S1 S2 S3 S4 Multivariable Calculus Workbook, Wiley, 2007 131

3. In each of the diagrams below, the thicker arrows represent values of a vector ﬁeld, F,~ while the thinner arrows represent unit vectors normal to the surface of S in the direction of the orientation that has been chosen for S. In each case, decide whether the ﬂux of F~ through the surface S is positive, negative, or zero.

(a) (c)

(b) (d) 132 Sonoma State University

4. A constant electric point charge, q, is placed at the origin in 3-space. The resulting electric ﬁeld E~ (~r) at the point with position vector ~r is given by

~r E~ (~r) = q , ~r 6= ~0. k~rk3

Find the ﬂux of E~ out of the sphere S of radius R centered at the origin. Multivariable Calculus Workbook, Wiley, 2007 133

5. Calculate R F~ · dA~ if F~ = x~i + (3y − 1)~j + z ~k and S S z is the portion of the plane x + y + z = 1 that lies in the ﬁrst octant, oriented upward (see diagram to the 1 right).

y 1

x 1 134 Sonoma State University

6. Let F~ = y~k, and let S be the portion of the paraboloid z = 25 − (x2 + y2) that lies above the xy-plane, oriented upward. Find the flux of F~ through S. Multivariable Calculus Workbook, Wiley, 2007 135 z 7. Find the flux of the vector field F~ = y~j + (3x + z)~k through the closed rectangular cube, of side length 1, shown to the right, oriented outward.

(0,0,0) y

x 136 Sonoma State University

8. Calculate the ﬂux of the vector ﬁeld F~ (x, y, z) = xz~i + yz~j + z3 ~k. through z the portion of the cylindrical surface x2 + y2 = 1 that is shown to the right, oriented inward. Note that S includes only the cylinder, not the circular disks 6 that form the top and bottom of the diagram to the right.

x 1 1 y Multivariable Calculus Workbook, Wiley, 2007 137

R ~ ~ ~ 2 ~ 2 2 2 9. Find S F · dA, where F = z k and S is the portion of the sphere x + y + z = 25 that lies in the 1st and 2nd octants, oriented outward. 138 Sonoma State University

R ~ ~ ~ ~ ~ ~ 2 2 10. Calculate S F · dA if F = x i + y j + z k and S is the portion of the surface y = x + z such that y ≤ 4, oriented in the positive y direction. Multivariable Calculus Workbook, Wiley, 2007 139 Section 20.1 – The Divergence of a Vector Field

2 2

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Geometric Definition of Divergence. The divergence, or flux density of a smooth vector field F~ is a scalar-valued function given by (x, y, z) S div F~ (x, y, z) = ,

where S is a sphere centered at (x, y, z), oriented outward, that contracts to the point (x, y, z) in the limit.

Notes. 1. div F~ (x, y, z) is when there is net ﬂow into the point (x, y, z) and when there is net ﬂow out of the point (x, y, z).

~ 2. If F~ = F1~i + F2~j + F3k, we can calculate the divergence of F~ as follows:

div F~ = .

This is called the coordinate deﬁnition of divergence.

3. Alternate Notation For Divergence: 140 Sonoma State University Example. Let F~ (x, y, z) = x2~i + y~j + z~k.

(a) Calculate the divergence of F~ at the point (0, 0, 0).

(b) Give a practical interpretation of the product div F~ (0, 0, 0) ∆V, where ∆V is the volume of a small cube centered at the point (0, 0, 0).

Examples and Exercises 1. Consider the vector ﬁelds shown below. F~ (~r) = ~r G~ (x, y) = x~j H~ (x, y) = −xy~j

2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0

-0.5 -0.5 -0.5

-1 -1 -1

-1.5 -1.5 -1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(a) Use graphical reasoning to decide whether each of the following quantities are positive, negative, or zero. (i) div F~ (0, 0) (ii) div G~ (1, 1) (iii) div H~ (1, 1) (iv) div H~ (−1, 1) Multivariable Calculus Workbook, Wiley, 2007 141

(b) Now, algebraically calculate the 4 quantities from part (a) to conﬁrm your graphical answers from above.

2. Let div F~ = x2 + 2y + 2z, where F~ represents the flow of a vector field in m3 per second. Estimate the flux out of a small sphere of radius 0.1 centered at each of the following points, including the proper units with each answer. (a) (1, 1, 1) (b) (0, 0, 3) (c) (0, −2, 1) 142 Sonoma State University

3. The magnetic field B~ around a long, straight and thin wire carrying constant current C y I in the positive z direction satisfies Ampère’sLaw, which states that Wire Z x B~ · d~r = kI, C where k is a constant and C is a closed curve surrounding the wire. Experiments show that B~ is tangent to every circle that is perpendicular to the wire and has center on the wire (see diagram to the right). Figure. Magnetic Field B~ circling the wire.

kI (a) Use Amp`ere’sLaw to show that kB~ k = everywhere along a circle of radius r centered along the 2πr wire, and perpendicular to the wire (as illustrated in the above ﬁgure).

−y x (b) Given that F~ = ~i + ~j is a field of unit vectors tangent to any circle like the one px2 + y2 px2 + y2 shown in the figure above, find a formula for B.~ (Hint: B~ = αF~ for some constant α.)

4. Verification that the Geometric and Coordinate Definitions of Divergence are Equivalent. ~ ~ Consider the cube shown to the right, and suppose that a vector field F = F1i+ ∆y ~ F2~j + F3k has been defined. Also assume that the shown cube is sufficiently small so that F~ is approximately constant on each face of the cube. Next, let S denote the entire outside surface (all 6 faces) of the cube, oriented outward, ∆z and let ∆V = ∆x∆y∆z (the volume of the cube). P=(x,y,z)

∆x

(a) Let S1 and S2 denote the front and the back faces of the cube (i.e., the faces of the cube on which x is R R F~ · dA~ + F~ · dA~ ∂F constant), oriented to agree with the orientation of S. Show that S1 S2 ≈ 1 . ∆V ∂x

(b) Let S3 and S4 denote the right and the left faces of the cube (i.e., the faces of the cube on which y is R R F~ · dA~ + F~ · dA~ ∂F constant), oriented to agree with the orientation of S. Show that S3 S4 ≈ 2 . ∆V ∂y 144 Sonoma State University

(c) Let S5 and S6 denote the top and the bottom faces of the cube (i.e., the faces of the cube on which z is R R F~ · dA~ + F~ · dA~ ∂F constant), oriented to agree with the orientation of S. Show that S5 S6 ≈ 3 . ∆V ∂z

(d) Use your answers to parts (a), (b), and (c) above to explain why the coordinate definition of divergence and the geometric definition of divergence are equivalent. Multivariable Calculus Workbook, Wiley, 2007 145 Section 20.2 – The Divergence Theorem Preliminary Illustration 1. Let S be the outside surface of the solid region shown in Figure 1, oriented outward, and let F~ (x, y, z) be a smooth vector field.

Figure 1. Solid region consisting of two cubes glued together.

Preliminary Illustration 2. Let S be the outside surface of the solid spherical region W shown in Figure 2, oriented outward, and let F~ (x, y, z) be a smooth vector ﬁeld.

Figure 2. Solid spherical region W divided into tiny cubes, with one “panel” of the outer surface S removed. 146 Sonoma State University

The Divergence Theorem. If W is a solid region whose boundary, S, is a piecewise smooth surface, and F~ is a smooth vector ﬁeld on an open region containing W and S, then Z Z F~ · dA~ = div F~ dV, S W where S is given the outward orientation. Notes.

1. The integral on the left-hand side of the above equation is a integral over a surface.

2. The integral on the right-hand side of the above equation is a integral over the .

3. This theorem is mainly useful for replacing a tedious ﬂux integral with a hopefully easier triple integral.

Examples and Exercises

1. Find the ﬂux of the vector ﬁeld F~ = y~j + (3x + z)~k through the closed z cube, of side length 1, shown in Figure 1 to the right, oriented outward.

(0,0,0) y

x Figure 1. Cube of side length 1. Multivariable Calculus Workbook, Wiley, 2007 147

~ R ~ ~ 2 2 2. Let F (~r) = ~r. Calculate S F · dA, where S is the surface of the solid region inside the cylinder x + y = 9 and between the xy-plane and the plane z = 4, oriented outward. 148 Sonoma State University

~ 3~ 3 ~ 3 ~ R ~ ~ 3. Let F (x, y, z) = x i + y j + z k. Calculate S F · dA, where S is the surface of the solid region above the cone z = px2 + y2 and below the plane z = 1, oriented outward. Multivariable Calculus Workbook, Wiley, 2007 149

4. According to Coulomb’s Law, a point charge Q located at the origin generates an electric ﬁeld εQ E~ (~r) = ~r, k~rk3

where ε is a constant that depends on the units used. It can be shown that div E~ = 0 away from the origin, a fact which you may assume and use freely throughout this problem.

(a) Let S1 be an arbitrary smooth, orientable, totally enclosed surface that contains the origin in its interior, and let S2 be a sphere of radius a centered at the origin that is contained inside S1. Use the Divergence R R Theorem to explain why E~ · dA~ = E~ · dA,~ where S1 and S2 are both oriented away from the origin. S1 S2 150 Sonoma State University

(b) Calculate R E~ · dA~ using the easiest method possible. S1 Multivariable Calculus Workbook, Wiley, 2007 151 Section 20.3 – The Curl of a Vector Field

Conceptual Description of Circulation. Let F~ (x, y, z) be a vector ﬁeld, and imagine placing a tiny paddlewheel (like the one shown to the right) in the Your eye vector ﬁeld at the point (x, y, z) so that the handle points in the direction of the axis that you want to measure the circulation of F~ around.

Notes. • The of the paddlewheel measures the strength of the circulation. (x,y,z) • The circulation is if the rotation is counterclockwise and if the rotation is clockwise (as you look down the handle of the paddlewheel).

y Preliminary Example. Let F~ be the vector ﬁeld -2 0 2 shown to the right. Decide whether the circulation of the vector ﬁeld at (0, 0, 0) around each of the following axes 2 would be positive, negative, or zero: (a) ~k (b) −~k (c) ~j

-2 -2 0 2 x

Definition. The circulation density of a smooth vector field F~ at P = (x, y, z) around the direction of a unit vector ~n is defined as

R F~ · d~r ~ C circ~nF (x, y, z) = lim , Area → 0 Area inside C where C is a circle centered at P and perpendicular to ~n. Note. The orientation of the circle C is determined by the right- hand rule: If you place your right thumb in the direction of ~n, your ﬁngers curl in the direction of C. 152 Sonoma State University 2 Example 1. Let F~ (x, y, z) = −y~i + x~j + 0~k. The cross-section of this vector ﬁeld for z = 0 is shown to the right. Calculate each 1 of the following.

(a) circ F~ (0, 0, 0) ~k y 0 ~ (b) circ−~kF (0, 0, 0) -1 ~ (c) circ~jF (0, 0, 0)

-1 0 1 2 x Multivariable Calculus Workbook, Wiley, 2007 153 y -2 0 2 Example 2. To the right, you are given the graph of a vector ﬁeld F~ (x, y, z). For approximately what value of ~n does circ F~ (0, 0, 0) appear to ~n 2 be the greatest? Explain. z0

-2 -2 0 2 x

Geometric Definition of Curl. The curl of a smooth vector field F,~ written curl F,~ is the vector field with the following properties: ~ ~ 1. The direction of curl F (x, y, z) is the direction of ~n for which circ~nF (x, y, z) is the greatest. 2. The magnitude of curl F~ (x, y, z) is the circulation density of F~ around that direction. If the circulation density is zero around every direction, then we define the curl to be ~0.

~ Coordinate Deﬁnition of Curl. If F~ = F1~i + F2~j + F3k, then ∂F ∂F ∂F ∂F ∂F ∂F curl F~ = 3 − 2 ~i + 1 − 3 ~j + 2 − 1 ~k ∂y ∂z ∂z ∂x ∂x ∂y

~i ~j ~k

= ∂ ∂ ∂ ∂x ∂y ∂z F1 F2 F3 = ∇ × F~

Examples and Exercises 1. Consider the two vector ﬁelds shown below. For each ﬁeld, draw in a possible vector representing curl F~ (0, 0, 0). F~ (x, y, z) F~ (x, y, z) y -2 0 2

-2 -2 0 2 x 154 Sonoma State University

2. Below, you are given a formula for three vector ﬁelds and a picture of the cross section of that ﬁeld that lies in the plane z = 0.

F~1(x, y, z) =~i + 2~j F~2(x, y, z) = |y|~i F~3(x, y, z) = −y~i + x~j 2 2 2

1 1 1

y 0 y 0 y 0

-1 -1 -1

-1 0 1 2 -1 0 1 2 -1 0 1 2 x x x

(a) For each vector ﬁeld, use graphical reasoning to describe the vectors curl F~ (0, 1, 0) and curl F~ (0, −1, 0). In particular, are they zero or nonzero? If they are nonzero, do they point in the ~k or the −~k direction?

(b) Conﬁrm your answers to part (a) by calculating curl F~ (0, 1, 0) and curl F~ (0, −1, 0) for each vector ﬁeld. Multivariable Calculus Workbook, Wiley, 2007 155

3. Using Green’s Theorem and the deﬁnition of curl, it can be shown that for any smooth vector ﬁeld F~ and any unit vector ~n, we have ~ ~ (∗) curl F ·~n = circ~nF,

a fact that you may use freely from here on out. Let S represent a small, two-dimensional solid square or disk centered at the point (x, y, z) and having area vector ∆A.~ Let C be the curve along the boundary of S, oriented ∆A ∆A to correspond to ∆A~ according to the right hand rule (see diagram to the right). Use equation (*) above to show that C S S C

Z (∗∗) curl F~ (x, y, z) · ∆A~ ≈ F~ · d~r. C 156 Sonoma State University

~ ~ ~ ~ R ~ 4. Suppose that F is a smooth vector ﬁeld and that curl F = −xi + zk. Estimate the circulation, C F · d~r, around a circle of radius 0.01 in each of the following situations. (a) Centered at (2, 0, 0), parallel to the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. (b) Centered at (0, 0, 3), parallel to the xy-plane, and oriented counterclockwise when viewed from the positive z-axis. (c) Centered at (0, 0, 3), contained in the yz-plane, and oriented counterclockwise when viewed from the positive x-direction. Multivariable Calculus Workbook, Wiley, 2007 157

5. Use the “curl test” to determine whether each of the following vector ﬁelds is conservative. If it is conservative, ﬁnd a function f such that F~ = ∇f. (a) F~ = yexy~i + xexy ~j + (2z + y)~k (b) F~ = 2xyz~i + x2z~j + (x2y + 1)~k 158 Sonoma State University Section 20.4 – Stokes’ Theorem

Preliminary Example. Let S be the surface shown below, oriented upward, and let C be the curve running along the boundary of S, oriented counterclockwise. Note that S has been divided into M small “patches”.

Stokes’ Theorem. Let S be a smooth oriented surface with piecewise smooth, oriented boundary C, and let F~ be a smooth vector ﬁeld on an open region containing S and C. Then

The orientation of C is determined from the orientation of S according to the right hand rule.

Notes. R ~ 1. The integral C F · d~r is a integral around the .

R ~ ~ 2. The integral S curl F · dA is a integral over .

3. Stokes’ Theorem is mainly useful to replace a tedious line integral with an easier ﬂux integral, or vice versa. Multivariable Calculus Workbook, Wiley, 2007 159 The Boundary of a Surface

Deﬁnition. The boundary of a surface, S, is the curve (or curves) C running around the edge of S (like the hem on clothing). Once an orientation for S has been chosen, the corresponding orientation for C is determined by the right-hand rule.

Examples. For each of the following surfaces, draw in or describe both possible orientations. Then, for each surface orientation, draw in the induced orientation of the boundary curve(s).

1. Cylinder (without top and bottom)

2. Hemisphere (with open bottom)

3. Punctured Yield Sign 160 Sonoma State University Exercises and Examples

1. Let F~ (x, y, z) = x2z~i + y2exz ~j + z2ey ~k, and let S be the portion of the paraboloid z = 9 − x2 − y2 that lies R ~ ~ above the xy-plane, oriented upward. Calculate S curl F · dA. Multivariable Calculus Workbook, Wiley, 2007 161

~ ~ ~ R ~ 2. Let F (x, y, z) = −2yi + 2xj. Use Stokes’ Theorem to ﬁnd C F · d~r, where C is a circle... (a) parallel to the yz-plane, of radius a, centered at a point on the x-axis, and oriented counterclockwise as viewed from the positive x-axis.

(b) parallel to the xy-plane, of radius a, centered at a point on the z-axis, and oriented counterclockwise as viewed from above. 162 Sonoma State University

3. Let F~ (x, y, z) = −y~i + x~j + zex+y ~k, and let E be the solid region that lies above the plane z = 0, below the plane z = 1, and inside z the half of the cylinder x2 + y2 = 1 for which x ≤ 0. Let S be the surface of E, EXCLUDING THE SEMICIRCULAR TOP PANEL that lies in the plane z = 1 (see diagram to the right). Calculate R ~ ~ S curl F · dA, where S is given the outward orientation. x y Multivariable Calculus Workbook, Wiley, 2007 163

4. Let F~ (x, y, z) = (2y + x2ez)~i + (4x + ey)~j + cos(x2yz)~k, and let S be the portion of the sphere x2 + y2 + z2 = 4 for which z ≥ 0, oriented upward. (a) Describe and draw a ﬂat surface T that has the same boundary curve as S. Also state the proper orientation for T so that it corresponds to the orientation on the boundary curve induced by S. R ~ ~ (b) Assuming the orientations for S and T speciﬁed above, explain why S curl F · dA must equal R ~ ~ T curl F · dA. R ~ ~ (c) Evaluate S curl F · dA. 164 Sonoma State University

~ 5. Let F~ = F1~i + F2 ~j + F3 k be a smooth vector ﬁeld, and let S be a piecewise smooth surface that completely encloses a solid region in R3; in other words, S is a closed surface. Let S have the outward orientation, and suppose that S has been divided into M small patches Si, each having boundary curve Ci and area vector ∆A~i, where the orientations of Si and Ci are chosen to correspond to the outward orientation of S (see diagram below).

M X Z (a) Explain why F~ · d~r must equal zero. i=1 Ci

C i

S i ∆Ai

(b) Let Pi be a point chosen on the surface Si for each value of i. Use the fact that Z (curl F~ )(Pi) · ∆A~i ≈ F~ · d~r Ci R ~ ~ and your answer to part (a) to calculate the value of S curl F · dA. Multivariable Calculus Workbook, Wiley, 2007 165

R ~ ~ (d) Use the Divergence Theorem to calculate S curl F · dA and conﬁrm your answer to part (b).