Calculus III - math.uaa.alaska.edu

Calculus III

Dr. Mark Fitch

December 13, 2014

Contents

1 Vectors 71.1 Journey vs. Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Points, Vectors, and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.1 Orthogonal Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6.2 Dot Product Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8.2 Representation of Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.9 Limits of Vector Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.10 Derivatives of Vector Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.11 Integrals of Vector Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.11.1 Integrating Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.11.2 Arclength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Surface Limits & Derivatives 192.1 Limits for Higher Dimensional Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.4 Maximum Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3

4 CONTENTS

2.7 Extrema with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Integration 313.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 Conversion of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Scalar Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.3 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Volumes with Non-rectangular Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.1 Double Integrals with Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Parametric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6.1 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6.3 Scalar Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7 Triple Integrals in Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Triple Integrals in Cylindrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.9 Triple Integrals in Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.10 Change of Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.11 Integrals in Action: Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.11.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.11.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Vector Fields 474.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Vector Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Vector Field Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Path Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.2 Fundamental Theorem of Vector Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.3 Closed Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.4 Conservative Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.1 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Surface Vector Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7.1 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

CONTENTS 5

4.8 Stoke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 CONTENTS

Chapter 1

Vectors

1.1 Journey vs. Location

Discovery 1.1.1 Suppose for each problem that the person begins at the corner of 15th and I St.

1. If Guido walks 2 blocks north, where is he?

2. If Guido walks 5 blocks north then 3 blocks east, where is he?

3. If Uwe walks 3 blocks east then 5 blocks north, where is he?

4. If Guido walks 2 blocks north then 1 block east then 1 block north then 2 blocks east, where is he?

5. If Uwe walks 3 blocks east then 3 blocks north, where is he?

6. If Wolfgang rides in a helicopter 3√

2 blocks northeast, where is he?

7. How many locations are 3 blocks east and 3 blocks north of 15th and I St?

8. How many ways was this location reached in the examples above? Note all of these use only linear travel (nocurves).

Discovery 1.1.2 Guido’s parents asked him “Where did you go at noon?” Guido’s reply was “I traveled 4 milesnorth.” Note Guido fails to tell his parents were he was at noon.

1. Where was Guido if he started at Elmore and Abbott?

2. Where was Guido if he started at Point Woronzof Park?

3. Why should his parents be worried if Guido states that he drove a car, and he started at Elmore and Abbott?

4. Draw a starting point. Sketch a path that travels 3 units north then 3 units to the east. Sketch the path fromthe starting point to the ending point. If Guido follows the first path, does he ever travel northeast?

Discovery 1.1.3 On another day Guido tells his parents that he traveled from 15th and I St to 13th and I St thento 13th and L St.

1. Do his parents know where he was at the three times mentioned?

2. Do his parents know which direction he traveled?

7

8 CHAPTER 1. VECTORS

1.2 Terminology and Notation

Definition 1 (Point) A point specifies a location.

Points are denoted with capital letters and coordinates (Cartesian) and are specified with ordered lists. Typicallypoints are represented graphically as a dot at the specified location. For example A = (1, 3). B = (−4, 7,−2).

Definition 2 (Vector) A vector specified a direction and a distance traveled in that direction.

Vectors are denoted with lower case letters with a right arrow above it and are specified with ordered lists.Typically vectors are represented graphically as an arrow beginning at the origin–unless otherwise specified–andending the specified number of units away. For example ~x = (1, 3). ~y = (−4, 7,−2).

Definition 3 (Directed Line Segment) A directed line segment specifies two locations and the direction traveledfrom one to the other.

Directed line segments are denotes by an ordered pair of letters–representing the two points–with a right arrowabove it. They are specified by stating the two points. The direction is from the first point toward the second.Directed line segments are typically represented graphically by an arrow beginning at the first point and ending at

the second point. For example−→AC A = (1, 3), C = (4,−5).

Please note that notations for points, vectors, and various linear objects vary from author to author. You mustbe careful to check each text before assuming meanings.

1.3 Arithmetic

Practice 1.3.1 If you have not learned the vector arithmetic used below and it is not immediately obvious, pleasesee the instructor, tutors, or someone else who can show you the details.

1. (2, 7) + (−3, 4)

2. (1, 1, 3) + (0, 1, 9)

3. 4(1, 1, 3)

4. ~v = (3, 7,−2) and ~w = (5,−2, 4).

(a) ~v + ~w

(b) ~v − ~w

(c) 3~v − 4~w

While points, vectors, and directed line segments have different interpretations, they can be mixed in arithmeticoperations. The result of the arithmetic is considered a point, vector, or other linear object based on context.

Practice 1.3.2 Use these points and vectors for the following problems.O = (0, 0), A = (3, 2), B = (−1, 4).~u = (3, 2), ~v = (−4, 2).

1. Calculate and sketch a graph for each of the following.

(a) ~u,~v, ~u+ ~v

(b)−→OA,

−→AB

2. What is the vector for−→AB?

3. What is the vector for−→BA?

1.4. POINTS, VECTORS, AND FUNCTIONS 9

Watch the provided video to learn how to sketch 3D points before continuing. Note sketching in 3D by hand willlead to a better understanding of functions and surfaces; it is not a skill for daily use.

Practice 1.3.3 Use the following points and vectors for the following problems.O = (0, 0, 0), A = (2,−3, 5), B = (4,−1,−3).~u = (4, 4, 3), ~v = (6, 2, 3).

1. Calculate and sketch a graph for each of the following.

(a) ~u,~v,−3~u+ 2~v

(b)−→OA,

−→AB

2. What is the vector for−→AB?

3. What point is at the end of B − 2~u?

1.4 Points, Vectors, and Functions

Notation 1 The notation Rn is used to express the set of all points or vectors with n entries. R = R1.

Definition 4 (Vector Valued Function) A function is vector valued if and only if the codomain (inclusive of therange) is Rn with n ≥ 2.

Note that the domain may be Rm for any m ≥ 1. Also note that in a cruel twist of fate for calculus students theexpression ‘vector’ valued function is used for functions whose output is vectors and for functions whose output ispoints. In this class a function with vector outputs will be denoted like a vector (arrow on top) and a function withpoint outputs will be denoted without the vector notation. For example ~v(x, y) = (2x, 2y) and `(t) = (0, 3) + (1, 1)t.

Discovery 1.4.1 1. For B = (5, 0,−4) and ~m = (1,−1, 2) calculate the following. B + ~m, B + 2~m, B + 3~m,B − ~m.

2. Are the objects in the previous problem points or vectors?

3. Calculate 5 points for the function `1(t) = (−1, 3, 2) + (1, 1, 1)t.

4. Sketch your 5 points.

5. Pretend you are very young and do what comes naturally with a set of dots.

6. Based on your sketches what type of function is this?

7. Sketch `2(t) = (1, 3,−2) + (1, 1, 1)t.

8. Sketch `3(t) = (1, 3,−2) + (1, 1,−2)t.

9. Compare and contrast `1(t) and `2(t).


11. Write a function for a line through the point (5, 2, 0) with the direction given by the vector (1, 1, 1).

12. Write a function for a line through the points (5, 2, 0) and (6, 2, 3).

13. Graph the function f(t) = (3 + t, 1 + t2, 2 + 5t).


1.5 Magnitude

You will remember that the absolute value is associated with the distance of a number from 0 on the number line or,simply stated, how big the number is without concern for which direction from 0. A similar definition and notationis used in higher dimensions.

Definition 5 The magnitude of a vector ~u, denoted ‖~u‖, is the distance between the origin and a point at the endof the vector (placed at the origin).

Practice 1.5.1 Let A = (2, 3), B = (1,−5), ~u = (2, 3), ~v = (1, 8), X = (5, 4, 7), Y = (1, 3,−4), ~w = (4, 1, 11).

1. How far is A from the origin?

2. Calculate ‖~u‖.

3. Calculate ‖~v‖.

4. Calculate ‖−→AB ‖.

5. Calculate ‖~w‖. Your instincts for this are most likely correct.

6. Calculate ‖−→XY ‖.

With numbers the absolute value could be used to determine the distance between two numbers. For example|5− 3| = 2, so 5 and 3 are 2 units apart. This also extends to higher dimensions.

Practice 1.5.2 Use the points and vectors from the previous problem sets for the following calculations.

1. Calculate the distance between A and B.

2. Calculate the distance between X and Y.

1.5.1 Spheres

Definition 6 (Circle) A circle is a curve (2D) consisting of the set points such that they are all equidistant froma fixed point called the center.

Definition 7 (Sphere) A sphere is the surface (2D in 3D) consisting of the set of points such that they are allequidistant from a fixed point called the center.

The coordinate planes are the xy-plane, xz-plane, and the yz-plane.

Discovery 1.5.1 First derive an equation for a sphere using point and vector notations.

1. Let P be a point on the sphere and C be the fixed point that is the center. Write an expression for the distancebetween P and C.

2. Let the fixed distance between points P on the sphere and the center C be the positive, real value r. Write anequation for a sphere using point/vector notation.

3. In this notation is there any difference between the equation for a circle and a sphere?

4. Let P = (x, y, z) and C = (cx, cy, cz) and the radius be r. Write the equation for the sphere using this notation.

5. Write the equation for a sphere with center C = (4, 7,−5) and radius 5. Describe the intersections of thissphere with the coordinate planes.

6. Write the equation for a sphere with center C = (1, 3, 2) that passes through the origin.

1.6. DOT PRODUCT 11

1.6 Dot Product

Definition 8 (Dot Product) The dot product of two vectors ~u = (u1, u2, . . . , un) and ~v = (v1, v2, . . . , vn), denoted~u · ~v, is the scalar u1v1 + u2v2 + . . .+ unvn.

Note the dot product is a function from a pair of vectors to a scalar. The following problems lead to a derivationof the dot product from a trigonometric concept.

x

y

αβ

θ

A=(x ,y )1 1

B=(x ,y )2 2

s 1

s 2

l 1

l = 2

Discovery 1.6.1 1. Write the ratio for cos θ in terms of right triangle ABO.

2. Write ratio = cos θ.

3. Re-write θ in terms of α and β.

4. Re-write the cos θ using an appropriate trig identity.

5. Replace each trig function from the identity with its appropriate ratio in terms of appropriate triangles. Youmay wish to sketch lines perpendicular to the x-axis from A and from B.

6. Simplify.

7. Identify the dot product in this equation and solve for it.

8. Note the lengths of the vectors are in the equation. Use the standard magnitude notation.

This relationship between the dot product and the cosine function provides an interpretation (and primary use)for the dot product. Discover this using the problems below.

Discovery 1.6.2 Use the following below. ~a1 = (1, 1, 1), ~b1 = (1, 1,−2), and ~b2 = (2, 2, 2).

1. Sketch `2(t) = (1, 3,−2) + ~a1t.

2. Sketch `3(t) = (1, 3,−2) +~b1t.

3. Sketch `4(t) = (1, 3,−2) +~b2t.

4. Calculate ~a1 ·~b1.

5. Calculate ~a1 ·~b2.




8. If ~a ·~b = 0, what is the (geometric) relationship between the two vectors?

9. If ~a ·~b = ‖~a‖‖~b‖, what is the (geometric) relationship between the two vectors?

10. If 0 < ~a ·~b < ‖~a‖‖~b‖, what is the relationship?

1.6.1 Orthogonal Projection

The following definition is based on the trigonometric relationship discovered above.

Definition 9 (Orthogonal Projection) The orthogonal projection of ~x on ~y is given by proj~y~x = ~x·~y‖~y‖ ·

~y‖~y‖ .

Discovery 1.6.3 For the following use ~x = (5, 1) and ~y = (1, 3).

1. Calculate ~z = proj~y~x.

2. Sketch ~x, ~y, and ~z.

3. Describe the relationship between ~x, ~y, and ~z.

4. Calculate the new vector ~z ′ = ~x− ~z.

5. Describe the relationship between ~y, ~z, and ~z ′.

Practice 1.6.1 For the following use ~a = (6, 1, 2), ~b = (7, 6, 2), ~c = (1, 2,−4).

1. Determine if any pair of these three vectors are orthogonal.

2. Calculate proj~a~b and a vector orthogonal to it.

3. Calculate proj~a~c and a vector orthogonal to it.

1.6.2 Dot Product Properties

Discovery 1.6.4 Discover and list properties of the dot product from each of these experiments.

1. Let ~a = (a1, a2, a3) and ~b = (b1, b2, b3).

(a) Expand |~a ·~b|.

(b) Expand ‖~a‖‖~b‖.

(c) Compare the two results and write an inequality.

2. Choose some ~a and ~b (probably in 2D). Draw ~a, and ~a +~b from the origin and ~b from the head of ~a. (This is

the usual demonstration of vector addition.) What is the relationship between ‖~a‖2 + ‖~b‖2 and ‖~a+~b‖2?

3. Now find a case such that ‖~a‖2 + ‖~b‖2 = ‖~a+~b‖2

4. Find specific vectors such that ~a ·~b = ~a · ~c but ~b 6= ~c and ~a 6= ~0.

1.7. CROSS PRODUCT 13

1.7 Cross Product

Definition 10 (Cross Product) The cross product of two vectors ~u = (u1, u2, u3) and ~v = (v1, v2, v3), denoted~u× ~v, is the vector (u2v3 − u3v2,−(u1v3 − u3v1), u1v2 − u2v1).

Note the cross product is a function from a pair of vectors to a vector.The following problems display properties of the cross product.

Discovery 1.7.1 Use ~a = (6, 1, 2), ~b = (7, 6, 2), ~c = (1, 2,−4).

1. Calculate ~d = ~a×~b.

2. Calculate ~a · ~d.

3. Calculate proj~b~d.

4. Calculate ~f = ~a× ~c.

5. Calculate ~f = ~c× ~a.

6. How many vectors are orthogonal to two vectors in 3D? 4D?

Discovery 1.7.2 For the following use ~x = (−1, 2,−1), ~y = (3, 3,−2), ~z = (2, 1, 0).

1. Determine if any pair of these vectors are orthogonal.

2. Produce a vector orthogonal to ~x and ~y.

3. Produce a vector orthogonal to ~z.

4. How many pairs of vectors are orthogonal to ~z?

5. If ~a×~b = ~a× ~c and ~a 6= ~0 must ~b = ~c?

1.8 Planes

1.8.1 Examples

Watch the provided video to learn how to sketch 3D surfaces before continuing. Note sketching in 3D by handwill lead to a better understanding of functions and surfaces; it is not a skill for daily use. This section illustratesrelationships between curves–including lines–and surfaces–including planes.

Discovery 1.8.1 Confirm by graphing that each of the following sets of lines lie on the same plane.

1. Plane 1

(a) `1(t) = (4, 4, 5) + (−1, 3,−4)t.

(b) `2(t) = (7, 2, 4) + (−1, 3,−4)t.

(c) `3(t) = (10, 0, 3) + (−1, 3,−4)t.

2. Plane 2

(a) `1(t) = (4, 4, 5) + (3,−2,−1)t.

(b) `2(t) = (3, 7, 1) + (3,−2,−1)t.

(c) `3(t) = (2, 10,−3) + (3,−2,−1)t.

3. Plane 3

(a) `1(t) = (4, 4, 5) + (−1, 3,−4)t.


(b) `2(t) = (4, 4, 5) + (3,−2,−1)t.

(c) `3(t) = (4, 4, 5) + (1, 4,−9)t.

4. Randomly pick three lines in 3D. Graph to check if they are in the same plane.

The following table is for illustration only. Adjust to match each problem below.

x−2 −1 0 1 2

−2−1

y 012

Practice 1.8.1 For each of the following functions of two variables complete a table of values. Use the followingvectors and points. ~a = (3, 1, 1), ~b = (2,−5,−3), and C = (1, 0, 1).

1. 2x+ 4y − 6z + 2 = 0.

2. 2x2 + 4y − 6z + 2 = 0.

3. X = ~at+~bs+ C.

Discovery 1.8.2 The following illustrate a fundamental concept of surfaces.

1. Graph each surface above.

2. On the graph of each surface above graph the following curves.

(a) The points of the surface above y = 0.

(b) The points of the surface above x = 0.

(c) The points of the surface above y = 2.

(d) The points of the surface above y = x.

(e) The points of the surface above y = x2.

3. For the planes what is true about these curves?

1.8.2 Representation of Planes

Discovery 1.8.3 Use the following illustration to develop an important property of planes. Use ~u1 = (−1, 3,−4),~u2 = (3,−2,−1), ~u3 = (1, 4,−9), and ~v = (11, 13, 7).

1. Use a textbook or similar object to represent a plane. Take a pencil or similar vector like object and place it sothat it is orthogonal to the textbook (plane).

2. Place a second vector object on the plane at a different point on the plane but also orthogonal to the plane.Compare the direction of the two orthogonal vectors.

3. Repeat this experiment with a non-planar surface. You can use a backpack, your leg, or many other items.Contrast the results to those of the plane.

4. Calculate ~ui · ~v for i = 1, 2, 3.

5. If ~xi · ~v = 0 for all i do the ~xi all lie in the same plane?

Discovery 1.8.4 When we were young we could represent a line in multiple ways.

• Slope intercept y = mx+ b

1.9. LIMITS OF VECTOR VALUED FUNCTIONS 15

• Linear ax+ by + c = 0.

1. List both analogous representations for planes.

2. In days gone by the b from y = mx+b had an unfortunate (misleading) name and a purpose. What informationdoes it provide? Note c from ax+ by + c = 0 provides the same information.

3. What provides the analogous information for the plane representations?

Orthogonality provides solutions to many problems. The distance between a point and a line or plane, thedistance between non-intersecting lines, the distance between a line and a plane, and similar distances are defined asthe shortest distance between some point on one and some point on the other. It can be shown (and will be later)that the distance between a point and a line or a plane is the length of a segment from the point that is orthogonalto the line or plane.

Discovery 1.8.5 Use the following to demonstrate that this also holds true for the distance between parallel lines orparallel planes.

1. Find the distance between the parallel lines `1(t) = (1, 3, 1)t+ (1, 2, 2) and `2(t) = (1, 3, 1)t+ (2, 2, 3).

2. Find a plane such that every point on the plane is equidistant from the points (1, 3, 1) and (1, 2, 2). Note the2D version of this was a theorem in high school geometry.

Challenge 1 (Orthogonality) Complete the following steps to discover how to express orthogonal vectors in 4D.

1. How many distinct vectors (not scalar multiples of each other) are orthogonal to one vector in 2D?


3. How many distinct vectors (not scalar multiples of each other) are orthogonal to two, distinct vectors in 3D?


5. How many distinct vectors (not scalar multiples of each other) are orthogonal to two, distinct vectors in 4D?

6. How many distinct vectors (not scalar multiples of each other) are orthogonal to three, distinct vectors in 4D?

7. Choose two, distinct, 4D vectors.

8. Write the coordinates for all the distinct, 4D vectors orthogonal to these two vectors. Hint, if the orthogonalvectors are hard to calculate, pick two easier 4D vectors.

1.9 Limits of Vector Valued Functions

Practice 1.9.1 Review the video on graphing 3D curves as needed then graph the following curves by hand. Graphingby hand should help you estimate the limits below.

1. C(t) = (5t+ 3, 7− 2t, 4t− 1).

2. D(t) =(

cos tt , sin t

t , 1 + πt

).

3. F (t) = (t, t2, e−t).

Discovery 1.9.1 Estimate based on your graphing the following limits.

1. limt→5

C(t)

2. limt→∞

D(t)

3. limt→∞

F (t)


Discovery 1.9.2 All limit definitions thus far are for scalar rather than vector valued functions. The following stepslead to a generalized definition. Note it is often convenient to reference individual elements of vectors. C1 means thefirst element of the point (or vector) C. The notation Cx refers to the x (or first) element of the point (or vector) C.

1. Calculate the following three limits. limt→5

C1(t), limt→5

C2(t), limt→5

C3(t).

2. Do these calculations match your estimate for the limit above?

3. Copy the formal definition (ε and δ expressions) of limt→a

f(t) = L. You may wish to look up the Illustrated Limit

available online.

4. Determine which parts of the definition must change for a vector valued function and write this definition.

Challenge 2 (Vector Valued Limit Proof) Use the following example and the definition from above to prove thelimit below.

Consider the following proof of a limit. limx→5

7x− 9 = 26.

|7x− 9− 26| < ε From the definition of a limit.|7x− 35| < ε Algebra

−ε < 7x− 35 < ε From the definition of absolute value.−ε < 7(x− 5) < ε Algebra, because we want |x− 5|− ε

7 < x− 5 < ε7 Algebra

|x− 5| < ε7 From the definition of absolute value.

δ = ε7 From the definition of a limit.

Thus for all ε > 0 there exists a δ > 0 namely δ = ε/7 that satisfies |x− 5| < δ implies |f(x)− 26| < ε.

Challenge 3 Now prove limt→5

C(t) using the formal definition and this example.

1.10 Derivatives of Vector Valued Functions

Definition 11 (Derivative) The derivative of the function f(x) at a value a denoted f ′(a) is defined by f ′(a) =

limx→a

f(x)− f(a)

x− a.

Discovery 1.10.1 This definition is for scalar rather than vector valued functions. The following steps lead to ageneralized definition.

1. Note that f(x)− f(a) provides the distance from a to x. How does it also include direction?

2. This is direction and distance which we call what?

3. Can the same calculation be performed for a vector valued function?

4. Can the division (by x− a) still be performed?

5. Write a definition for the derivative of a vector valued function.

6. If f(x) represented the position of an object at time x, then f ′(x) represented the velocity of the object. Doesthis still work with vector valued functions?

7. How are the vector valued function definitions of limit and derivative similar?

Practice 1.10.1 Calculate the following.

1. ~r(t) = (t2 + 1, 3t, 5). ~r′(t) =

2. P (t) = (t, cos t, sin t). P ′(t) =

3. If P represents Guido’s position at time t, what is P ′?

1.11. INTEGRALS OF VECTOR VALUED FUNCTIONS 17

1.11 Integrals of Vector Valued Functions

Just as limits and derivatives can be extended to vector valued functions, so can integrals. Before extending the def-inition of Riemann integration, consider the following discrete problems which provide motivation for these integralsand hint at later problems.

1.11.1 Integrating Vectors

In this section consider how integrals can be applied when vector valued functions output vectors (as opposed topoints.) In winds aloft images the arrows indicate the direction of the wind. The flags indicate the speed. Each stemis 10 mph, and each triangle is 50 mph.

Discovery 1.11.1 1. If a kite were dropped off above Anchorage, where would the winds shown above take it?

2. If a kite were dropped off above Adak, where would the winds shown above take it?

3. If a kite were dropped off above Yakutat, where would the winds shown above take it?

Note that calculating the actual path of the kite requires topics presented later.

Discovery 1.11.2 Suppose the path of the kite is already known; in particular suppose the winds affecting the kiteat time t in minutes are given by ~w(t) = (2t, 3t2).

1. Calculate the wind force be at times t = 0, 1, 2, 3, 4. Based on these calculations where will the kite be after 4minutes?

2. The wind does not blow in straight lines for segments before turning sharply, rather it turns continuously. Howcould the location estimate above be improved? Can this improvement be repeated?

3. How can the exact location be calculated? How does this match with the definition of a Riemann integral (shownbelow)?


Definition 12 (Riemann Integral) The Riemann integral of a function f over the interval [a, b] is∫ b

a

f(x) dx = lim‖δx‖→0

∑xi∈P

(f(xi+1)− f(xi))δx

if it exists where P is a partition of the interval [a, b].

Discovery 1.11.3 Modify this definition to work for vector valued functions such as w(t) above.

1.11.2 Arclength

In this section consider how integrals can be applied when vector valued functions output points. It is possible toadd the areas beneath curves, but that will be done later. Note in the following examples the functions providepoints, but you must calculate distances.

Discovery 1.11.4 1. As a big ball is pushed around it ends up at the following locations in order. (0, 0), (3, 1),(5, 2), (9, 3), (12, 5). How far has the ball traveled?

2. A car is spinning across an icy surface. At time t the car is at position (sin t, t). At time t = π how far has thecar spun?

Because the distance is calculated these problems can be considered to be summing the length of the vectors fromone point to the next (if only finitely many points are considered). Thus arclength of a vector valued function P (t)over t ∈ [a, b] can be calculated by ∫ b

a

‖P ′(t)‖ dt.

Practice 1.11.1 1. Calculate the arclength of P (t) = (cos t, sin t) using the formula.

2. Calculate the arclength again using some other method.

Chapter 2

Surface Limits & Derivatives

The previous section developed definitions for limits, derivatives, and integrals of vector valued functions. Thesedefinitions were developed for curves which can be parameterized in one variable. In the notation of analysis theseare functions f : R → Rn for n ≥ 1 some integer. These functions are curves (topologically one dimensional). Thissection will develop definitions for limits, derivatives, and integrals on functions of multiple variables (f : Rn → R).These functions are surfaces for the case f : R2 → R (topologically two dimensional).

2.1 Limits for Higher Dimensional Functions

Discovery 2.1.1 As preparation for the limit questions below graph the following surfaces by constructing a table ofvalues and plotting by hand. Review the video for Section 1.8 as needed.

1. x+ 2y + 3z − 5 = 0. Use x and y values from {−2,−1, 0, 1, 2}.

2. z = x2 + y2. Use x and y values from {−2,−1, 0, 1, 2}.

3. z = sin(x2+y2)x2+y2 . Use x and y values from

{−π2 ,−

π4 , 0,

π4 ,

π2

}.

4. Curves (1D) consist of what geometric objects? Reminder: you just used this principle to graph them.

5. Surfaces (2D) therefore consist of what geometric objects? Reminder: you just used this principle to graphthem.

Discovery 2.1.2 Suppose Guido is romping on the surface z = x2 + y2.

1. If his path (path’s shadow in the xy-plane) is x = t, y = 1 what is the equation P (t) of the curve (3D) he isfollowing?

2. If he never steps off this path, for all practical purposes is Guido romping on a surface or a curve?

3. Calculate limt→0

P (t).

4. If his path is x = 0, y = t+ 1 what is the equation P (t) of the curve (3D) he is following?


P (t).

6. If his path is x = t, y = t+ 1 what is the equation P (t) of the curve (3D) he is following?


P (t).

8. If his path is x = t2, y = t+ 1 what is the equation P (t) of the curve (3D) he is following?


P (t).

19

20 CHAPTER 2. SURFACE LIMITS & DERIVATIVES

2.1.1 Derivation

Discovery 2.1.3 Review as necessary the floor function bxc. Evaluate the following limits.

1. limx→2.5

bxc

2. limx→−2.5

bxc

3. limx→3.3

bxc

4. limx→4.7

bxc

5. limx→4bxc

6. limx→0

sin

(1

x

)Examining the graph may be helpful for this limit.

Discovery 2.1.4 Extending the definition of limit to higher dimensions requires considering the conditions on thecurrent definition of limit.

1. What two conditions cause a limit not to exist? Note these are illustrated in the examples above.

2. For functions f : R → R the domain is one dimensional. From how many directions can a limit approach avalue?

3. For a limit to exist what must be true about these directional limits?

4. For functions f : R2 → R the domain is two dimensional. From how many directions can a limit approach avalue?

5. For a limit to exist what must be true about these directional limits?

Practice 2.1.1 1. Suppose Guido is romping on the surface f(x, y) = x2−y2x2+y2 .

(a) If Guido’s route (2D) is along the path p(t) = (t, 0), what is the curve (3D) he is following?

(b) As t approaches 0 (which is (x, y)→ (0, 0)), what is the limit?

(c) What is the limit as t approaches 0, if the path is p(t) = (0, t)?

(d) What is the limit as t approaches 0, if the path is p(t) = (t, t)?

(e) What is the limit as t approaches 0, if the path is p(t) = (t,mt) for any real number m?

(f) What is true about lim(x,y)→(0,0)

f(x)?

2. Suppose Guido is romping on the surface f(x, y) = x2yx4+y2 .

(a) What is the limit as t approaches 0, if the path is p(t) = (t, 0)?

(b) What is the limit as t approaches 0, if the path is p(t) = (0, t)?

(c) What is the limit as t approaches 0, if the path is p(t) = (t, t)?

(d) What is the limit as t approaches 0, if the path is p(t) = (t,mt) for any real number m?

(e) What is the limit as t approaches 0, if the path is p(t) = (t, t2)?

(f) What is true about lim(x,y)→(0,0)

f(x)?

3. Calculate lim(x,y)→(1,2)

y

(x− 1)2.

2.2. PARTIAL DERIVATIVES 21

2.1.2 Continuity

Discovery 2.1.5 These problems lead to an extension of the definition of limit and continuity.

1. Consider the surface function f(x, y) = sin(x2+y2)x2+y2 .

(a) Calculate lim(x,y)→(0,0)

f(x)?

(b) Does f(x, y) appear to be continuous everywhere?

2. Deriving a definition

(a) In Section 2.1.1 above what is the definition of lim(x,y)→(a,b)

f(x, y)? Note this will be in words not symbols.

(b) In practice what did we do to calculate a limit? How does this differ from the definition?

(c) Lookup the formal definition of continuity.

(d) Does the definition of continuity need to be changed for surface functions? Explain.

2.2 Partial Derivatives

2.2.1 Derivation

Discovery 2.2.1 Suppose Guido is romping on the surface z = x2 + y2.

1. If his path (path’s shadow in the xy-plane) is x = t, y = 1 what is the equation P (t) of the curve (3D) he isfollowing?

2. If he never steps off this path, for all practical purposes is Guido romping on a surface or a curve?

3. Calculate the instantaneous rate of change of his altitude (z) at x = 5, x = −3, and x.

4. If his path is x = t, y = 4 what is the equation P (t) of the curve (3D) he is following?

5. Calculate the instantaneous rate of change of his altitude (z) at x = 5, x = −3, and x.

6. If his path is x = −3, y = t what is the equation P (t) of the curve (3D) he is following?

7. Calculate the instantaneous rate of change of his altitude (z) at y = 4, y = −2, and t.

8. If his path is x = 7, y = t what is the equation P (t) of the curve (3D) he is following?

9. Calculate the instantaneous rate of change of his altitude (z) at y = 4, y = −2, and t.

10. If Guido is moving parallel to an axis, in how many dimensions does the rate of change problem exist?

11. How many variables are in use as a result?

12. How does this help us use Calculus I to solve multivariable problems?

2.2.2 Notation

• First derivatives

– δfδx is the derivative of f with y held constant (i.e., y = k).

– δfδy is the derivative of f with x held constant (i.e., x = k).

• Higher derivatives

– δ2fδxδy is the derivative of δf

δy with respect to x (i.e., first y then x).


– δ2fδx2 is the derivative with respect to x both times.

– δ3fδyδxδy with respect to y then x then y again.

• Note a common alternate notation uses the reversed notation fxy = δ2fδyδx .

Practice 2.2.1f(x, y) = x3 + x2y + xy2 + y3.r(u, e) = u sin e.

1. δfδx =

2. δ2fδyδx =

3. δfδy =

4. δ2fδxδy =

5. δ2fδx2 =

6. δ2rδuδe =

The following theorem can be proved.

Theorem 1 (Clairaut) If f is defined on the points ‖P − C‖ ≤ r for some number r and δ2fδxδy and δ2f

δyδx are

continuous, then at the point C, δ2fδxδy = δ2f

δyδx .

2.3 Differentiability

2.3.1 Motivation

For 2-dimensional functions (f : R→ R), differentiable is defined as the derivative existing. For higher dimensionalfunctions simply extending this definition to all partial derivatives exist is insufficient.

Discovery 2.3.1 The following steps review the geometric characterization of differentiable, 2D functions.

1. f(x) = x2 is differentiable at x = 0. Graph f(x) for x ∈ [−a, a] with a view window of x ∈ [−a, a] and y ∈ [0, a]and let a approach 0.001 (zoom in).

2. What does f(x) look increasingly like as a smaller portion is viewed?

3. g(x) = x2/3 is not differentiable at x = 0. Graph g(x) for x ∈ [−a, a] with a view window of x ∈ [−a, a] andy ∈ [0, a] and let a approach 0.001 (zoom in).

4. How does g(x) differ from f(x) under this same experiment?

5. Calculate and graph the line tangent to f(x) at x = 0. How does this compare to the graph of f(x) zoomed in?

2.3.2 Concept

The relationship between the tangent line and the differentiable curve provides a means to extend the definition ofdifferentiable to surfaces and higher dimensions. First consider the extension of the concept of tangent to a surface.For a curve (one dimensional object) the tangent is a one dimensional, “flat” object that locally contacts the curveat only one point. This is a line. For a surface (two dimensional object) the tangent is a two dimensional, “flat”object, that locally contacts the surface at only one point. This is a plane.

Discovery 2.3.2 The following illustrate how to calculate a tangent plane and how tangent planes help define dif-ferentiable.

2.3. DIFFERENTIABILITY 23

1. Use the surface f(x, y) = x2 − y2 for the following.

(a) If Guido is walking along this surface following the path P (t) = (t, 0), what curve (3D) is he following?

(b) Calculate the tangent line to this curve at the point (0, 0).

(c) If Guido is walking along this surface following the path P (t) = (0, t), what curve (3D) is he following?

(d) Calculate the tangent line to this curve at the point (0, 0).

(e) Both of these curves are in the surface. Does it make sense for their tangent lines to be in the tangentplane?

(f) Calculate the equation of the plane containing these two lines.

(g) If Guido is walking along this surface following the path P (t) = (t, t), what curve (3D) is he following?

(h) Calculate the tangent line to this curve at the point (0, 0).

(i) Should this tangent line also be in the tangent plane?

(j) Check if this tangent line is also in the tangent plane.

2. Use the surface g(x, y) =

{0 if(x, y) = (0, 0)xy

x2+y2 elsewisefor the following.

(a) Why can’t you use derivative rules to calculate δfδx at (0, 0)? The answer is not that you cannot plug in

the values.

(b) Write the path in g(x, y) for y = 0.

(c) Calculate the tangent derivative of this path at (0, 0).

(d) Write the path in g(x, y) for x = 0.

(e) Calculate the tangent derivative of this path at (0, 0).

(f) Using this information what is δfδx at (0, 0).

(g) Using this information what is δfδy at (0, 0).

(h) Write the equation of the plane defined by these two vectors.

(i) Compare this plane to the surface by graphing.

3. Which of these surfaces appears to be differentiable at (0, 0)?

2.3.3 Calculation

The next section explains how to conveniently calculate the tangent plane before formally defining differentiable.

Discovery 2.3.3 Use the surface f(x, y) = x2 − y2 for the following.

1. Calculate δfδx and δf

δy .

2. If Guido is walking along this surface following the path P (t) = (t, 0), what curve (3D) is he following?

3. Calculate the slope of the tangent line to this curve at the point (x, 0).

4. If Guido is walking along this surface following the path P (t) = (0, t), what curve (3D) is he following?

5. Calculate the slope of the tangent line to this curve at the point (0, y).

6. Where do the partial derivatives from the first step show up in the slopes?

7. Calculate the vector orthogonal to these two tangent line slopes.

8. Write the equation for the plane tangent to f at (x, y).

9. Where do the partial derivatives from the first step show up in this plane equation?


Definition 13 (Differentiable) A surface z = f(x, y) is differentiable if and only if δz = f(a + ∆x, b + ∆y) −f(a, b) = ∆x δf(a,b)

δx + ∆y δf(a,b)δy + ∆xε1 + ∆yε2 and ε1, ε2 → 0 as (∆x,∆y)→ (0, 0).

Note this means that the plane is a good approximation of the surface near the point of tangency. The followingconditions are also sufficient for differentiability and are easier to check.

Theorem 2 (Differentiable) For a surface f(x, y) if δfδx and δf

δy exist in a neighborhood around (a, b) and are

continuous at (a, b) then f is differentiable at (a, b).

Practice 2.3.1 Use the surface f(x, y) = x2 + xy + y2 − 16x+ 2 for the following.

1. Find the equation of the plane tangent to f at (0, 0).

2. Find the equation of the plane tangent to f that is parallel to the plane 2x+ 3y − z + 1 = 0.

2.4 Directional Derivatives

2.4.1 Motivation

Discovery 2.4.1 Suppose Guido is again romping on the surface z = x2 + y2.

1. If his path is x = t+ 2, y = 1 what is the equation P (t) of the curve (3D) he is following?

2. Calculate the instantaneous rate of change of his altitude (z) at t = 0.

3. If his path is x = 2, y = t+ 1 what is the equation P (t) of the curve (3D) he is following?


5. If his path is x = t+ 2, y = t+ 1 what is the equation P (t) of the curve (3D) he is following?


7. What is the instantaneous rate of change of Guido’s altitude (z) at (2, 1) if he is facing ~d1 = (1, 0), ~d2 = (0, 1),

and ~d3 = (1, 1)?

8. What is the instantaneous rate of change of Guido’s altitude (z) at (2, 1) if he is facing ~d1 = (1, 1), ~d2 = (2, 2),

and ~d3 = (√

2,√

2)?

2.4.2 Notation

Definition 14 (Gradient) For a function f : Rn → R the vector of all partial derivatives is called the gradient and

is denoted ∇f =(δfδx1

, δfδx2, . . . δfδxn

).

Practice 2.4.1 Calculate the gradients for the following functions.

1. f(x, y) = x2 + xy + y2.

2. g(x, y, z) = xyz.

2.5. CHAIN RULE 25

2.4.3 Derivation

Discovery 2.4.2 Use the following to connect the old concept of derivative with the concept of directional derivatives.

1. Copy the following definition for derivatives of single variate functions.

f ′(x) = limh→0

f(x+ h)− f(x)

h.

2. Review how the definition for derivative of vector valued functions is constructed in Section 1.10.

3. Modify the definition of derivative to work with surface functions f(x, y).

The following can be proved and is helpful for calculating directional derivatives.

Theorem 3 (Directional Derivative) If f is differentiable at (x, y), then the directional derivative of f in thedirection of a unit vector ~u = (a, b) is

D~uf(x, y) = ∇f · ~u.

Practice 2.4.2 Calculate the directional derivative of f(x, y) = sin(x+ y) in the following directions.

1. ~u = (1, 0)

2. ~u = (1, 1)

3. ~u = (1, 2)

4. ~u = (2, 1)

2.4.4 Maximum Gradient

Discovery 2.4.3 Use the function f(x, y) = x2 + xy + y2 for the following.

1. Calculate ∇f(1, 3).

2. Calculate the directional derivative of f at (1, 3) in the directions (−7, 5), (5, 6), (5, 7).

3. On a plane graph ∇f(1, 3) and the three directions.

4. Explain in which direction the directional derivative will be greatest for any function f(x, y).

2.5 Chain Rule

On surfaces and in higher dimensions the derivative of a function f : Rn → R is dependent on the direction. Asshown in Section 2.4.1 this direction can derive from a path in the surface (or other function). This can be generalizedto provide a means for calculating derivatives involving composition of functions.

Discovery 2.5.1 Guido is romping on the surface f(x, y) = x2 + xy+ y2. He is following the path P (t) = (t2, sin t).

1. Calculate ∇f(x, y).

2. Calculate P ′(t).

3. Consider time t = 0.

(a) Where is Guido?

(b) What is his direction (2D)?

(c) What is his instantaneous change of altitude?

4. Consider time t = π/2.

(a) Where is Guido?




5. Consider time t.

(a) Where is Guido?



Theorem 4 (Chain Rule 1) If f(x, y) is differentiable and x(t) and y(t) are differentiable functions, then

δf

δt=

(δf

δx,δf

δy

)·(dx

dt,dy

dt

).

Note this could also be written ∇f(x(t), y(t)) · (x′(t), y′(t)). This can be extended to higher dimensional cases.

Theorem 5 (Chain Rule 2) If f(x, y) is differentiable and x(s, t) and y(s, t) are differentiable functions, then

δf

δs=

(δf

δx,δf

δy

)·(dx

ds,dy

ds

)δf

δt=

(δf

δx,δf

δy

)·(dx

dt,dy

dt

).

Practice 2.5.1 1. Why would this be called the ‘chain rule’?

2. Calculate δfdt

for f(x, y) = sinx+ sin yx(t) = cos t, y(t) = sin t.

3. Hot and ColdLet T (x, y) = xy−2 represent the temperature at the point (x, y). Suppose Guido is walking around an elliptical

path given by x2

8 + y2

2 = 1.

(a) Find a parametric representation of the ellipse. Call this vector valued function e(t).

(b) Calculate δT (e(t))dt .

(c) Use this to find the lowest and highest temperature Guido experiences on this walk.

2.5.1 Implicit Differentiation

Discovery 2.5.2 1. Review implicit differentiation as needed.

2. Differentiate y2 = x2 + 2y with respect to x (remember y is implicitly a function of x).

3. Using the chain rule we can show the following.If F (x, y) = 0 and the partials of F are continuous around (x, y) then y is a function of x and

dy

dx= −δF/δx

δF/δy.

4. Use the theorem above to calculate dydx for 1 + xey + x2ey2 = 0.

2.6 Extrema

Just as curves have high and low points, surfaces have high and low points. Finding these extrema in 3D is similarto finding extrema in 2D, but there are additional options. First, review the process of finding extrema in 2D.

2.6. EXTREMA 27

2.6.1 Review

Discovery 2.6.1 Consider the function f(x) = 3x4 − 52x3 + 282x2 − 420x+ 925.

1. Calculate f ′(x).

2. Determine where extrema might occur using f ′(x).

3. Identify maxima and minima.

4. Why does the first derivative identify extrema?

2.6.2 Experiment

Discovery 2.6.2 For the following questions use the surface f(x, y) = y3 − y + x3 − x.

1. Calculate δfδx and δf

δy .

2. Determine where extrema might occur using δfδx and δf

δy .

3. Identify the maxima and minima by comparing your results above to a graph.

4. Can maxima and minima be identified using only the first partials?

2.6.3 Method

Definition 15 (Maximum) A function f : Rn → R has a maximum at A ∈ Rn if and only if for some δ,‖X −A‖ < δ implies f(A) ≥ f(X).

Discovery 2.6.3 1. Explain how this definition matches the definition for maximum on 2D functions.

2. Write the definition for a minimum.

Theorem 6 (Critical Values) If a function f(x, y) has an extreme value at (x, y) then fx(x, y) = 0 and fy(x, y) =0.

Theorem 7 (Extrema) If the second partials are continuous about (a, b), and fx(a, b) = 0, fy(a, b) = 0 then forD(a, b) = fxx(a, b)fyy(a, b)− [fxy(a, b)]2

• If D > 0 and fxx(a, b) > 0, a minimum occurs at (a, b)

• If D > 0 and fxx(a, b) < 0, a maximum occurs at (a, b)

• If D < 0 a saddle point occurs at (a, b).

Discovery 2.6.4 1. How could D(a, b) be re-written under the conditions of Clairaut’s Theorem in Section 2.2.1?

2. Devise a mnemonic to remember D based on the result above.

Practice 2.6.1 Use this theorem to solve the following.

1. Identify extrema for f(x, y) = x2 + xy + y2.

2. Identify extrema for f(x, y) = x3 − x+ y3 − y.

3. Identify extrema for f(α, β) = sinα+ sinβ.

4. Identify extrema for f(x, y) = sin(x2 + y2).

5. Identify all high and low areas for f(x, y) = sin(x2 + y2).


2.7 Extrema with Constraints

Sometimes rather than knowing the highest or lowest point it is important to know the highest or lowest point undercertain constraints. For example knowing the highest point of the mountain is unimportant if your hike does not takeyou over the mountain’s highest point. Instead it is interesting to know the highest point on the mountain (surface)that is also on your path (constraint).

2.7.1 Derivation

If you are unfamiliar with contour maps such as topographical maps, you may wish to learn about them before tryingthis experiment.

- 3

- 2

- 1

0

1

2

3- 3

- 2

- 1

0

1

2

3

- 2

- 1

0

1

2

3

2

1

0

1-

2-

3-

32101-2-3-

2

1.6

21.

80.

40.

0

40.-

40.-

40.-

40.-

80.-

80.-

80.-

80.-

21.-

21.-

21.-

21.-

61.-

61.-

61.-

61.-

Discovery 2.7.1 Guido is walking along the path marked in gray above. The map is a contour map with approximateheights listed. We want to know when Guido is at the highest and lowest altitudes.

1. Could he have been at an extreme altitude anywhere his path crosses from pink to violet?

2. Could he have been at an extreme altitude anywhere his path crosses from violet to blue?

3. Could he have been at an extreme altitude anywhere his path changes from gray to black?

4. Could a high or low occur where some contour line is crossed?

5. Where could a high or low on a path occur?

The problems above illustrate where extrema on a surface can occur when constrained by a curve. The nextproblems illustrate two means for calculating these constrained surface extrema.

The max and min temperature question in Section 2.5 demonstrated one means for calculating extrema along acurve on a surface by parameterizing the curve and using single variate extrema techniques.

Discovery 2.7.2 The following illustrates a second technique, which is extensible to higher dimensions.

1. If a path osculates (just touches) a contour line what is true of the tangents of both curves there?

2. If a path osculates (just touches) a contour line what is true of the normals of both curves there?

3. Remember the following.

(a) From Calculus 1 what does f ′(x) > 0 and f ′(x) < 0 imply about the curve at x?

(b) Why then does f ′(x) = 0 at a maximum or minimum?

(c) From Section 2.4.4 what direction does ∇f point?

(d) Therefore what is the relation between ∇f and the contour lines?

4. If f(x, y) is a surface and g(x, y) = k is some path, what does ∇f = λ∇g have to do with the above statements?Note this technique is called Lagrange Multipliers.

5. Use the idea above to determine the following. Suppose Guido is walking a path given by g(x, y) = x2 + y2 = 1.If the surface on which he is walking is given by f(x, y) = x2 − y2, where is Guido at the highest and lowestaltitude?

2.7. EXTREMA WITH CONSTRAINTS 29

6. Suppose Guido is flying such that his position is always on g(x, y, z) = x2 + y2 + z2 = 1. If the temperature ateach point is given by f(x, y, z) = 2x+ 4y + 6z, what is the hottest temperature experienced by Guido?

7. Suppose Guido is flying such that his position is always on g(x, y, z) = x2 + y2 + z2 = 1, and Swen is flyingsuch that his position is always on h(x, y, z) = x2 − y2 + z2 = 1. If the temperature at each point is given byf(x, y, z) = 2x+ 4y + 6z, what is the hottest temperature experienced by both pilots?

8. Suppose Guido is flying such that his position is always on g(x, y, z) = x2 + y2 + z2 = 1, and Swen is flyingsuch that his position is always on h(x, y, z) = x2 + y2 = 1. If the temperature at each point is given byf(x, y, z) = 2x+ 4y + 6z, what is the hottest temperature experienced by both pilots?

Chapter 3

Integration

3.1 Coordinate Systems

3.1.1 Definition

In two dimension some curves are easier to represent in polar coordinates than in Cartesian coordinates. This is truein three dimensions as well. There are two ways to extend polar coordinates to three dimensions. Watch the threevideos provided for definitions of the three coordinate systems to be used in this course.

• Cartesian coordinates

• Cylindrical coordinates

• Spherical coordinates

Practice 3.1.1 1. Graph by hand each of the points below. Note sketching in 3D by hand will lead to a betterunderstanding of of each coordinate system and its inherent strengths. Graphing by hand is not a skill for dailyuse.

(a) (x, y, z) = (−2, 3, 5)

(b) (r, θ, z) = (√

3, π/3, 4)

(c) (r, θ, z) = (√

3, π/3, 6)

(d) (r, θ, z) = (1, π/2, 4)

(e) (ρ, θ, φ) = (2, 0, 0)

(f) (ρ, θ, φ) = (2, 0, π/6)

(g) (ρ, θ, φ) = (2, π/4, 0)

(h) (ρ, θ, φ) = (2, π/4, π/6)

2. Compare and contrast cylindrical coordinates (3D) to polar coordinates (2D).

3. Compare and contrast spherical coordinates (3D) to polar coordinates (2D).

3.1.2 Conversion of Coordinates

A coordinate system is chosen for a problem, because it is convenient for that problem. However, sometimes it isnecessary to convert from one coordinate system to another. One such area is integration which is covered in a latersection.

Discovery 3.1.1 1. Cylindrical to Cartesian

(a) How are polar coordinates converted to Cartesian coordinates (2D)? List by name the two principles used.

31

32 CHAPTER 3. INTEGRATION

(b) How can cylindrical coordinates be converted to Cartesian coordinates?

(c) Convert the points above in cylindrical coordinates to Cartesian.

(d) Convert the point above in Cartesian coordinates to cylindrical.

2. Spherical to Cylindrical

(a) How can θ be converted?

(b) How can ρ and φ be used to calculate z?

(c) How can ρ and φ be used to calculate r?

(d) Convert one point above in cylindrical coordinates to spherical coordinates.

(e) Convert one point above in spherical coordinates to cylindrical coordinates.

3. Spherical to Cartesian

(a) How can spherical coordinates by converted to Cartesian coordinates?

(b) Convert the points above in spherical coordinates to Cartesian.

(c) Convert the point above in Cartesian coordinates to spherical.

Discovery 3.1.2 1. For cylindrical coordinates what range of values are needed for θ?

2. For spherical coordinates what range of values are needed for θ?

3. For spherical coordinates what range of values are needed for φ?

Practice 3.1.2 Graph each function or equation below by hand if directed or using technology. If using technologygraphing only part of the θ or φ ranges may help in studying the graph. Identify each of graphs as a curve or surface.

1. Cartesian x = 0 (by hand)

2. Cartesian (0, 1, z) (by hand)

3. Cartesian x2 + y2 + z2 = 1.

4. Cylindrical r = 1. (by hand)

5. Cylindrical (1, θ, 0) (by hand)

6. Cylindrical r = 1 + cos θ

7. Cylindrical r = z(1 + cos θ)

8. Spherical ρ = 1 (by hand)

9. Spherical (1, θ, π/2) (by hand)

10. Spherical (1, 0, φ) (by hand)

11. Spherical ρ = 1 + cos(4θ)

12. Spherical ρ = φ

13. Why are cylindrical coordinates called ‘cylindrical’?

14. Why are spherical coordinates called ‘spherical’?

3.2 Scalar Line Integrals

Surfaces consist of many curves. Sections 2.1 and 2.4.1 presented the concept of many curves passing through asurface and how this applies to limits and derivatives. Integrals can also be defined in terms of the curves of a surfaceor other function.

3.2. SCALAR LINE INTEGRALS 33

3.2.1 Derivation

Discovery 3.2.1 Use the function h(x, y) = 200− x2 − y2 for the following.

1. Example 1

(a) Parameterize the curve y = 0. Write as (x(t), y(t)).

(b) Write f(t) = h(x(t), y(t)).

(c) Calculate (x(t), y(t)) for t = 0, 1, 2, 3. What is the distance between each of these points?

(d) Setup an integral to calculate the area between this curve and the xy-plane.

2. Example 2

(a) Parameterize the curve x = 0. Write as (x(t), y(t)).




3. Example 3

(a) Parameterize the curve y = x. Write as (x(t), y(t)).




4. Example 4

(a) Parameterize the curve y = sinx. Write as (x(t), y(t)).




3.2.2 Method

Definition 16 (Scalar Line Integral) For a a smooth, parameterized curve C given by (x(t), y(t)) for t ∈ [a, b]the scalar line integral of a function f : R2 → R is

∫C

f(x, y) ds = lim∆s→0

n∑i=0

f(x∗, y∗)∆s.

For a continuous function f the scalar line integral can be calculated using

∫C

f(x, y) ds =

∫ b

a

f(x(t), y(t))√

(x′)2 + (y′)2 dt.


-Π

-3 Π

4 -Π

2-

Π

4 0Π

4 Π

2 3 Π

4 Π

x

-Π

-3 Π

4

-Π

2

-Π

4

0

Π

4

Π

2

3 Π

4

Π

y

0

1

2

3

4

z

Practice 3.2.1 1. Integrate f(x, y) = sin(x+ y) along the curve y = 4x− 1 for x ∈ [0, 5].

2. Integrate f(x, y) = 5x+ 3y + 7 along the unit circle.

3. Integrate f(x, y) = x3 along the curve y = x3 for x ∈ [−1, 1].

4. Integrate f(x, y) = x sin y along the triangle with vertices (0, 0), (3, 0), (0, 4).

3.3 Double Integrals

The previous sections have discussed surfaces and some of their properties. The next few sections cover propertiesof solids. This includes finding volumes.

3.3.1 Review

Discovery 3.3.1 Consider the curve in 2D f(x) = 1−x2 on the domain [−1, 1]. Use the partition {−1,−1/2, 0, 1/2, 1}of this domain (ordered set) for the following questions. Use the provided video to review the following.

1. Make a table of values using the partition for the x values and calculating the y = f(x) values.

2. Graph each of these points.

3. Sketch the curve through the points.

4. For each point graphed draw a line from the point, orthogonally down to the x-axis (domain).

3.3. DOUBLE INTEGRALS 35

5. Derivation 1

(a) Explain how rectangles can be used to approximate the area between a curve and the x-axis. Your graphfrom above will help.

(b) Explain the role a limit plays in calculating this area.

(c) Write the definition of Riemann integrals and connect each part of the expression to your explanationabove.

6. Derivation 2

(a) You graphed 5 points on the curve. How many points are there on the curve over this domain?

(b) Imagine drawing the lines from all the points orthogonally down to the x-axis. What is the relationship(visually) between these lines and the area between the curve and the x-axis?

(c) A 2D shape (filled) consists of what geometric objects?

(d) Based on your conclusion above, the area between the curve and the x-axis could be calculated using whatmeasurement of the geometric objects?

3.3.2 Volumes

Discovery 3.3.2 Next consider the surface in 3D g(x, y) = 2− x2 − y2 on the domain x ∈ [−1, 1] y ∈ [−1, 1]. Usethe partition {−1,−1/2, 0, 1/2, 1} of both the x and y domains for the following questions. Use the provided video toreview the following.

1. Make a table of values using the partition for the x and y values and calculating the z = g(x, y) values.

2. Graph each of these points.

3. Sketch the surface through these points by sketching the curves parallel to the x and y axes.

4. For each point graphed draw a line from the point, orthogonally down to the xy-plane (domain).

5. Derivation 1

(a) In the 2D example above the lines produce rectangles (2D objects). What shape do the lines in this exampleproduce? Can you calculate the volume of these?

(b) Explain the role a limit plays in calculating this volume.

(c) Write a definition of Riemann integration that will calculate this volume.

6. Derivation 2 will be used below to find an easier way to calculate these volumes.

3.3.3 Calculation

While extending the definition of Riemann integral from the case that calculates area to the case the calculatesvolume is not difficult. Actually calculating these limits, in 2D or 3D, is difficult. The following steps illustrate atheorem that makes calculation easier.

Discovery 3.3.3 Consider the surface in 3D g(x, y) = 2− x2 − y2 on the domain x ∈ [−1, 1] y ∈ [−1, 1].

1. Graph g(x, y) with y = −1 a fixed value.

2. Calculate the area between the line y = −1 and this curve using an integral.

3. Graph g(x, y) with y = 0 a fixed value.

4. Calculate the area between the line y = 0 and this curve using an integral.

5. Graph g(x, y) with y = c a fixed value.


6. Calculate the area between the line y = c and this curve using an integral.

7. Of how many of these curves, which define an area, does the surface, which defines a volume, consist?

8. Graph g(x, y) with x = −1 a fixed value.

9. Calculate the area between the line x = −1 and this curve using an integral.

10. Graph g(x, y) with x = 0 a fixed value.

11. Calculate the area between the line x = 0 and this curve using an integral.

12. Graph g(x, y) with x = a a fixed value.

13. Calculate the area between the line x = a and this curve using an integral.

14. Of how many of these curves, which define an area, does the surface, which defines a volume, consist?

15. Describe in general terms how the areas so calculated could be used to calculate the volume.

Theorem 8 (Fubini) If f is continuous on the rectangular domain R defined by x ∈ [a, b] and y ∈ [c, d] then∫∫R

f(x, y) dA =

∫ b

a

∫ d

c

f(x, y) dydx =

∫ d

c

∫ b

a

f(x, y) dxdy

This theorem still works if the domain is not rectangular and any discontinuities are limited to a finite numberof smooth curves.

Practice 3.3.1 1.

∫ 1

0

∫ 1

0

ex+y dxdy

2.

∫ 1

−1

∫ 1

−1

x2 − y2 dxdy

3. Calculate the volume between the xy-plane and the surface 2 + sinx + sin y for x ∈ [−π/2, π/2] and y ∈[−π/2, π/2].

3.4 Volumes with Non-rectangular Bases

In 2D integrals the regions of integration (1D domain) are all intervals. When an interval is extended to 3D problemsthis type of region of integration becomes a rectangle (2D domain). However, not all shapes have rectangular bases.The problems below illustrate how to extend the method above to these non-rectangular domains.

Remember that the sections above developed volume as a sum of areas of the curves that constitute the surface.Keep this concept in mind as the integrals over rectangular and non-rectangular domains are compared below.

Discovery 3.4.1 Use f(x, y) = 2− x2 − y2 for the following problems.

1. Use the rectangular domain x ∈ [−1, 1] and y ∈ [−1, 1] for the following.

(a) Setup the area integral for the curve from f(x, y) with y = −1.

(b) What are the left and right boundaries when y = −1? Note these are the limits of integration of the area.

(c) Setup the area integral for the curve from f(x, y) with y = 0.

(d) What are the left and right boundaries when y = 0? Note these are the limits of integration of the area.

(e) Setup the area integral for the curve from f(x, y) with y = 1.

(f) What are the left and right boundaries when y = 1? Note these are the limits of integration of the area.

(g) In general what are the left and right boundaries for any curve from f(x, y) with a fixed y value?

2. Use the non-rectangular domain x2 + y2 = 2 for the following.

3.4. VOLUMES WITH NON-RECTANGULAR BASES 37

(a) Setup the area integral for the curve from f(x, y) with y = −1.

(b) What are the left and right boundaries when y = −1? Note these are the limits of integration of the area.

(c) Setup the area integral for the curve from f(x, y) with y = 0.

(d) What are the left and right boundaries when y = 0? Note these are the limits of integration of the area.

(e) Setup the area integral for the curve from f(x, y) with y = 1.

(f) What are the left and right boundaries when y = 1? Note these are the limits of integration of the area.

(g) Setup the area integral for the curve from f(x, y) with y =√

2.

(h) What are the left and right boundaries when y =√

2? Note these are the limits of integration of the area.

(i) In general what are the left and right boundaries for any curve from f(x, y) with a fixed y value? Note tryrepeating the process above with y = yi (using y as a constant).

Practice 3.4.1 To setup these volume integrals use the process above for finding the x boundaries given a fixed yvalue. Note Fubini’s theorem implies you can swap the roles of x and y if it is convenient.

1. Find the volume between the surface s(x, y) = 7x + 2y + 6 and the xy-plane over the region x ∈ [−1, 1] andy ∈ [−1, 1].

2. Find the volume between the surface s(x, y) = 7x+ 2y+ 6 and the xy-plane over the region between x = 1− y2

and x = 0.

3. Find the volume between the surface s(x, y) = sinx+ sin y and the xy-plane over the region enclosed by y = 1,x = 0, and y = x.

4. Determine the solid whose volume is calculated by

∫ 1

−2

∫ 1−x2

x−1

x2 + y2 dy dx

5.

∫ 4

1

∫ 2

√y

sin

(x3

3− x)dx dy

3.4.1 Double Integrals with Cylindrical Coordinates

The double integrals developed in Section 3.3.2 calculate volumes by multiplying the area of a rectangle in the domainby the height of the surface thus calculating the volume of a cuboid (3D rectangle). This same method can be usedfor volumes using cylindrical coordinates.

Discovery 3.4.2 Use the surface z = r for the following problems. The partition for r is {0, 1, 2, 3}. The partitionfor θ is {0, π/2, π, 3π/2}. Use the provided video to review the following.

1. Construct a table of values for z using the partitions above.

2. Sketch lines from these points orthogonally down to the xy-plane.

3. Review how to calculate an area using polar coordinates.

4. Write the integral that will calculate the volume between this surface and the xy-plane over the region r ≤ 3.

To find the volume of one of the shapes complete the following steps.

Discovery 3.4.3 1. Note the object in Figure 3.1 consists of two slices of concentric circles.

2. Calculate the area of the smaller, circular wedge. If you use an integral, you are working far too hard.

3. Calculate the area of the larger, circular wedge.

4. Calculate the area between the smaller and larger circular wedges.

5. Simplify by dividing, collecting, factoring, and using ∆θ and ∆r where appropriate.


6. Calculate the limit of this as ∆θ → 0 and ∆r → 0.

7. This is what is integrated in place of the area of a rectangle (i.e., dx dy).

Practice 3.4.2 Note Fubini’s theorem works regardless of the coordinate system. It states that under the conditionsiterated integrals can be used instead of the double integral for two variables without respect to what those variablesrepresent.

1. Find the volume enclosed between z = 1 + cos r and the xy-plane for r ∈ [0, 2π].

2. Find the volume enclosed between z = 1 and z = r.

3. Find the volume enclosed between z = 1 + sin θ and the xy-plane for r ∈ [0, 2].

4. Check if the following limit exists. limr→0

1 + sin θ.

3.5 Parametric Surfaces

Previous sections introduced parametric curves. These are topologically one dimensional relations that give a contin-uous set of points given a single parameter. Section 1.8 introduced planes, a type of surface which is a topologicallytwo dimensional relation that requires two parameters. Planes are not the only surfaces that can be parameterized.

Discovery 3.5.1 Graph the following parametric surfaces using technology then answer the questions.

1. P (t, u) = (1, 2, 2)t+ (−1, 1,−1)u+ (7, 2, 4). Why does adding (1, 2, 2)t to (−1, 1,−1)u+ (7, 2, 4) turn a line intoa plane?

2. S(t, θ) = (cos θ, sin θ, 0) + (−1, 1,−1)t+ (7, 2, 4). Why does adding (cos θ, sin θ, 0) to (−1, 1,−1)t+ (7, 2, 4) turna line into a (leaning) cylinder?

3. F (t, θ) = (0, cos θ, sin θ) + (t, t2, 0). Why does adding (0, cos θ, sin θ) to (t, t2, 0) turn a parabola into a pipesegment (parabolic cylinder)?

3.6 Surface Area

3.6.1 Illustration

Discovery 3.6.1 This section illustrates a formula. Use the parameterized surface S(t, u) = (et, eu, tu). Theseproblems illustrate constructing a plane tangent to a parametric surface.

1. Calculate δSδt =

(δSx

δt ,δSy

δt ,δSz

δt

).

2. Calculate δSδu =

(δSx

δu ,δSy

δu ,δSz

δu

).

r1

r2θ1

θ2

Figure 3.1: Cylindrical Volume

3.6. SURFACE AREA 39

3. Evaluate both vectors above at (t, u) = (0, 0).

4. Use these vectors to construct the equation of a plane. Call it P (t, u).

5. Graph the surface and the plane together.

Discovery 3.6.2 This set of problems illustrate why the plane constructed above is a tangent plane at that point.

1. S(t, u) is a surface. What is S(t, k) for some constant value k?

2. What is δSδt with respect to the previous problem?

3. S(t, u) is a surface. What is S(k, u) for some constant value k?

4. What is δSδu with respect to the previous problem?

5. Use this to explain why the plane P (t, u) is tangent to S(t, u) at (0, 0).

Discovery 3.6.3 These problems illustrate the formula.

1. Calculate the cross product δSδt and δS

δu .

2. Calculate the norm of this cross product.

3. Evaluate P (t, u) (the tangent plane from the previous problems) at the points (0, 0), (1, 0), (1, 1), and (0, 1).

4. Calculate the area of the rectangle formed by these four points.

5. Compare these two scalar results.

6. How could the total surface area of this surface for t ∈ [0, 10] and u ∈ [0, 10] be calculated?

3.6.2 Calculation

For a smooth surface S(t, u) the surface area can be calculated as∫∫ ∥∥∥∥δSδt × δS

δu

∥∥∥∥ dA.Practice 3.6.1 For each of the following surfaces. Graph the surface using technology, then setup the integral forits surface area. Finally use technology to calculate the result.

1. S(θ, φ) = (cos θ sinφ, sin θ sinφ, cosφ) for θ ∈ [0, 2π] and φ ∈ [0, π].

2. S(θ, z) = (cos θ, sin θ, z) for θ ∈ [0, 2π] and z ∈ [0, 1].

3. S(θ, z) = (z cos θ, z sin θ, z) for θ ∈ [0, 2π] and z ∈ [0, 1].

4. f(x, y) = x2 + y2 for x ∈ [−1, 1] and y ∈ [−1, 1].

3.6.3 Scalar Surface Integrals

Section 3.2 presented a method for calculating the scalar integral along a parameterized curve. Such an integral canbe interpreted as the total affect of a function along a curve. This idea can be extended to calculating the scalarintegral over a parameterized surface. Such an integral can be interpreted as the total affect of a function over asurface.

Discovery 3.6.4 1. Write the formula for the scalar line integral for a function f along a curve C(t).

2. Explain the role of f(C(t)) in this integral.

3. Explain the role of√C ′(t) in this integral.


4. If the temperature in an oven is uniformly 375◦ F at each point, how much heat is absorbed by the curveC(t) = (cos t, sin t, 1)?

5. Based on the pattern of the scalar line integral write the formula for the scalar surface integral for a functionf over a surface S(t, u).

6. Explain the role of f(S(t, u)) in this integral.

7. Explain the role of∣∣ δSδt ×

δSδu

∣∣ in this integral.

8. If the temperature in an oven is uniformly 375◦ F at each point, how much heat is absorbed by the surface~s(t, s) = (2t+ 3s+ 1, t+ s+ 1, t− s+ 1) for t ∈ [0, 1], s ∈ [0, 1]?

For a smooth surface S(t, u) the scalar surface integral can be calculated as∫∫f(S(t, u))

∥∥∥∥δSδt × δS

δu

∥∥∥∥ dA.

3.7 Triple Integrals in Cartesian

To find the volume between a surface and the xy-plane, a double integral partitions the domain (xy-plane) intosquares which are the bases for cuboids (rectangular boxes). To calculate the same volume a triple integral partitionsthe x, y, and z axes. The result divides a volume into cubes. See Figure 3.2.

-1

-

3

4-

1

2-

1

40

1

41

23

41

x

-1

-

3

4

-

1

2

-

1

4

0

1

4

1

2

3

4

1

y

0.0

0.5

1.0

1.5

2.0

Figure 3.2: Triple Integral in Cartesian

For double integrals the cuboids’ volumes are height times width times depth which become the f(x, y)×dx×dy.The limits of integration determine the domain for the shape. For triple integrals the cubes’ volumes are dx×dy×dz.The limits of integration determine the domain for the shape including the vertical. As a result the process of settingup triple integrals is the same as double integrals, but there is a third pair of limits of integration. Watch the providedvideo for an illustration of setting up a triple integral for volume.

Practice 3.7.1 Use triple integrals to calculate each of the following.

1. Find the volume enclosed between f(x, y) = 36− 4x2 − 9y2 and the xy-plane.

2. Find the volume enclosed between f(x, y) = 1− x2 − y2 and g(x, y) = 1− 2x.

3. Find the volume of x2 + y2 + z2 = 1.

3.8. TRIPLE INTEGRALS IN CYLINDRICAL 41

3.8 Triple Integrals in Cylindrical

For double integrals in Cartesian coordinates the integral (over a rectangular region) looks like∫ b

a

∫ d

c

f(x, y) dy dx.

The triple integral in Cartesian for the same calculation looks like∫ b

a

∫ d

c

∫ f(x,y)

0

dz dy dx.

Notice how the integrand and limits of integration are changed. The change is similar for non-rectangular regions,since they do not change the z coordinates.

Discovery 3.8.1 For double integrals in cylindrical coordinates the integral (over a circular region) looks like∫ b

a

∫ d

c

z(r, θ)r dr dθ.

1. Write the general form of the triple integral for the volume given by the double integral above.

Practice 3.8.1 1. Find the volume enclosed between z = 1− r and the xy-plane using a triple integral.

2. Find the volume enclosed between z = 1 + cos θ and the xy-plane using a triple integral.

3. Find the volume enclosed between z = 1− r and the xy-plane over the region r = cos θ using a triple integral.

3.9 Triple Integrals in Spherical

The fundamental shapes for integrating in each coordinate system along with the formula are shown in Figure 3.3.The derivation of the volume formula for the spherical shape is found in Section 3.10.

dx dy dz r dr dθ dz ρ2 sinφ dρ dφ dθ

Figure 3.3: Fundamental Shapes for Triple Integrals

Practice 3.9.1 Setup the following triple integrals. Use technology as needed for showing all steps of the calculations.

1. Find the volume inside ρ = sin θ.

2. Find the volume inside ρ = φ.

3. Find the volume inside ρ = cosφ.

3.10 Change of Variable

While Cartesian, cylindrical, and spherical coordinates enable integration of some functions over some intervals, theyare not convenient for all functions over and all intervals. Sometimes it is possible to adjust the function or intervalusing a carefully chosen change of variable.


3.10.1 Review

Discovery 3.10.1 First integrate the following.

1.

∫ 3

0

e3x dx

2.

∫ 1

0

sin(x2)2x dx

3.

∫ π/2

0

sin57 θ cos θ dθ

Note that your substitutions are all smooth functions.

3.10.2 Terminology

A function that maps points to points is called a planar transformation. Most graphics effects you use in photoediting software fall in this category (e.g., scaling, reflection, translation, fish eye lens). Following is one example.Consider for P = (x, y), C(P ) = 1

x2+y2 (x, y). Figure 3.4 shows the original and transformed versions of an image.

Figure 3.4: Transformation Example

3.10.3 Experiment

The following examples will lead to and understanding of how the transformations works and when they are useful.

Discovery 3.10.2 • u(x, y) = −y and v(x, y) = x

• u(x, y) = 2x and v(x, y) = 3y.

• u(x, y) = x and v(x, y) = y + x2.Use the grid in Figure 3.5 for these problems. For each of the transformations above complete the following steps.

1. Map each of the nine labeled points (x, y) to the transformed points (u, v).

2. Sketch the transformed points (on u and v axes).

3. Describe the affect of the transformation on shape and size.

4. Calculate the area of the original square.

5. For each transformation calculate the area of the new shape.

6. Compute

[ δuδx

δvδx

δuδy

δvδy

]

3.10. CHANGE OF VARIABLE 43

H0,0L

H0,1L

H0,2L

H1,0L

H1,1L

H1,2L

H2,0L

H2,1L

H2,2L

Figure 3.5: Grid for Transformations

7. Compute D = δuδx

δvδy −

δuδy

δvδx .

8. Compare the area of the original square, the area of the transformed square, and the value D.

Discovery 3.10.3 One more property is needed. Consider x(u, v) = 12 (u+ v) and y(u, v) = 1

2 (u− v).

1. Calculate

∣∣∣∣ δxδu

δxδv

δyδu

δyδv

∣∣∣∣ .2. Reading carefully, calculate

∣∣∣∣ δxδv

δxδu

δyδv

δyδu

∣∣∣∣3. Reading carefully, calculate

∣∣∣∣ δyδu

δyδv

δxδu

δxδv

∣∣∣∣ .4. Compare the results.

5. Can any transformation make area negative?

3.10.4 Method

Definition 17 (Jacobian) For a transformation T given by x = t1(u, v) and y = t2(u, v) the Jacobian is∣∣∣∣ δxδu

δxδv

δyδu

δyδv

∣∣∣∣For any smooth transformation x(u, v), y(u, v), the double integral over a region S is given by∫∫

S

f(x(u, v), y(u, v))

∣∣∣∣ δxδu

δxδv

δyδu

δyδv

∣∣∣∣ du dvwhere ∣∣∣∣ δx

δuδxδv

δyδu

δyδv

∣∣∣∣ =δx

δu

δy

δv− δx

δv

δy

δu.


For any smooth transformation x(u, v, w), y(u, v, w), z(u, v, w), the triple integral over a region S is given by∫∫∫S

f(x(u, v, w), y(u, v, w), z(u, v, w))

∣∣∣∣ δ(x, y, z)δ(u, v, w)

∣∣∣∣ du dv dwwhere

∣∣∣ δ(x,y,z)δ(u,v,w)

∣∣∣ is the determinant of the 3× 3 gradient matrix (Jacobian) given below.

∣∣∣∣∣∣δxδu

δxδv

δxδw

δyδu

δyδv

δyδw

δzδu

δzδv

δzδw

∣∣∣∣∣∣ =

δx

δu

[(δy

δv

)(δz

δw

)−(δz

δv

)(δy

δw

)]−δxδv

[(δy

δu

)(δz

δw

)−(δz

δu

)(δy

δw

)]+δx

δw

[(δy

δu

)(δz

δv

)−(δz

δu

)(δy

δv

)]Practice 3.10.1 Integrate the following functions using the specified change of coordinates and intervals.

1. f(x, y) = x+ y. r ∈ [0, 1] and θ ∈ [0, 2π] with transformation x(r, θ) = r cos θ and y(r, θ) = r sin θ.

2. f(x, y, z) = x + y + z. x(ρ, φ, θ) = ρ cos θ sinφ, y(ρ, φ, θ) = ρ sin θ sinφ, z(ρ, φ, θ) = ρ cosφ for ρ ∈ [0, 1],φ ∈ [0, π], and θ ∈ [0, 2π].

3. f(x, y) = xy over the convex region with vertices (1, 1), (1,−5), (9, 1), and (11,−5) using the transformationx = 1 + 2v and y = 1− 3u.

4.∫ 1

0

∫√1−y20

ex2+y2 dx dy using a useful transformation of your choosing.

3.11 Integrals in Action: Centroid

3.11.1 Derivation

In childhood you may have learned how to balance on a teeter-totter with someone significantly lighter or heavierthan you. This illustrates a basic statement about the physics behind levers. Each of the images in Figures 3.6 to3.8 show balanced conditions.

Discovery 3.11.1 1. For Figure 3.6 complete the following steps.

(a) Suppose the coordinate of the center is 0.

(b) Write coordinates for each of the weights (balls) based on this center.

(c) Use these coordinates to prove this is balanced.

(d) Suppose the coordinate of the center is 5.

(e) Write coordinates for each of the weights (balls) based on this center.

(f) Use these coordinates to prove this is balanced.

2. For Figure 3.7 complete the following steps.

(a) Label the center point c.

(b) Label each location xi where i = 1, 2, 3.

(c) Write the equation that shows the balance using these (variable) coordinates.

(d) Solve this equation for c.

3.11. INTEGRALS IN ACTION: CENTROID 45

3. For Figure 3.8 complete the following steps.

(a) Label the center point (c, d).

(b) Label each location (xi, yi) where i = 1, 2, 3, 4..

(c) Write the equation that shows the balance using these (variable) coordinates.

(d) Solve this vector equation for (c, d).

126

100 50

Figure 3.6: Centroid 1 (1D)

12

6

100 5050

9


3.11.2 Definition

Equation 1 (Centroid) The centroid (geometric center) of a region D is given by

(x, y) =

(∫∫D

x dA,

∫∫D

y dA

)/

(∫∫D

1 dA

)Note the similarity of this equation to an average. A centroid can be thought of as the average location of an

region. For many averaging problems not every point is of equal value. In these cases a weight is assigned to eachpoint and a weighted average is calculated. For centroids the weighted average is the center of mass.

Equation 2 (Center of Mass) The center of mass of a region D is given by

(x, y) =

(∫∫D

xρ(x, y) dA,

∫∫D

yρ(x, y) dA

)/

(∫∫D

ρ(x, y) dA

)


100

50

50

50

9

6

12

6

6 6

12

12


Practice 3.11.1 Note that these equations can be extended to higher (and lower) dimensions.

1. Find the centroid of the triangle with vertices (0, 0), (5, 0), and (0, 5).

2. Find the center of mass of the triangle with vertices (0, 0), (5, 0), and (0, 5) and density function ρ(x, y) = x+y.

3. Find the center of mass of the region enclosed between y = 1−x2 and y = 0 with density function ρ(x, y) = x2+y.

4. Find the centroid of the object enclosed within f(x, y) = 1− x2 − y2 and z = 0.

5. Find the center of mass of the object enclosed within f(x, y) = 1 − x2 − y2 and z = 0 with density functionρ(x, y, z) = x2 + y2.

6. Find the center of mass of the region enclosed within r = cos θ and ρ(r, θ) = r.

7. Find the center of mass of the region enclosed within ρ = φ (spherical coordinates) and density functiond(ρ, φ, θ) = ρ.

8. Find the center of mass of the wire structure (space curve) given by the parametric function x(t) = cos t,y(t) = sin t, and z(t) = sin(2t) with density function ρ(x, y, z) = 1.

Chapter 4

Vector Fields

4.1 Vector Fields

The calculus of functions f : R → R like f(x) = x3 + 3x2 − 5x + 2 is the topic of Calculus 1. These functionsmap (1D) points to (1D) points. Chapter 1 presented the vectors and points and how they are related. Thecalculus of functions P : R → Rn like P (t) = (t2, t3) followed. These functions map a parameter (not graphed)to points or vectors (n-dimensional). The following chapters presented the calculus of functions g : Rn → R likeg(x, y, z) = x3 + y3 + z3 +xyz. These functions map points (2D, 3D, or higher) to points (1D). This chapter presentsthe calculus of functions ~v : Rn → Rm. These functions map points (n-dimensional) to vectors (m-dimensional).Because the outputs represent vectors integration is significantly different.

4.1.1 Presentation

Practice 4.1.1 Because a vector field maps a point to a vector, vector fields are represented by graphing arrows(vectors) at each point. Use technology to graph the following vector fields.

1. ~u(x, y) = (y, x).

2. ~v(x, y) = (y,−x).

3. ~w(x, y) = (2x− y,−x+ 2y).

4. ~z(x, y, z) = (yz, xz, xy).

4.1.2 Interpretation

Vector fields appear in many applications. The problems below illustrate vector fields with common concepts.

Discovery 4.1.1 1. If a leaf drops in the stream in Figure 4.1 at (x, y) = (0, π) where will it go?

2. If a leaf drops in the stream in Figure 4.1 at (x, y) = (0, 0) where will it go?

3. If a leaf drops in the stream in Figure 4.2 at (x, y) = (0, π) where will it go?

4. Suppose a neutrally buoyant ball with radius 0.1 has its center at (0, 0, 0.5) in the stream in Figure 4.3. Whatis the force at the top of the ball? What is the force at the bottom of the ball? What will this cause the ball todo?

4.2 Vector Line Integrals

Section 3.2 presented integrals of point valued functions along a fixed curve. Section 4.1.2 presented the effect of avector field on an inactive object. This section develops the effect of a vector field on an object traveling a knowncurve not determined by the field.

47

48 CHAPTER 4. VECTOR FIELDS

0 Π

2Π 3 Π

22 Π 5 Π

23 Π 7 Π

24 Π

0

Π

2

Π

3 Π

2

2 Π

x

y

Figure 4.1: A Stream (sin(3π sin(x)/16 + π/4), sin(π sin(y)/4)

0 Π

2Π 3 Π

22 Π 5 Π

23 Π 7 Π

24 Π

0

Π

2

Π

3 Π

2

2 Π

x

y

Figure 4.2: Another Stream (1, cosx)

4.2.1 Derivation

Discovery 4.2.1 1. Guido guides a boat in a river in the direction (5, 0). The current pushes the boat (2√

2, 2√

2).

(a) Sketch these vectors.

(b) Calculate the portion of the current that is beneficial to Guido. It is only beneficial if it helps him maintainhis course.

2. Guido guides a boat in a river along the path (t, cos t). The current has a constant value of (2, 0).

(a) What direction is the boat heading at t = 0? t = π/2?

(b) How much help is the current at t = 0? t = π/2?

3. Under what circumstance will a vector field most help/hinder motion along a path?

4.2. VECTOR LINE INTEGRALS 49

0

1

2

x

-1

0

1

y

0.0

0.5

1.0

z

Figure 4.3: 3D Stream (z(2− y2), 0, 0)

4.2.2 Evaluation

Definition 18 (Vector Line Integral) For a continuous vector field ~F and a smooth curve C given in parametricby ~r(t) ∫

C

~F · d~r =

∫ b

a

~F (~r(t)) · ~r′(t) dt.

Practice 4.2.1 Evaluate the following vector line integrals.

1. ~F (x, y) = (5x, 3y) along ~r(t) = (2t, 3t) for t ∈ [0, 2].

2. ~F (x, y) = (5x, 3y) along the line from (0, 0) to (5, 1)

3. ~F (x, y) = (5x, 3y) along the line from (5, 1) to (0, 0)

4. ~F (x, y) = (5x, 3y) along the unit circle

5. ~F (x, y) = (5x, 3y) along the triangle with vertices (0, 0), (3, 0), and (0, 4).


4.3 Vector Field Theorems

The following sections will develop properties of nice vector fields and how that affects vector line integrals.

4.3.1 Path Independence

The following problems illustrate a property of nice vector fields.

Discovery 4.3.1 For the following problems use ~F (x, y) = (2x, 2y). The vector field and curves are shown in Figure4.4.

1. Calculate the vector line integral of ~F along each of the following curves.

(a) C1: x = t, y = t, t ∈ [0, 1].

(b) C2: x = t, y = t2, t ∈ [0, 1].

(c) C3: x = t, y = t3, t ∈ [0, 1].

(d) C4: x = 1− cos t, y = sin t, t ∈ [0, π/2].

2. Using the graphs determine list the curves in order of length.

3. Do the integral values calculated above vary with the length of the curve?

4. Using the graphs explain your answer.

5. Evaluate

∫C

(2x−y, 2y−x) ·d~r for C the sequence of lines along the following points. (0, 0), (1, 1), (5, 6), (0, 7).

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 4.4: Nice Vector Field with Curves

4.3.2 Fundamental Theorem of Vector Line Integrals


Discovery 4.3.2 For the following problems use F (x, y) = x2 + y2.

4.3. VECTOR FIELD THEOREMS 51

1. Calculate ∇F. Calculate F (x, y) at the beginning and end of the following curves.

(a) C1: x = t, y = t, t ∈ [0, 1].

(b) C2: x = t, y = t2, t ∈ [0, 1].

(c) C3: x = t, y = t3, t ∈ [0, 1].

(d) C4: x = 1− cos t, y = sin t, t ∈ [0, π/2].

2. Compare the results of the line integrals in the previous section to the evaluation of F.

3. What function was integrated in the line integrals (be general)?

4. Where did you see this in Calculus 1?

4.3.3 Closed Curves


Discovery 4.3.3 For the following problems use ~F (x, y) = (2x, 2y) and the curves from the sections above.

1. Calculate without the Fundamental Theorem of Line Integrals the vector line integrals of ~F over the followingcurves.

(a) C4 but change t ∈ [0, 2π].

(b) Out C1 and back by C2.

(c) Out C2 and back by C3.

(d) C5: x = 5 cos t, y = 3 sin t, t ∈ [0, 2π].

2. What is the result? How does this result depend on the result of Section 4.3.1?

3. If the Fundamental Theorem of Line Integrals applies, would it have predicted this result? Explain.

4. Explain the result using the graphs in Figure 4.4.

4.3.4 Conservative Fields

The previous sections illustrate properties of nice vector fields. These are properties of “conservative” vector fields.

Discovery 4.3.4 The following problems illustrate the definition of conservative fields.

1. For each of the following fields calculate the vector line integral of the given field over C1 : (t, 1− t) for t ∈ [0, 1]and C2 : (cos t, sin t) for t ∈ [0, π/2]. Do not use the Fundamental Theorem of Line Integrals as it does notapply to some of the fields. Technology may be used for the calculations.

(a) ~F (x, y) = (y, x).

(b) ~G(x, y) = (y,−x).

(c) ~H(x, y) = (2x− y, 2y − x).

(d) ~J(x, y) = (2x− y, 2y).

2. For each of the functions above check if it is the gradient of some function. (Hint: calculate∫~Fx dx and∫

~Fy dy and compare.)

3. From Section 2.2.1 what is true for nice functions f(x, y) about δ2fδxδy and δ2f

δyδx?

4. Because conservative fields are gradients, what should be true about δ ~Fx

δy andδ ~Fy

δx .

Practice 4.3.1 Integrate each function along the specified curve. Use the most efficient technique.


1. ~f(x, y) = (x− y, x− 2) along the unit circle.

2. ~g(x, y) = (cos y,−x sin y) along the triangle with vertices (0, 0), (5, 0), (6, 0).

3. ~h(x, y) = (|x|, |y|) along the line y = x for x ∈ [−1, 1].

4. ~j(x, y, z) = (2x+ yz, 2y + xz, 2z + xy) along the curve (t, cos(πt/5), e3t) for t ∈ [0, 7].

4.4 Divergence

4.4.1 Illustration

Consider ~F (x, y, z) = (P (x, y, z), Q(x, y, z), R(x, y, z)). Note this has 3 inputs and three outputs. For convenienceδ ~Fx

δx = δP (x,y,z)δx .

Discovery 4.4.1 1. If people arrive at the gas station at a rate of 6 every 15 minutes and leave at a rate of 3every 15 minutes what will be the result?

2. If people are arriving at lover’s leap at a rate of 1 person per hour and leaving at a rate of 1 per minute, whatwill be the result?

3. In terms of rates what information do δ ~Fx

δx ,δ ~Fy

δy , and δ ~Fz

δz provide?

4. As a result how can δ ~Fx

δx +δ ~Fy

δy + δ ~Fz

δz be described?

4.4.2 Definition

Definition 19 (Div) For a vector field F , the divergence is

div ~F =δ ~Fxδx

+δ ~Fyδy

+δ ~Fzδz

.

Practice 4.4.1 Calculate the divergence as directed and compare the result to the vector field images.

1. ~F (x, y, z) = (−x2,−y2,−z2)

(a) div(~F )(0, 0, 0)

(b) div(~F )(−1,−1,−1)

(c) div(~F )(1, 1, 1)

2. ~G(x, y, z) = (x3, y3, 3)

(a) div(~G)(0, 0, 0)

(b) div(~G)(−1,−1,−1)

(c) div(~G)(1, 1, 1)

3. ~H(x, y, z) = (x, xy, 0)

(a) div( ~H)(0, 0, 0)

(b) div( ~H)(−2,−1,−1)

(c) div( ~H)(2, 1, 1)

4. ~J(x, y, z) = (z3y, x2z, xy)

(a) div( ~J)(0, 0, 0)

(b) div( ~J)(−1,−1,−1)

(c) div( ~J)(1, 1, 1)

4.5. CURL 53

4.5 Curl

4.5.1 Definition

Definition 20 (Curl) For a 3D vector field ~F the curl at a point is

curl ~F =

(δ ~Fzδy− δ ~Fy

δz,−

(δ ~Fzδx− δ ~Fx

δz

),δ ~Fyδx− δ ~Fx

δy

).

Discovery 4.5.1 1. Review Clairaut’s Theorem in Section 2.2.1.

2. Compare Clairaut’s Theorem to the curl formula.

3. Review the criteria for a vector field to be conservative.

4. Write a new condition for a field to be conservative using these conclusions.

Practice 4.5.1 Calculate the curl at the indicated points. Determine which fields are conservative.

1. ~F (x, y, z) = (−x2,−y2,−z2)

(a) curl(~F )(0, 0, 0)

(b) curl(~F )(−1,−1,−1)

(c) curl(~F )(1, 1, 1)

2. ~G(x, y, z) = (x3, y3, 3)

(a) curl(~G)(0, 0, 0)

(b) curl(~G)(−1,−1,−1)

(c) curl(~G)(1, 1, 1)

3. ~H(x, y, z) = (x, xy, 0)

(a) curl( ~H)(0, 0, 0)

(b) curl( ~H)(−2,−1,−1)

(c) curl( ~H)(2, 1, 1)

4. ~J(x, y, z) = (z3y, x2z, xy)

(a) curl( ~J)(0, 0, 0)

(b) curl( ~J)(−1,−1,−1)

(c) curl( ~J)(1, 1, 1)

4.6 Green’s Theorem

The following theorem provides a simplification technique for some vector line integrals.

Theorem 9 (Green’s Theorem) For a simple, closed region D inside a curve C parameterized so that D is on

the left of C and a vector field ~F = (M(x, y), N(x, y)) whose derivatives are continuous in the region∫C

~F · d~r =

∫ b

a

~F (x(t), y(t)) ·(dx

dt,dy

dt

)dt

=

∫ b

a

M(x(t), y(t)) dx+N(x(t), y(t)) dy

=

∫∫D

(δN

δx− δM

δy

)dx dy.


Practice 4.6.1 1. Confirm the theorem by completing the following steps. ~F (x, y) = (xy, y2) enclosed withiny = x, y = x2.

(a) Parameterize the curves.

(b) Integrate the line integral∫ ba~F (x(t), y(t)) ·

(dxdt ,

dydt

)dt

(c) Integrate∫∫D

(δNδx −

δMδy

)dx dy

(d) Compare.

2. Integrate ~F (x, y) = (2xy + 2, x2 + 3x + 1) along the triangle with vertices (−3, 1), (3, 1), and (1, 3) traversedcounterclockwise. Do not use Green’s Theorem.

3. Repeat the previous problem using Green’s Theorem.

4.7 Surface Vector Integrals

Section 4.2 presented how to calculate the affect of a vector field on a path. This section presents how to calculatethe affect of a vector field on a surface.

4.7.1 Illustration

-5

0

5

-5

0

5

-5

0

5

Discovery 4.7.1 The following problems illustrate how a field affects a surface.

1. Rank the three surfaces according to the effect of the field on them. It may help to think of the surfaces as sailsand the field as wind.

2. Why is the effect greater on some surfaces?

3. Locate a point on the plane (red surface) where the affect of the field is maximum.

4. Explain the relationship of the field vector to the plane at that point.

5. Locate a point on the plane (red surface) where the affect of the field is not maximal.

6. Explain the relationship of the field vector to the plane at that point.

7. Explain how you could calculate the portion of the field vector that is maximally affecting the surface. Thisis the same as calculating the portion of the field vector that is in the same direction as a vector that wouldmaximally affect the surface (like the previous questions).

4.8. STOKE’S THEOREM 55

4.7.2 Calculation

For a continuous vector field F (x, y, z) and an parameterized oriented surface S(t, u) the surface integral of F overS is ∫∫

~F · ~n dS =

∫∫~F (S(t, u)) ·

(δS

δt× δS

δu

)dA.

Practice 4.7.1 Calculate the vector surface integral for each field over the specified surface.

1. F (x, y, z) = (yz, xz, xy) over the cone S(θ, h) = (h cos θ, h sin θ, h) for θ ∈ [0, 2π] and h ∈ [0, 1].

2. F (x, y, z) = (y + z, x+ z, x+ y) over the helicoid S(θ, u) = (u cos θ, u sin θ, θ) for θ ∈ [0, π] and u ∈ [0, 1].

4.8 Stoke’s Theorem

The following is an integral identity that is useful for some calculations.For a oriented, smooth surface D with positive orientation inside a simple, closed curve C and a vector field

~F (x, y, z) whose derivatives are continuous in the surface

∫C

~F · d~r =

∫ b

a

~F (x(t), y(t), z(t)) ·(dx

dt,dy

dt,dz

dt

)dt

=

∫∫D

curl ~F · d~S

=

∫∫D

curl ~F · ~n dS.

Discovery 4.8.1 1. Compare Green’s and Stoke’s Theorems below.

Green’s Theorem

∫C

~F · d~r =

∫∫D

(δN

δx− δM

δy

)dx dy.

Stoke’s Theorem

∫C

~F · d~r =

∫∫D

curl ~F · ~n dS.

Practice 4.8.1 1. Calculate the effect of the vector field F (x, y, z) = (yz, xz, 0) on the outer edge of the spi-ral ramp S(r, θ) = (r cos θ, r sin θ, θ) with r ∈ [0, 10] and θ ∈ [0, 2π]. The parameterized path is ~r(θ) =(10 cos θ, 10 sin θ, θ).

2. Use Stoke’s Theorem to calculate the effect.

Calculus III - math.uaa.alaska.edu

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