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### Calculus III

#### Contents1 3-D Geometry and Vectors1.1 Three-Dimensional Coordinate Systems1.2 Vectors1.3 Dot Product1.4 Cross Product1.5 Equations of Lines and Planes1.6 Cylinders and Quadric Surfaces1.7 Vector Functions and Space Curves1.8 Arc Length and Curvature1.9 Velocity and Acceleration1.10 Cylindrical and Spherical Coordinates2 Partial Derivatives2.1 Functions of Several Variables2.2 Limits and Continuity2.3 Partial Derivatives2.4 Tangent Planes and Differentials2.5 The Chain Rule2.6 Directional Derivatives and the Gradient Vector2.7 Maximum and Minimum Values2.8 Lagrange Multipliers3 Multiple Integrals3.1 Double Integrals over Rectangles3.2 Iterated Integrals3.3 Double Integrals over General Regions3.4 Double Integrals in Polar Coordinates3.5 Applications of Double Integrals3.6 Surface Area3.7 Triple Integrals3.8 Triple Integrals in Cylindrical and Spherical Coordinates3.9 Change of Variable in Multiple Integrals4 Vector Calculus4.1 Vector Fields4.2 Line Integrals4.3 Fundamental Theorem for Line Integrals4.4 Greens' Theorem4.5 Curl and Divergence4.6 Parametric Surfaces and Their Areas4.7 Surface Integrals4.8 Stokes' Theorem4.9 The Divergence Theorem

Recommended Textbooks:

Useful Resources:
• Collected Topic and Course Resources
• Lectures
• Practice Problems
• InterAct Math: Similar to MyMathLab or Mastering Math, choose a Calculus book from the drop down list to access practice problems. Many have explanations and solutions.
• Exams

### 3-D Geometry and Vectors

Corresponding Chapters
Briggs [1e]     - Ch. 11
Stewart [5e]     - Ch. 12
Stewart [6e]     - Ch. 13

#### Three-Dimensional Coordinate Systems

Stewart [5e]: 12.1
Stewart [6e]: 13.1

Problems:
Stewart [6e], 13.1
#5, 9, 11, 13, 15, 21, 27, 37, 38, 41.

#### Vectors

Stewart [5e]: 12.2
Stewart [6e]: 13.2

Problems:
Briggs 11.1: Vectors in the Plane
#15, 18, 19, 21, 26, 29, 31, 32, 35, 37.

Briggs 11.2: Vectors in 3D
#2, 3, 5, 8, 11, 19, 22, 24, 29, 32, 35.

Stewart 6e, 13.2
#3, 7, 11, 17, 19, 21, 25, 29, 31, 33, 37, 43.

#### Dot Product

Briggs: 11.3
Stewart [5e]: 12.3
Stewart [6e]: 13.3

Problems:
Briggs 1e, 11.3
#4, 10, 11, 14, 18, 19, 23, 26, 33.

Stewart 6e, 13.3
#1, 7, 9, 11, 19, 23, 27, 31, 39, 47, 53, 59.

#### Cross Product

Briggs: 11.4
Stewart [5e]: 12.4
Stewart [6e]: 13.4

Problems:
Briggs [1e], 11.4
#4, 8, 9, 11, 13, 17, 18, 20, 21, 24, 25, 26, 27, 29, 31, 33, 35, 45, 55, 56.

Stewart [6e], 13.4
#5, 7, 9, 11, 15, 23, 25, 29, 35, 39, 43, 45.

#### Equations of Lines and Planes

Briggs: 11.5?
Stewart [5e]: 12.5
Stewart [6e]: 13.5

Problems:
Stewart [6e], 13.5
#3, 5, 9, 15, 19, 25, 31, 39, 43, 45, 51, 53, 61, 63, 65, 67.

Stewart [5e]: 12.6
Stewart [6e]: 13.6

Problems:
Briggs [1e], 11.5
#9, 11, 13, 15, 19, 35, 41, 43, 47, 51.

Stewart [6e], 13.6
#1, 9, 19, 21, 23, 27, 29, 35, 41, 43, 45.

#### Vector Functions and Space Curves

Stewart [5e]: 13.1
Stewart [6e]: 14.1
Derivatives and Integrals - Stewart [6e]: 14.2

Problems:

Briggs [1e], 11.5
#9, 11, 13, 15, 19, 35, 41, 43, 47, 51.

Stewart [6e], 14.1
#1, 5, 11, 15, 19, 25, 33, 35, 39.

Derivatives and Integrals
Stewart [6e], 14.2
#5, 13, 16, 19, 21, 25, 31, 35, 39, 49.

#### Arc Length and Curvature

Briggs: 11.8 & 11.9
Stewart [5e]: 13.3
Stewart [6e]: 14.3
Problems:

Stewart [6e], 14.3
#3, 11, 13, 19, 27, 29, 37, 39, 55.

Arc Length
Briggs [1e], 11.8
#7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 27, 31, 36, 39.

Curvature & Normal Vectors
Briggs [1e], 11.9
#3, 7, 9–17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 38, 39, 40, 41, 54, 56, 63, 75.

#### Velocity and Acceleration

Briggs: 11.7
Stewart [5e]: 13.4
Stewart [6e]: 14.4

Problems:

Briggs [1e], 11.7
#7, 9, 11, 12, 13, 14, 16, 22, 25, 28, 35, 51, 53, 57 (a,b).

Stewart [6e], 14.4
#5, 7, 11, 15, 19, 25, 33, 39.

#### Cylindrical and Spherical Coordinates

Stewart [5e]: 12.7

Problems:
???

### Partial Derivatives

Corresponding Chapters
Briggs [1e]    - Ch. 12
Stewart [5e]     - Ch. 14
Stewart [6e]     - Ch. 15

#### Functions of Several Variables

Stewart [5e]: 14.1
Stewart [6e]: 15.1

Problems:

Stewart [6e], 15.1
#1, 3, 5, 9, 13, 17, 23, 29, 30, 31, 43, 55, 61, 63.

#### Limits and Continuity

Briggs: 12.3
Stewart [5e]: 14.2
Stewart [6e]: 15.2

Problems:

Briggs [1e], 12.3
#8, 11–23, 25, 27, 29, 30, 31–45 ,48–59.

Stewart [6e], 15.2
#1, 3, 7, 11, 15, 19, 23, 35, 37, 39.

#### Partial Derivatives

Briggs: 12.4
Stewart [5e]: 14.3
Stewart [6e]: 15.3

Problems:

Briggs 12.4
#3, 7–37, 43, 45, 47, 49, 51, 53, 55, 65.
(Highlights: #3, 7, 10, 11, 13, 15, 17, 19, 21, 26, 35, 36, 43, 47, 53.)

Stewart [6e], 15.3
#1, 3, 5, 7, 17, 33, 37, 39, 41, 45, 53, 61, 65, 67, 79, 89.

Several sources schedule exams either immediately before or after this section.

#### Tangent Planes and Differentials

Briggs: 12.7
Stewart [5e]: 14.4
Stewart [6e]: 15.4
Problems:
Briggs 12.7
#3, 4, 9, 10, 11, 15, 17, 18, 21, 23, 24, 25, 42, 43, 44, 47, 52.

Stewart [6e], 15.4
#1, 11, 17, 21, 23, 29, 35, 37.

#### The Chain Rule

Briggs: 12.5
Stewart [5e]: 14.5
Stewart [6e]: 15.5
Problems:

Briggs 12.5
#7–19, 21, 27, 28, 29, 30, 45, 53, 55, 57

Stewart [6e], 15.5
#4, 10, 16, 18, 22, 30, 32, 36, 38, 48.
#2, 4, 8, 16, 22, 34, 36, 40, 48, 54.

#### Directional Derivatives and the Gradient Vector

Briggs: 12.6
Stewart [5e]: 14.6
Stewart [6e]: 15.6
Problems:

Briggs 12.6
#6, 7, 9, 10–29, 33, 34, 35, 37, 39, 45, 47, 49, 51, 53, 56, 59, 61, 63.

#### Maximum and Minimum Values

Briggs: 12.8
Stewart [5e]: 14.7
Stewart [6e]: 15.7

Problems:

Briggs [1e], 12.8
#9, 11, 13, 18, 19, 20, 21, 23, 25, 29, 31, 34, 37, 39, 40, 45, 47, 52.

Briggs [1e], 12.9:
#4, 5, 7, 9, 10.

Stewart [6e], 15.7
#2, 4, 6, 18, 30, 38, 44, 48, 52, 54.

#### Lagrange Multipliers

Briggs: 12.9
Stewart [5e]: 14.8
Stewart [6e]: 15.8

Paul's Online Notes

Problems:

Briggs [1e], 12.9:
#11, 12, 16, 18, 20, 22, 25, 32, 42.

Stewart [6e], 15.8
#4, 8, 16, 18, 22, 24, 32, 36.

### Multiple Integrals

Corresponding Chapters
Briggs [1e]    - Ch. 13
Stewart [5e]    - Ch. 15
Stewart [6e]    - Ch.16

Supplementary Resources:
Khan Academy: Double and Triple Integrals

#### Double Integrals over Rectangles

Briggs: 13.1
Stewart [5e]: 15.1
Stewart [6e]: 16.1

Problems:

Briggs[1e], 13.1
#3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 23, 24, 35, 37, 39, 41.

Briggs [1e], 13.2:
#5, 6, 7, 9, 13, 14.

Stewart [6e], 16.1
#3, 5, 9, 13.

#### Iterated Integrals

Stewart [5e]: 15.2
Stewart [6e]: 16.2
Problems:

Stewart [6e], 16.2
#3, 7, 17, 21, 23, 29, 33.

#### Double Integrals over General Regions

Briggs: 13.2
Stewart [5e]: 15.3
Stewart [6e]: 16.3

Problems:

Briggs 13.2
#7–15, 16, 19–25, 29, 30, 31, 32, 35, 37, 39–45, 49–63, 69.

Stewart [6e], 16.3
#3, 7, 13, 17, 19, 31, 37, 43, 49, 55.

#### Double Integrals in Polar Coordinates

Briggs: 13.3
Stewart [5e]: 15.4
Stewart [6e]: 16.4
Problems:

Briggs [1e], 13.3
#11, 13, 15, 17, 19, 21, 23–29, 35, 37, 39, 40, 41, 47, 49, 53.

Stewart [6e], 16.4
#3, 5, 7, 13, 17, 25, 29, 37.

#### Applications of Double Integrals

Briggs: 13.6
Stewart [5e]: 15.5
Stewart [6e]: 16.5

Problems:

Briggs [1e], 13.6
#9, 13, 16, 19, 24, 27, 34, 37.

Stewart [6e], 16.5
#1, 5, 15, 23.

#### Surface Area

Stewart [5e]: 15.6

Problems:
???

Paul's Online Notes: Surface Area

#### Triple Integrals

Briggs: 13.4
Stewart [5e]: 15.7
Stewart [6e]: 16.6
Problems:

Briggs [1e], 13.4
#7–17, 20, 23, 25, 28, 29–37, 39, 47, 49, 57.

Briggs [1e], 13.5:
#12, 15, 16, 18, 21.

Stewart [6e], 16.6
#3, 9, 11, 17, 25, 31, 37, 43.

#### Triple Integrals in Cylindrical and Spherical Coordinates

Briggs: 13.5
Stewart [5e]: 15.8
Stewart [6e]:
16.7 (Cylindrical),
16.8 (Spherical)

Problems:

Briggs [1e], 13.5
#11–21, 23, 25–31, 35–43, 42, 44, 46, 47, 49, 51, 55, 57, 63, 65, 67.

Cylindrical
Stewart [6e], 16.7
#1, 3, 7, 11, 13.

Spherical
Stewart [6e], 16.8
#1, 4, 7, 9, 13, 15.

Several sources schedule exams either immediately before or after this section.

#### Change of Variable in Multiple Integrals

Briggs: 13.7
Stewart [5e]: 15.9
Stewart [6e]: 16.9

Problems:

Briggs [1e], 13.7
#5, 7, 11, 13, 24, 27, 29, 31, 33, 34, 42.

Stewart [6e], 16.9
#3, 5, 7, 9, 11, 19, 23.

### Vector Calculus

Corresponding Chapters
Briggs [1e]    - Ch. 14
Stewart [5e]    - Ch. 16
Stewart [6e]    - Ch. 17

#### Vector Fields

Briggs: 14.1
Stewart [5e]: 16.1
Stewart [6e]: 17.1

Problems:

Briggs 14.1
#2, 6, 7, 8, 11, 13, 16, 17, 18, 19, 25–35, 41, 43.

Stewart [6e], 17.1
#1, 11, 17, 23, 25, 33.

#### Line Integrals

Briggs: 14.2
Stewart [5e]: 16.2
Stewart [6e]: 17.2

Problems:

Briggs [1e], 14.2
#11–19, 25–29, 33–39, 43, 47, 49, 54, 55.

Stewart [6e], 17.2
#1, 15, 17, 21, 33, 37, 39, 43, 47.

#### Fundamental Theorem for Line Integrals

Stewart [5e]: 16.3
Stewart [6e]: 17.3

Paul's Notes
PatrickJMT

Problems:

Stewart [6e], 17.3
#1, 3, 15, 19, 23, 33.

#### Greens' Theorem

Briggs: 14.4
Stewart [5e]: 16.4
Stewart [6e]: 17.4
Problems:

Briggs 14.4
#11, 13, 17, 19, 23, 25, 29, 30, 32, 35, 37, 41

Stewart [6e], 17.4
#1, 5, 13, 17, 21, 23.

#### Curl and Divergence

Briggs: 14.5
Stewart [5e]: 16.5
Stewart [6e]: 17.5

Problems:

Briggs 14.5
#9–15, 16, 17, 19, 21, 23–33, 38, 47.

Stewart [6e], 17.5
#1, 5, 11, 12, 15, 19, 25, 37.

#### Parametric Surfaces and Their Areas

Stewart [5e]: 16.6
Stewart [6e]: 17.6

Problems:

Stewart [6e], 17.6
#1, 3, 5, 15, 21, 25, 35, 45, 59.

#### Surface Integrals

Briggs: 14.6
Stewart [5e]: 16.7
Stewart [6e]: 17.7

Problems:

Briggs 14.6
#11–29, 30, 32, 35, 41, 43, 45, 47, 61.

Briggs [1e], 14.7:
#5, 8, 11, 15, 18, 22.

Stewart [6e], 17.7
#3, 7, 11, 15, 21, 23, 41, 47.

#### Stokes' Theorem

Briggs: 14.7
Stewart [5e]: 16.8
Stewart [6e]: 17.8

Problems:

Briggs [1e], 14.7
#7, 11–19, 27, 29, 39.

Stewart [6e], 17.8
#3, 7, 15, 17, 19.

#### The Divergence Theorem

Briggs: 14.8
Stewart [5e]: 16.9
Stewart [6e]: 17.9

Problems:

Briggs [1e], 14.8
#9, 13–25, 33, 35, 39.

Stewart [6e], 17.9
#1, 7, 11, 13, 17, 19, 23, 31, 32.