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Active Calculus - Multivariable
Steve Schlicker, Mitchel T. Keller, Nicholas Long
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Front Matter
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Colophon
Features of the Text
Vector Calculus Preface
Acknowledgments
Active Calculus - Multivariable: our goals
How to Use this Text
9
Multivariable and Vector Functions
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9.1
Functions of Several Variables and Three Dimensional Space
9.1.1
Functions of Several Variables
9.1.2
Representing Functions of Two Variables
9.1.3
Some Standard Equations in Three-Space
9.1.4
Traces
9.1.5
Contour Maps and Level Curves
9.1.6
A gallery of functions
9.1.7
Summary
9.1.8
Exercises
9.2
Vectors
9.2.1
Representations of Vectors
9.2.2
Equality of Vectors
9.2.3
Operations on Vectors
9.2.4
Properties of Vector Operations
9.2.5
Geometric Interpretation of Vector Operations
9.2.6
The Magnitude of a Vector
9.2.7
Summary
9.2.8
Exercises
9.3
The Dot Product
9.3.1
The Dot Product
9.3.2
The angle between vectors
9.3.3
The Dot Product and Orthogonality
9.3.4
Work, Force, and Displacement
9.3.5
Projections
9.3.6
Summary
9.3.7
Exercises
9.4
The Cross Product
9.4.1
Computing the cross product
9.4.2
The Length of
u
×
v
9.4.3
The Direction of
u
×
v
9.4.4
Torque is measured by a cross product
9.4.5
Comparing the dot and cross products
9.4.6
Summary
9.4.7
Exercises
9.5
Lines and Planes in Space
9.5.1
Lines in Space
9.5.2
The Parametric Equations of a Line
9.5.3
Planes in Space
9.5.4
Summary
9.5.5
Exercises
9.6
Vector-Valued Functions
9.6.1
Vector-Valued Functions
9.6.2
Summary
9.6.3
Exercises
9.7
Derivatives and Integrals of Vector-Valued Functions
9.7.1
The Derivative
9.7.2
Computing Derivatives
9.7.3
Tangent Lines
9.7.4
Integrating a Vector-Valued Function
9.7.5
Projectile Motion
9.7.6
Summary
9.7.7
Exercises
9.8
Arc Length and Curvature
9.8.1
Arc Length
9.8.2
Parameterizing With Respect To Arc Length
9.8.3
Curvature
9.8.4
Summary
9.8.5
Exercises
10
Derivatives of Multivariable Functions
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10.1
Limits
10.1.1
Limits of Functions of Two Variables
10.1.2
Continuity
10.1.3
Summary
10.1.4
Exercises
10.2
First-Order Partial Derivatives
10.2.1
First-Order Partial Derivatives
10.2.2
Interpretations of First-Order Partial Derivatives
10.2.3
Using tables and contours to estimate partial derivatives
10.2.4
Summary
10.2.5
Exercises
10.3
Second-Order Partial Derivatives
10.3.1
Second-Order Partial Derivatives
10.3.2
Interpreting the Second-Order Partial Derivatives
10.3.3
Summary
10.3.4
Exercises
10.4
Linearization: Tangent Planes and Differentials
10.4.1
The Tangent Plane
10.4.2
Linearization
10.4.3
Differentials
10.4.4
Summary
10.4.5
Exercises
10.5
The Chain Rule
10.5.1
The Chain Rule
10.5.2
Tree Diagrams
10.5.3
Summary
10.5.4
Exercises
10.6
Directional Derivatives and the Gradient
10.6.1
Directional Derivatives
10.6.2
Computing the Directional Derivative
10.6.3
The Gradient
10.6.4
The Direction of the Gradient
10.6.5
The Length of the Gradient
10.6.6
Applications
10.6.7
Summary
10.6.8
Exercises
10.7
Optimization
10.7.1
Extrema and Critical Points
10.7.2
Classifying Critical Points: The Second Derivative Test
10.7.3
Optimization on a Restricted Domain
10.7.4
Summary
10.7.5
Exercises
10.8
Constrained Optimization: Lagrange Multipliers
10.8.1
Constrained Optimization and Lagrange Multipliers
10.8.2
Summary
10.8.3
Exercises
11
Multiple Integrals
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11.1
Double Riemann Sums and Double Integrals over Rectangles
11.1.1
Double Riemann Sums over Rectangles
11.1.2
Double Riemann Sums and Double Integrals
11.1.3
Interpretation of Double Riemann Sums and Double integrals.
11.1.4
Summary
11.1.5
Exercises
11.2
Iterated Integrals
11.2.1
Iterated Integrals
11.2.2
Summary
11.2.3
Exercises
11.3
Double Integrals over General Regions
11.3.1
Double Integrals over General Regions
11.3.2
Summary
11.3.3
Exercises
11.4
Applications of Double Integrals
11.4.1
Mass
11.4.2
Area
11.4.3
Center of Mass
11.4.4
Probability
11.4.5
Summary
11.4.6
Exercises
11.5
Double Integrals in Polar Coordinates
11.5.1
Polar Coordinates
11.5.2
Integrating in Polar Coordinates
11.5.3
Summary
11.5.4
Exercises
11.6
Surfaces Defined Parametrically and Surface Area
11.6.1
Parametric Surfaces
11.6.2
The Surface Area of Parametrically Defined Surfaces
11.6.3
Summary
11.6.4
Exercises
11.7
Triple Integrals
11.7.1
Triple Riemann Sums and Triple Integrals
11.7.2
Summary
11.7.3
Exercises
11.8
Triple Integrals in Cylindrical and Spherical Coordinates
11.8.1
Cylindrical Coordinates
11.8.2
Triple Integrals in Cylindrical Coordinates
11.8.3
Spherical Coordinates
11.8.4
Triple Integrals in Spherical Coordinates
11.8.5
Summary
11.8.6
Exercises
11.9
Change of Variables
11.9.1
Change of Variables in Polar Coordinates
11.9.2
General Change of Coordinates
11.9.3
Change of Variables in a Triple Integral
11.9.4
Summary
11.9.5
Exercises
12
Vector Calculus
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12.1
Vector Fields
12.1.1
Examples of Vector Fields
12.1.2
Mathematical Vector Fields
12.1.3
Plotting Vector Fields
12.1.4
Gradient Vector Fields
12.1.5
Summary
12.1.6
Exercises
12.1.7
Notes to the Instructor
12.2
The Idea of a Line Integral
12.2.1
Orientations of Curves
12.2.2
Line Integrals
12.2.3
Properties of Line Integrals
12.2.4
The Circulation of a Vector Field
12.2.5
Summary
12.2.6
Exercises
12.2.7
Notes to Instructors and Dependencies
12.3
Using Parametrizations to Calculate Line Integrals
12.3.1
Parametrizations in the Definition of
∫
C
F
⋅
d
r
12.3.2
Alternative Notation for Line Integrals
12.3.3
Independence of Parametrization for a Fixed Curve
12.3.4
Summary
12.3.5
Exercises
12.3.6
Notes to Instructors and Dependencies
12.4
Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals
12.4.1
Path-Independent Vector Fields
12.4.2
Line Integrals Along Closed Curves
12.4.3
What other vector fields are path-independent?
12.4.4
Summary
12.4.5
Exercises
12.4.6
Notes to Instructors and Dependencies
12.5
Line Integrals of Scalar Functions
12.5.1
Defining line integrals of scalar functions
12.5.2
Using Parameterizations to Calculate Scalar Line Integrals
12.5.3
Properties of Scalar Line Integrals
12.5.4
Visualizations of Scalar Line Integrals as Area Under a Curve
12.5.5
Summary
12.5.6
Exercises
12.6
The Divergence of a Vector Field
12.6.1
Definition of the Divergence of a Vector Field
12.6.2
Measuring the Change in Strength of a Vector Field
12.6.3
Summary
12.6.4
Exercises
12.6.5
Notes to Instructors and Dependencies
12.7
The Curl of a Vector Field
12.7.1
Measuring the Circulation Density of Vector Field in
R
2
12.7.2
Measuring Rotation in Three Dimensions
12.7.3
Circulation Density in Three Dimensions
12.7.4
Interpretation and Usage of Curl
12.7.5
Summary
12.7.6
Exercises
12.7.7
Notes to Instructors and Dependencies
12.8
Green’s Theorem
12.8.1
Circulation
12.8.2
Green’s Theorem
12.8.3
What happens when vector fields are not smooth?
12.8.4
Summary
12.8.5
Exercises
12.8.6
Notes to Instructors and Dependencies
12.9
Flux Integrals
12.9.1
The Idea of the Flux of a Vector Field through a Surface
12.9.2
The Details of Measuring the Flux of a Vector Field through a Surface
12.9.3
Summary
12.9.4
Exercises
12.9.5
Notes to Instructors and Dependencies
12.10
Surface Integrals of Scalar Valued Functions
12.10.1
Defining surface integrals of scalar functions
12.10.2
Properties of Scalar Surface Integrals
12.10.3
Summary
12.10.4
Exercises
12.10.5
Notes to Instructors and Dependencies
12.11
Stokes’ Theorem
12.11.1
Circulation in three dimensions and Stokes’ Theorem
12.11.2
Verifying and Applying Stokes’ Theorem
12.11.3
Practice with Surfaces and their Boundaries
12.11.4
Summary
12.11.5
Exercises
12.11.6
Notes to Instructors and Dependencies
12.12
The Divergence Theorem
12.12.1
The Divergence Theorem
12.12.2
Summary
12.12.3
Exercises
12.12.4
Notes to Instructors and Dependencies
Back Matter
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Index
Active Calculus - Multivariable
Steve Schlicker
Department of Mathematics
Grand Valley State University
schlicks@gvsu.edu
Mitchel T. Keller
Department of Mathematics
University of Wisconsin-Madison
mitch.keller@wisc.edu
Nicholas Long
Department of Mathematics and Statistics
Stephen F. Austin State University
longne@sfasu.edu
Contributing Authors
David Austin
Department of Mathematics
Grand Valley State University
austind@gvsu.edu
Matt Boelkins
Department of Mathematics
Grand Valley State University
boelkinm@gvsu.edu
April 29, 2025
Colophon
Features of the Text
Vector Calculus Preface
Acknowledgments
Active Calculus - Multivariable: our goals
How to Use this Text
🔗
