Skip to main content
Lessons for Multivariable Calculus
Mark Fitch
x
Search Results:
No results.
☰
Contents
You!
Choose avatar
▻
✔️
You!
😺
👤
👽
🐶
🐼
🌈
Font family
▻
✔️
Open Sans
AaBbCc 123 PreTeXt
Roboto Serif
AaBbCc 123 PreTeXt
Adjust font
▻
Size
12
Smaller
Larger
Width
100
narrower
wider
Weight
400
thinner
heavier
Letter spacing
0
/200
closer
f a r t h e r
Word spacing
0
/50
smaller gap
larger gap
Line Spacing
135
/100
closer
together
further
apart
Light/dark mode
▻
✔️
default
pastel
twilight
dark
midnight
Reading ruler
▻
✔️
none
underline
L-underline
grey bar
light box
sunrise
sunrise underline
Motion by:
✔️
follow the mouse
up/down arrows - not yet
eye tracking - not yet
<
Prev
^
Up
Next
>
🔍
Front Matter
Colophon
Preface
1
Vectors
1.1
Journey vs Location
1.1.1
Understand Location and Direction
1.1.2
Terminology and Notation
1.1.3
Vector Arithmetic
1.1.3
Exercises
1.1.4
Points, Vectors, and Functions
1.2
Vector Operations
1.2.1
Distance
1.2.2
Magnitude
1.2.3
Unit Vectors
1.2.4
Spheres
1.2.5
Dot Product
1.2.6
Orthogonal Projection
1.2.7
Dot Product Properties
1.2.8
Cross Product
1.3
Planes
1.3.1
Expressions for Planes
1.3.2
Lines and Planes
1.3.3
Some Geometric Questions
1.4
Calculus on Space Curves
1.4.1
Defining a Limit of Vector Valued Functions
1.4.2
Defining a Derivative of Vector Valued Functions
1.4.2
Exercises
1.4.3
Integrals of Vector Valued Functions
1.4.3.1
Arclength
1.4.3.1
Exercises
1.5
Space Curve Properties
1.5.1
Tangents for Space Curves
1.5.2
Unit Tangent Vector for a Space Curve
1.5.3
Normal Vector for a Space Curve
1.5.4
Binormal Vector for a Space Curve
1.5.5
Curvature
1.6
Parameterizing Curves
1.6.1
Effect of Parameterization
1.6.2
Parameterizing by Arclength
1.7
Project: Orthogonal Vectors
2
Surfaces
2.1
Limits in Higher Dimensions
2.1.1
Limits for Higher Dimensional Functions
2.1.2
Derivation
2.1.2
Exercises
2.1.3
Continuity
2.2
Partial Derivatives
2.2.1
Derivation
2.2.2
Notation and Calculation
2.3
Differentiability
2.3.1
Motivation
2.3.2
Concept
2.3.3
Calculation
2.4
Directional Derivatives
2.4.1
Motivation
2.4.2
Notation
2.4.3
Derivation
2.4.4
Maximum Gradient
2.5
Chain Rule
2.5.1
Chain Rule
2.5.2
Implicit Differentiation
2.6
Extrema
2.6.1
Review
2.6.2
Experiment
2.6.3
Method
2.6.3
Exercises
2.7
Extrema with Constraints
2.7.1
Derivation
2.8
Project: Types of Surface Extrema
2.9
Project: Following the Gradient
3
Double Integration
3.1
Coordinate Systems
3.1.1
Coordinate System Definitions
3.1.1
Exercises
3.1.2
Conversion of Coordinates
3.1.2
Exercises
3.2
Scalar Line Integral
3.2.1
Coordinate System Definitions
3.2.2
Derivation
3.2.3
Method
3.2.4
Exercises
3.3
Double Integrals
3.3.1
Review of Riemann Integration
3.3.2
Volumes
3.3.3
Calculation
3.3.4
Exercises
3.4
Volumes with Non-rectangular Bases
3.4.1
Derivation
3.4.1
Exercises
3.4.2
Double Integrals with Cylindrical Coordinates
3.4.2
Exercises
3.5
Parametric Surfaces
3.5.1
Recognizing Surfaces in Parametric Form
3.5.2
Tangent Planes for Parametrized Surfaces
3.6
Surface Area
3.6.1
Calculation
3.6.1
Exercises
3.6.2
Surface Area in Cylindrical Coordinates
3.6.3
Scalar Surface Integrals
3.7
Project: Improper but Correct
4
Triple Integration
4.1
Triple Integrals in Cartesian
4.1
Exercises
4.2
Triple Integrals in Cylindrical
4.2
Exercises
4.3
Triple Integrals in Spherical
4.3
Exercises
4.4
Change of Variable
4.4.1
Review
4.4.2
Illustration
4.4.3
Method
4.4.4
Exercises
4.5
Centroid
4.5.1
Derivation of Centroid
4.5.2
Definition
4.5.2
Exercises
5
Vector Calculus
5.1
Vector Fields
5.1.1
Presentation
5.1.2
Interpretation
5.2
Vector Line Integrals
5.2.1
Derivation
5.2.2
Evaluation
5.2.2
Exercises
5.3
Vector Field Theorems
5.3.1
Path Independence
5.3.2
Fundamental Theorem of Vector Line Integrals
5.3.3
Closed Curves
5.3.4
Conservative Fields
5.3.5
Exercises
5.4
Divergence
5.4.1
Illustration
5.4.2
Definition
5.4.3
Exercises
5.5
Curl
5.5.1
Definition
5.5.2
Interpretation
5.5.3
Curl and Conservative Fields
5.5.4
Exercises
5.6
Green’s Theorem
5.6
Exercises
5.7
Divergence Theorem
5.8
Surface Vector Integrals
5.8.1
Illustration
5.8.2
Calculation
5.8.3
Exercises
5.9
Stoke’s Theorem
5.9.1
Statement
5.9.2
Exercises
🔗
Chapter
3
Double Integration
3.1
Coordinate Systems
3.2
Scalar Line Integral
3.3
Double Integrals
3.4
Volumes with Non-rectangular Bases
3.5
Parametric Surfaces
3.6
Surface Area
3.7
Project: Improper but Correct
