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Calculus 3 (Math 221) Notes and Study Guide
Brenton LeMesurier
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Front Matter
1
Parametric Equations and Polar Coordinates
1.1
Parametric Equations
1.2
Calculus of Parametric Curves
1.3
Polar Coordinates
1.4
Area and Arc Length in Polar Coordinates
2
Vectors in Space
2.1
Vectors in the Plane
2.2
Vectors in Three Dimensions
2.2.1
Three-Dimensional Coordinate Systems
2.2.2
Vectors in
\(\mathbb{R}^3\)
2.3
The Dot Product (a.k.a. Scalar Product)
2.4
The Cross Product (a.k.a. Vector Product)
2.4.1
Definition and a Derivation
2.4.2
Determinants
2.4.3
Computing Some Areas and Volumes
2.4.4
Algebraic Properties of the Cross Product
2.5
Equations of Lines and Planes in Space
2.6
Quadric Surfaces: Omitted for now
2.7
Cylindrical and Spherical Coordinates
2.7.1
Cylindrical Coordinates
2.7.2
Spherical Coordinates
3
Vector-valued Functions
3.1
Vector-Valued Functions and Space Curves
3.2
Calculus of Vector-Valued Functions
3.3
Arc Length and Curvature
3.3.1
Arc Length
3.3.2
Curvature
3.3.3
Normal and Binormal Vectors, and the Osculating Plane
3.4
Motion in Space
4
Differentiation of Functions of Several Variables
4.1
Functions of Several Variables
4.2
Limits and Continuity
4.3
Partial Derivatives
4.3.1
Partial Derivatives of
\(f(x,y)\)
at a point
4.3.2
Partial Derivatives as Functions
4.3.3
Notations
4.3.4
Geometrical Meaning
4.3.5
Functions of More than Two Variables
4.3.6
Second Partial Derivatives
4.3.7
Does the Order of Derivatives Matter?
4.3.8
Higher Derivatives
4.3.9
Partial Differential Equations
4.4
Tangent Planes and Linear Approximations
4.5
The Chain Rule and Implicit Differentiation
4.6
Directional Derivatives and the Gradient
4.7
Maxima/Minima Problems
4.8
Lagrange Multipliers
4.8.1
Finding the Extrema Values on a Curve of a Function of Two Variables
4.8.2
Finding the Extrema Values on a Surface of a Function of Three Variables
4.8.3
Finding Extreme Values under Several Constraints
5
Multiple Integration
5.1
Double Integrals over Rectangular Regions, and Iterated Integrals
5.1.1
Double Integrals over Rectangles
5.1.2
Properties of Double Integrals Over Rectangles
5.1.3
Iterated Integrals
5.2
Double Integrals over General Regions
5.2.1
The Double Integral Over a Bounded Domain
5.2.2
Iterated Integrals over Non-rectangular Regions
5.2.3
A Strategy for Evaluating Double Integrals
5.2.4
Properties of Double Integrals
5.2.5
Domains of More Complicated Shapes: Divide and Conquer
5.2.6
Changing the Order of Integration
5.2.7
The Average Value of a Function over a Region
5.2.8
Optional Topic: Improper Double Integrals
5.3
Double Integrals in Polar Coordinates
5.3.1
Disks, Annuli, Sectors, and Polar Rectangles
5.3.2
Integration Over a Polar Rectangle
5.3.3
Integrals in Polar Coordinates Over Other Domains
5.3.4
Calculating Areas and Volumes using Polar Coordinates
5.3.5
Optional Topic: Improper Double Integrals Using Polar Coordinates
5.4
Triple Integrals
5.4.1
Triple Integrals over a Box
5.4.2
Triple Integrals over Bounded Regions
5.4.3
Iterated Integral Form for Type
\(dz\)
-
\(dy\)
-
\(dx\)
Regions in Space
5.4.4
Changing the Order of Integration
5.4.5
Volumes and Averages
5.5
Triple Integrals in Cylindrical and Spherical Coordinates
5.5.1
Preview: Double Integrals in Polar Coordinates Revisited
5.5.2
Triple Integrals in Cylindrical Coordinates
5.5.3
Triple Integrals in Spherical Coordinates
5.6
Calculating Centers of Mass and Moments of Inertia (Omitted)
5.7
Change of Variables in Multiple Integrals
5.7.1
Changing Variables in 1D Integrals
5.7.2
Transformations: Changes of Coordinates in 2D (and then 3D)
5.7.3
Transformations and Double Integals
5.7.4
Triple Integals
6
Vector Calculus (complete to Section 6.7, on Stokes’ Theorem)
6.1
Vector Fields
6.1.1
Definitions
6.1.2
Gradient Vector Fields, or Conservative Vector Fields
6.1.3
The Cross-Partial Property and Non-Conservative Vector Fields
6.2
Line Integrals
6.2.1
Scalar Line Integrals: Integrating With Respect to Arc Length Along a Curve
6.2.1.1
Scalar Line Integrals in The Plane
6.2.1.2
Scalar Line Integrals in Space
6.2.1.3
Integrals Along
Paths
: Piecewise Smooth Curves
6.2.2
Vector Line Integrals: Integrating With Respect to Position Coordinates
6.2.2.1
Line Integrals in the Plane with Respect to the Coordinates,
\(x\)
and
\(y\)
6.2.2.2
Line Integrals in Space Coordinates
6.2.2.3
Paired (and Tripled) Line Integrals and Vector Line Integrals
6.2.2.4
Vector Line Integrals in Terms of the Arc-length Differential
\(ds\text{:}\)
Circulation and Flux
6.2.3
Reversing the Orientation of a Curve
6.2.4
Properties of Vector Line Integrals
6.3
Conservative Vector Fields
6.3.1
The Fundamental Theorem for Path Integrals
6.3.2
Independence of Path for Integrals of Gradient Fields
6.3.3
Closed Paths
6.3.4
Independence of Path Implies that a Field is Conservative
6.3.5
Testing if a Vector Field is Conservative
6.3.6
Conservation of Energy
6.4
Green’s Theorem
6.4.1
Simple Closed Curves, Positive Orientation, and Green’s Theorem
6.4.2
Partial Proof of Green’s Theorem
6.4.2.1
Verifying Equation (6.4.1) on Type I regions
6.4.2.2
Verifying Equation (6.4.1) on more general domains
6.4.2.3
Verifying Equation (6.4.2)
6.4.3
An Application of Green’s Theorem: When the Cross-Partials Condition Implies That A Vector Field is Conservative
6.4.4
The Flux Form of Green’s Theorem
6.4.4.1
Source-free Vector Fields and Their Stream Functions
6.4.5
Green’s Theorem for Non-simply Connected Domains
6.5
Divergence and Curl
6.5.1
Divergence
6.5.1.1
The Divergence of a Two Dimensional Vector Field
6.5.1.2
The Divergence of a Three Dimensional Vector Field
6.5.2
Curl
6.5.3
Some Connections Between Div, Curl and Grad
6.5.4
Some Fundamental Differential Equations of Physics
6.6
Surface Integrals
6.6.1
Parametric Surfaces
6.6.2
Surface Area of a Parametric Surface
6.6.3
Surface Integral of a Scalar-Valued Function
6.6.4
Oriented Surfaces
6.6.5
Surface Integral of a Vector-Valued Function
6.7
Stokes’ Theorem
6.8
The Divergence Theorem (just the statement for now)
Appendices
A
Rules for Derivatives and Integrals
A.1
Rules for Derivatives
A.2
Rules for Integrals
B
Reduction Formulas For Integrals
B.1
Integrals Involving Exponential or Trigonometric Functions
B.2
Integrals Involving Inverse Trigonometric Functions
B.3
Integrals Involving
\(\sqrt{a + bu}\)
C
Strategy for Evaluating Integrals
C.1
A few general tactics for integration
C.2
A detailed strategy for integration
C.2.1
Use tables of integrals and known integrals
C.2.2
Do basic simplifications
C.2.3
Substitution
C.2.4
Choosing a substitution function
\(u(x)\)
C.2.5
Integration by Parts
C.2.6
Inverse Substitution, especially with trigonometric functions
C.2.7
Special simplifications and substitutions for products of trigonometric functions
C.2.8
Integration of rational functions (ratios of polynomials)
C.2.9
Final steps: make sure that you answer the original question
D
Some Formulas Worth Knowing
E
Some Trigonometry
Calculus 3 (Math 221) Notes and Study Guide
Brenton LeMesurier
Department of Mathematics
College of Charleston
Charleston, South Carolina
lemesurierb@cofc.edu
Version of April 18, 2024 at 17:38:57 (-04:00).
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