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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
2.1.1
Using Vectors as "Bundled Coordinates"
2.1.2
The
Length
or
Magnitude
of a Vector
2.1.3
Adding Vectors
2.1.4
Scalar Multiples of Vectors
2.1.5
Vector Subtraction and Vector-Scalar Division
2.1.6
Basis Vectors
2.1.7
Unit Vectors
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)
2.3.1
Properties
2.3.2
Geometric Characterization
2.3.3
Direction Angles and Direction Cosines
2.3.4
Projections
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
2.5.1
Lines
2.5.2
Planes
2.6
Quadric Surfaces: Omitted
2.7
Cylindrical and Spherical Coordinates (Postponed)
2.7.1
Cylindrical Coordinates
2.7.2
Spherical Coordinates
3
Vector-Valued Functions
3.1
Vector-Valued Functions and Space Curves
3.1.1
Limits of Vector-Valued Functions
3.1.2
Continuity of Vector-Valued Functions
3.1.3
Space Curves
3.1.4
The Visual Meaning of Continuity
3.2
Calculus of Vector-Valued Functions
3.2.1
Derivatives of Vector-Valued Functions
3.2.2
Properties of the Derivative of Vector-Valued Functions
3.2.3
Tangent Vectors and the Principle Unit Tangent Vector
3.2.4
Integrals 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
3.4.1
Velocity, Speed and Acceleration
3.4.2
Force and Motion: Position from Acceleration
3.4.3
Tangential and Normal Components of Acceleration
3.4.4
Acceleration Components in Terms of Velocity and Acceleration Vectors
3.4.5
Another Way to Compute
\(\N\)
4
Differentiation of Functions of Several Variables
4.1
Functions of Several Variables
4.1.1
Definitions and Notation
4.1.2
Graphs
4.1.3
Level Curves and Contour Plots
4.1.4
Functions of Three Variables
4.1.5
Level Surfaces
4.1.6
Even More Variables, and Vector Valued Arguments
4.2
Limits and Continuity
4.2.1
Limits
4.2.2
Continuity
4.2.3
Functions of Three or More Variables
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? Clairaut’s Theorem
4.3.8
Higher Derivatives
4.3.9
Partial Differential Equations
4.4
Tangent Planes and Linear Approximations
4.4.1
Tangent Lines Revisited
4.4.2
Big-O and little-o notation
4.4.3
Tangent Planes
4.4.4
The Tangent Plane as a Linear Approximation to
\(f\)
4.4.5
Linear Approximations and Differentiable Functions
4.4.6
A Criterion for Differentiability
4.4.7
Differentials
4.4.7.1
Estimating Error with Differentials
4.4.8
Functions of Three Variables
4.4.9
Functions of Many Variables
4.5
The Chain Rule and Implicit Differentiation
4.5.1
The Chain Rule
4.5.2
Implicit Differentiation Made Easy
4.5.3
The Implicit Function Theorem
4.5.4
Implicitly Defined Functions of Several Variables
4.6
Directional Derivatives and the Gradient
4.6.1
Directional Derivatives
4.6.2
The Gradient Vector
4.6.3
Tangent Lines to Level Curves
4.6.4
Maximizing the Directional Derivative
4.6.5
The Implicit Function Theorem in Terms of The Gradient
4.6.6
Functions of Three Variables
4.7
Maxima/Minima Problems
4.7.1
Local Extrema and Critical Points
4.7.2
Absolute Maximum and Minimum Values
4.7.3
Functions of More Than Two Variables
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
5.1.1.1
The Definite Integral and Area Under the Graph of
\(f(x)\)
5.1.1.2
Double Integrals and Volume under the Graph of
\(f(x,y)\)
5.1.1.3
The Midpoint Rule
5.1.1.4
Exact Volume as a Limit
5.1.1.5
The Double Integral
5.1.2
Properties of Double Integrals Over Rectangles
5.1.3
Iterated Integrals and Fubini’s Theorem
5.1.3.1
Evaluating Partial Integrals
5.1.3.2
A Product Rule
5.1.3.3
Iterated Integrals Over Other Regions
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
Type 1, 2 and 3 Regions
5.4.6
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
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
Divergence-free fields
6.5.1.3
The Divergence of a Three Dimensional Vector Field
6.5.2
Curl
6.5.2.1
Curl and Rotation in a Fluid
6.5.2.2
A Short-hand Notation For the Curl
6.5.3
Some Connections Between Div, Curl and Grad
6.5.3.1
Gradients are “Curl Free”, and vice versa on Simply Conected Domains
6.5.3.2
Curls are “Divergence Free”
6.5.3.3
The Remaining Pairs: The Laplacian, etc.
6.5.3.4
Summary
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
6.7.1
Orientation of a Surface and its Boundary Curve
6.7.2
The Theorem, With a Partial Proof
6.7.3
Velocity Fields, Circulation, and Curl
6.7.4
The Connection Between “Curl-free” and Conservative
6.8
The Divergence Theorem
6.8.1
Statement of the Theorem
6.8.2
Overview of the Proof
6.8.3
Verifying Equation (6.8.5) on a type 1 region.
6.8.4
Verifying Equation (6.8.5) for general regions.
6.8.5
The Flux of an Electric Field Through a Surface
6.8.6
Gauss’s Law
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@charleston.edu
Version of April 28, 2025 at 12:26:52 (-04:00)
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