Calculus of One and Several Variables

21.6.1 The Chain Rule For Functions Of One Variable

First recall the chain rule for a function of one variable. Consider the following picture.

Here I and J are open intervals and it is assumed that g

⊆ J. The chain rule says that if f′
exists and g′
exists for x ∈ I, then the composition, f ∘ g also has a derivative at x and

Recall that f ∘ g is the name of the function defined by f ∘ g

≡ f
. In the notation of this chapter, the chain rule is written as

(21.8)

Part I  Real Valued Functions of One Variable

Chapter 1 Numbers and Algebra

1.1 Numbers and Simple Algebra
1.2 Exercises
1.3 Set Notation
1.4 Order
1.5 Exercises
1.6 The Binomial Theorem
1.7 Well Ordering Principle Math Induction
1.8 Exercises
1.9 Completeness of ℝ
1.10 Existence Of Roots
1.11 Dividing Polynomials
1.12 The Complex Numbers
1.13 Polar Form Of Complex Numbers
1.14 Roots Of Complex Numbers
1.15 The Quadratic Formula
1.16 Exercises

Chapter 2 Functions

2.1 General Considerations
2.2 Graphs of Functions and Relations
2.3 Examples of Functions Used in Calculus
2.3.1 The Circular Functions
2.3.2 Reference Angles And Other Identities
2.3.3 The sin(x)∕x Inequality
2.3.4 The Area Of A Circular Sector
2.4 Exercises
2.5 Exponential and Logarithmic Functions
2.6 The Function bx
2.7 Applications
2.7.1 Interest Compounded Continuously
2.7.2 Exponential Growth And Decay
2.7.3 The Logistic Equation
2.8 Exercises

Chapter 3 Sequences and Compactness

3.1 Exercises
3.2 The Limit Of A Sequence
3.3 The Nested Interval Lemma
3.4 Exercises
3.5 Compactness
3.5.1 Sequential Compactness
3.5.2 Closed And Open Sets
3.6 Exercises

Chapter 4 Continuous Functions and Limits of Functions

4.1 Equivalent Formulations Of Continuity
4.2 Exercises
4.3 The Extreme Values Theorem
4.4 The Intermediate Value Theorem
4.5 Continuity of the Inverse
4.6 Exercises
4.7 Uniform Continuity
4.8 Examples of Continuous Functions
4.9 Uniform Convergence of Continuous Functions
4.10 Exercises
4.11 Limit Of A Function
4.12 Exercises

Chapter 5 The Derivative

5.1 The Definition Of The Derivative
5.2 Finding The Derivative
5.3 Derivatives Of Inverse Functions
5.4 Circular Functions and Inverses
5.5 Exponential Functions and Logarithms
5.6 The Complex Exponential
5.7 Local Extreme Points
5.8 Exercises
5.9 Mean Value Theorem
5.10 Exercises
5.11 First and Second Derivative Tests
5.12 Exercises
5.13 Taylor Series Approximations
5.14 Exercises
5.15 L’Hôpital’s Rule
5.16 Interest Compounded Continuously
5.17 Exercises

Chapter 6 The Integral

6.1 Fundamental Considerations
6.2 Properties of the Integral
6.3 Uniform Convergence And The Integral
6.4 Exercises

Chapter 7 Methods for Finding Anti-derivatives

7.1 The Method of Substitution
7.2 Exercises
7.3 Integration By Parts
7.4 Exercises
7.5 Trig. Substitutions
7.6 Exercises
7.7 Partial Fractions
7.8 Rational Functions Of Trig. Functions
7.9 Exercises

Chapter 8 A Few Standard Applications

8.1 Lengths Of Curves And Areas Of Surfaces Of Revolution
8.1.1 Lengths
8.1.2 Surfaces Of Revolution
8.2 Exercises
8.3 Force On A Dam And Work
8.3.1 Force On A Dam
8.3.2 Work
8.4 Exercises

Chapter 9 Improper Integrals and Stirling’s Formula

9.1 Stirling’s Formula
9.2 The Gamma Function
9.3 Laplace Transforms
9.4 Exercises

Chapter 10 Infinite Series

10.1 Basic Considerations
10.2 Absolute Convergence
10.3 Exercises
10.4 More Tests For Convergence
10.4.1 Convergence Because Of Cancellation
10.4.2 Ratio And Root Tests
10.5 Double Series
10.6 Exercises
10.7 Series Of Functions
10.8 Weierstrass Approximation Theorem∗
10.9 Exercises

Chapter 11 Power Series

11.1 Functions Defined In Terms Of Series
11.2 Operations On Power Series
11.3 Power Series for Some Known Functions
11.4 The Binomial Theorem
11.5 Exercises
11.6 Multiplication Of Power Series
11.7 Exercises
11.8 Some Other Theorems

Chapter 12 Polar Coordinates

12.1 Graphs In Polar Coordinates
12.2 The Area In Polar Coordinates
12.3 The Acceleration In Polar Coordinates
12.4 The Fundamental Theorem Of Algebra
12.5 Exercises

Chapter 13 Algebra and Geometry of ℝn

13.1 ℝn
13.2 Algebra in ℝn
13.3 Geometric Meaning Of Vector Addition In ℝ3
13.4 Lines
13.5 Distance in ℝn
13.6 Geometric Meaning Of Scalar Multiplication In ℝ3
13.7 Exercises

Chapter 14 Vector Products

14.1 The Dot Product
14.2 Geometric Significance Of The Dot Product
14.2.1 The Angle Between Two Vectors
14.2.2 Work And Projections
14.3 Exercises
14.4 The Cross Product
14.4.1 The Box Product
14.5 Proof of the distributive law
14.5.1 Torque
14.5.2 Center Of Mass
14.5.3 Angular Velocity
14.6 Vector Identities And Notation
14.7 Planes
14.8 Exercises

Chapter 15 Sequences, Compactness, and Continuous Functions

15.1 Sequences of Vectors
15.2 Open And Closed Sets
15.3 Cartesian Products
15.4 Sequential Compactness
15.5 Vector Valued Functions
15.6 Continuous Functions
15.7 Sufficient Conditions For Continuity
15.8 Limits Of A Function of Many Variables
15.9 Vector Fields
15.10 Exercises
15.11 The Extreme Value Theorem And Uniform Continuity
15.12 Convergence of Functions
15.13 Fundamental Theorem of Algebra
15.14 Exercises

Chapter 16 The Derivative And Integral, One Variable

16.0.1 Geometric And Physical Significance Of The Derivative
16.0.2 Differentiation Rules
16.0.3 Leibniz’s Notation
16.1 Arc Length And Orientations
16.2 Arc Length and Parametrizations∗
16.2.1 Hard Calculus
16.2.2 Independence Of Parametrization
16.3 Exercises
16.4 Motion on a Space Curves
16.4.1 Some Simple Techniques
16.5 Geometry Of Space Curves∗
16.6 Exercises

Chapter 17 Some Physical Applications

17.1 Spherical and Cylindrical Coordinates
17.2 Exercises
17.3 Planetary Motion
17.3.1 The Equal Area Rule, Kepler’s Second Law
17.3.2 Inverse Square Law, Kepler’s First Law
17.3.3 Kepler’s Third Law
17.4 The Angular Velocity Vector
17.5 Angular Velocity Vector on Earth
17.6 Coriolis Force and Centripetal Force
17.7 Coriolis Force on the Rotating Earth
17.8 The Foucault Pendulum∗
17.9 Exercises

Part II  Functions of Many Variables

Chapter 18 Matrices

18.1 Systems Of Equations, Algebraic Procedures
18.1.1 Elementary Operations
18.1.2 Gauss Elimination
18.2 Matrices and Linear Transformations
18.3 Rotations in ℝ2
18.4 Multiplication of Matrices
18.5 The Transpose
18.6 The Inverse of a Matrix
18.6.1 The Identity And Inverses
18.6.2 Finding The Inverse Of A Matrix
18.6.3 Rotations About A Particular Vector∗
18.7 MATLAB And Matrix Arithmetic
18.8 Exercises
18.9 Subspaces Spans and Bases
18.10 Linear Independence
18.11 Exercises

Chapter 19 Eigenvalues and Eigenvectors

19.1 Definition of Eigenvalues
19.2 An Introduction to Determinants
19.2.1 Cofactors And 2 × 2 Determinants
19.2.2 The Determinant Of A Triangular Matrix
19.2.3 Properties Of Determinants
19.2.4 Finding Determinants Using Row Operations
19.3 MATLAB And Determinants
19.4 Applications
19.4.1 A Formula For The Inverse
19.4.2 Finding Eigenvalues Using Determinants
19.5 MATLAB And Eigenvalues
19.6 Matrices and The Dot Product
19.7 Distance and Orthogonal Matrices
19.8 Diagonalization of Symmetric Matrices
19.9 Exercises

Chapter 20 Functions Of Many Variables

20.1 Review Of Limits
20.2 Exercises
20.3 The Directional Derivative And Partial Derivatives
20.3.1 The Directional Derivative
20.3.2 Partial Derivatives
20.4 Exercises
20.5 Mixed Partial Derivatives
20.6 Partial Differential Equations
20.7 Exercises

Chapter 21 The Derivative Of A Function Of Many Variables

21.1 The Derivative Of Functions Of One Variable
21.2 Exercises
21.3 The Derivative Of Functions Of Many Variables
21.4 Exercises
21.5 C1 Functions
21.6 The Chain Rule
21.6.1 The Chain Rule For Functions Of One Variable
21.6.2 The Chain Rule For Functions Of Many Variables
21.7 Exercises
21.7.1 Related Rates Problems
21.7.2 The Derivative Of The Inverse Function
21.7.3 Proof Of The Chain Rule
21.8 Exercises
21.9 The Gradient
21.10 The Gradient And Tangent Planes
21.11 Exercises

Chapter 22 Optimization

22.1 Local Extrema
22.2 Exercises
22.3 The Second Derivative Test
22.4 Exercises
22.5 Lagrange Multipliers
22.6 Exercises
22.7 Proof Of The Second Derivative Test∗

Chapter 23 Line Integrals

23.1 Line Integrals And Work
23.2 Another Notation For Line Integrals
23.3 Exercises

Chapter 24 The Riemannn Integral On ℝn

24.1 Methods For Double Integrals
24.1.1 Density And Mass
24.2 Exercises
24.3 Methods For Triple Integrals
24.3.1 Definition Of The Integral
24.3.2 Iterated Integrals
24.4 Exercises
24.4.1 Mass And Density
24.5 Exercises

Chapter 25 The Integral In Other Coordinates

25.1 Polar Coordinates
25.2 Exercises
25.3 Cylindrical And Spherical Coordinates
25.3.1 Volume and Integrals in Cylindrical Coordinates
25.3.2 Volume And Integrals in Spherical Coordinates
25.4 Exercises
25.5 The General Procedure
25.6 Exercises
25.7 The Moment Of Inertia And Center Of Mass
25.8 Exercises

Chapter 26 The Integral On Two Dimensional Surfaces In ℝ3

26.1 The Two Dimensional Area In ℝ3
26.1.1 Surfaces Of The Form z = f (x,y)
26.2 Flux Integrals
26.3 Exercises

Chapter 27 Calculus Of Vector Fields

27.1 Divergence And Curl Of A Vector Field
27.1.1 Vector Identities
27.1.2 Vector Potentials
27.1.3 The Weak Maximum Principle
27.2 Exercises
27.3 The Divergence Theorem
27.3.1 Coordinate Free Concept Of Divergence
27.4 Applications Of The Divergence Theorem
27.4.1 Hydrostatic Pressure
27.4.2 Archimedes Law Of Buoyancy
27.4.3 Equations Of Heat And Diffusion
27.4.4 Balance Of Mass
27.4.5 Balance Of Momentum
27.4.6 Frame Indifference
27.4.7 Bernoulli’s Principle
27.4.8 The Wave Equation
27.4.9 A Negative Observation
27.4.10 Volumes Of Balls In ℝn
27.4.11 Electrostatics
27.5 Exercises

Chapter 28 Stokes And Green’s Theorems

28.1 Green’s Theorem
28.2 Exercises
28.3 Stoke’s Theorem From Green’s Theorem
28.3.1 The Normal And The Orientation
28.3.2 The Mobeus Band
28.4 A General Green’s Theorem
28.4.1 Conservative Vector Fields
28.4.2 Some Terminology
28.5 Exercises
A.1 The Function sgn
A.2 The Determinant
A.2.1 The Definition
A.2.2 Permuting Rows Or Columns
A.2.3 A Symmetric Definition
A.2.4 The Alternating Property Of The Determinant
A.2.5 Linear Combinations And Determinants
A.2.6 The Determinant Of A Product
A.2.7 Cofactor Expansions
A.2.8 Formula For The Inverse
A.2.9 The Cayley Hamilton Theorem
A.2.10 Cramer’s Rule
B.1 More Continuous Partial Derivatives
B.2 The Method Of Lagrange Multipliers
B.3 The Local Structure Of C1 Mappings∗
C.1 An Important Warning
C.2 Basic Definition
C.3 Basic Properties
C.4 Which Functions Are Integrable?
C.5 Iterated Integrals
C.6 The Change Of Variables Formula
C.7 Some Observations
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