Senior volumes · eight books · a visual companion
Little Blue Book, Visualized · Senior
Seven books are seven districts and their chapters are stations; the 20 problem-solving methods of the Methods volume are metro lines running through every district. Drag, take apart, guess — then go back to the book.
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Map too small? Here is the same content as a list. 104 of 104 stops are open.
Vol. 1SetsSubsets you can see · 9 stops
- Ch. 1 Elements and Sets Concept
- Ch. 2 Operations on Sets Concept
- Ch. 3 Counting the Elements of a Finite Set Skill
- Ch. 4 Partitions of a Set Concept
- Ch. 5 Families of Subsets Concept
- Ch. 6 Properties of Sets Concept
- Ch. 7 The Principle of Classification Method
- Ch. 8 The Extremal Principle Method
- Ch. 9 Inclusion–Exclusion Method
Vol. 2Functions and Functional EquationsCorrespondences that move · 17 stops
- §1.1 Mappings Concept
- §1.2 Functions Concept
- §2.1 Odd and Even Functions Concept
- §2.2 Monotonicity Concept
- §2.3 Periodicity Concept
- §3.1 Quadratic Functions Concept
- §3.2 Power, Exponential and Logarithmic Functions Concept
- §3.3 The Function f(x) = x + a²/x Concept
- Ch. 4 Maxima and Minima of Functions Skill
- Ch. 5 Solving Problems by Constructing a Function Method
- §6.1 Iteration of a Function Concept
- §6.2 Finding the n-th Iterate Skill
- §6.3 Applications of Iteration Method
- §7.1 Functional Equations: Substitution Skill
- §7.2 Functional Equations: Plugging in Values Skill
- §7.3 Functional Equations: Cauchy's Method Method
- §7.4 Functional Equations: Recursion Method
Vol. 3Trigonometric FunctionsThe unit circle, turning · 12 stops
- Ch. 1 The Unit Circle and the Trigonometric Functions Concept
- Ch. 1 Graphs and Their Transformations Concept
- Ch. 1 Period, Monotonicity and Symmetry Concept
- Ch. 2 Sum, Double-Angle and Half-Angle Formulas Skill
- Ch. 2 Sum-to-Product and Product-to-Sum Skill
- Ch. 2 The Auxiliary Angle and Evaluating Expressions Skill
- Ch. 3 The Laws of Sines and Cosines Concept
- Ch. 3 Identities and Area in a Triangle Skill
- Ch. 4 Inverse Trigonometric Functions Concept
- Ch. 4 Simple Trigonometric Equations Skill
- Ch. 5 Trigonometric Inequalities Method
- Ch. 6 Trigonometric Substitution and Applications Method
Vol. 4Complex Numbers and VectorsNumbers that rotate · 9 stops
- Ch. 1 Complex Numbers and Their Algebra Concept
- Ch. 2 Modulus and Argument (I) Concept
- Ch. 3 Complex Numbers and Equations Concept
- Ch. 4 Adding and Subtracting Vectors Concept
- Ch. 5 The Inner Product Concept
- Ch. 6 Vectors in Space Concept
- Ch. 7 Roots of Unity and Their Uses Concept
- Ch. 8 Modulus and Argument (II) Method
- Ch. 9 Applications of Complex Numbers and Vectors Method
Vol. 5Sequences and Mathematical InductionOne step at a time · 15 stops
- Ch. 1 The First Principle of Induction Method
- Ch. 2 The Second Principle of Induction Method
- Ch. 3 Well-Ordering and Infinite Descent Method
- Ch. 4 General Terms and Sums Skill
- Ch. 5 Arithmetic and Geometric Sequences Concept
- Ch. 6 Higher-Order Arithmetic Sequences and Differences Skill
- Ch. 7 Recursive Sequences Skill
- Ch. 8 Periodic Sequences Concept
- Ch. 9 The Fibonacci Sequence Concept
- Ch. 10 Several Proofs of the AM–GM Inequality Method
- Ch. 11 Choosing the Right Step Size Method
- Ch. 12 Choosing What to Induct On Method
- Ch. 13 Changing the Statement Method
- Ch. 14 Guess, Then Prove Method
- Ch. 15 Existence Problems for Sequences Method
Vol. 6The Mean Inequalities and the Cauchy InequalityWhen does equality hold? · 15 stops
- §1.1 The Mean Inequalities Concept
- §1.2 Proofs of the Mean Inequalities Concept
- §2.1 Proving Inequalities with AM–GM Skill
- §2.2 Extrema with AM–GM Skill
- §2.3 AM–GM in Geometric Inequalities Skill
- §2.4 Variants of AM–GM Skill
- §2.5 AM–GM with Parameters Skill
- §3.1 The Cauchy Inequality and Its Proofs Concept
- §3.2 Variants and Generalizations of Cauchy Concept
- §4.1 Proving Inequalities with Cauchy Skill
- §4.2 Cauchy for Systems and Extrema Skill
- §4.3 Cauchy and Fractional Inequalities Skill
- §4.4 Cauchy in Combinatorial Estimates Method
- §4.5 Cauchy with Parameters Skill
- §4.6 AM–GM and Cauchy Together Method
Vol. 7Analytic GeometryCurves drawn by equations · 27 stops
- §2.1 Directed Segments Concept
- §2.2 Basic Problems of the Coordinate Method Skill
- §2.3 Choosing a Good Coordinate System Method
- §2.4 Curves and Equations Concept
- §3.1 Equations of a Line Concept
- §3.2 The Normal Form of a Line Concept
- §3.3 Relations between Points and Lines Skill
- §3.4 Families of Lines Skill
- §4.1 The Circle Concept
- §4.2 The Ellipse Concept
- §4.3 The Hyperbola Concept
- §4.4 The Parabola Concept
- §4.5 Tangents and Normals of Conics Concept
- §5.1 Simplifying by Translating the Axes Skill
- §5.2 Simplifying by Rotating the Axes Skill
- §5.3 The General Quadratic Equation Concept
- §5.5 Families of Conics Skill
- §6.1 Polar Coordinates Concept
- §6.3 Equations of Curves in Polar Coordinates Concept
- §6.5 Simplifying Calculations and Proofs with Polar Coordinates Method
- §7.1 Parametric Equations of Curves Concept
- §7.2 Eliminating and Choosing Parameters Skill
- §7.4 Simplifying Calculations and Proofs with Parametric Equations Method
- §8.1 How Curves Divide the Plane into Regions Concept
- §8.2 On Extremum Problems Method
- §9.1 How Curves and Equations Correspond Concept
- §9.2 Methods for Locus Problems Method
Vol. 8Problem-Solving Methods and StrategiesMethod lines across the whole map · 20 lines
- M1 Reduction
- M2 Proof by Contradiction
- M3 Mathematical Induction
- M4 The Pigeonhole Principle
- M5 Inclusion–Exclusion
- M6 The Extremal Principle
- M7 Parity
- M8 The Area Method
- M9 Looking at the Whole
- M10 Choosing Good Notation
- M11 Combining Numbers and Figures
- M12 Correspondence and Pairing
- M13 Recursion
- M14 Coloring
- M15 Assigning Values
- M16 Counting in Two Ways
- M17 Step-by-Step Adjustment
- M18 Construction
- M19 Invariants and Monovariants
- M20 Graph-Theoretic Methods