Calculus I, II, & III

Cheatsheet Content

### Foundations of Calculus: A Rigorous Approach #### 1. Real Number System and Set Theory - **Axioms of Real Numbers:** Field axioms, order axioms, completeness axiom (least upper bound property). - **Sets and Functions:** Basic set operations, injectivity, surjectivity, bijectivity, cardinality (countable vs. uncountable sets). - **Metric Spaces:** Generalization of distance. Definition, open/closed sets, convergence of sequences. - **Topology of $\mathbb{R}^n$:** Open/closed sets, limit points, interior/closure, compactness (Heine-Borel theorem), connectedness. #### 2. Sequences and Series of Real Numbers - **Sequences:** Convergence, divergence, monotonic sequences, bounded sequences. - **Cauchy Sequences:** Definition, completeness of $\mathbb{R}$ (every Cauchy sequence converges). - **Series:** Convergence tests (comparison, ratio, root, integral, alternating series test), absolute vs. conditional convergence. - **Power Series:** Radius and interval of convergence. Taylor and Maclaurin series with remainder theorems. #### 3. Limits and Continuity - **$\epsilon-\delta$ Definition of Limit:** Rigorous definition for functions of one variable. - **Properties of Limits:** Limit laws proven from $\epsilon-\delta$. - **Continuity:** $\epsilon-\delta$ definition, equivalent definitions, properties of continuous functions (Intermediate Value Theorem, Extreme Value Theorem). - **Uniform Continuity:** Definition, relationship to continuity on compact sets. #### 4. Differentiation - **Definition of Derivative:** Limit definition, differentiability implies continuity. - **Rules of Differentiation:** Proofs of sum, product, quotient, and chain rules. - **Mean Value Theorem:** Statement and proof, applications (e.g., constant function theorem, Rolle's Theorem). - **Taylor's Theorem:** Generalization of MVT with higher-order derivatives and remainder terms. #### 5. Integration (Riemann Integral) - **Partitions and Riemann Sums:** Upper and lower sums, definition of integrability. - **Properties of the Riemann Integral:** Linearity, monotonicity, additivity. - **Fundamental Theorem of Calculus:** Rigorous proof of both parts, connection between differentiation and integration. - **Improper Integrals:** Convergence criteria, comparison tests. ### Multivariable Calculus: Advanced Perspective #### 1. Topology of $\mathbb{R}^n$ - **Open and Closed Sets in $\mathbb{R}^n$:** Generalization from $\mathbb{R}$. - **Compactness in $\mathbb{R}^n$:** Heine-Borel theorem. - **Connectedness and Path-Connectedness:** Definitions and properties. - **Functions of Several Variables:** Limits, continuity, uniform continuity. #### 2. Differentiation in $\mathbb{R}^n$ - **Partial Derivatives:** Definition, geometric interpretation. - **Differentiability:** Definition using linear approximation (Jacobian matrix), relation to partial derivatives (not all partials imply differentiability). - **Chain Rule for Multivariable Functions:** General formulation. - **Implicit Function Theorem:** Conditions for locally defining a function implicitly. - **Inverse Function Theorem:** Conditions for a local inverse function. - **Higher-Order Partial Derivatives:** Clairaut's Theorem (equality of mixed partials). - **Taylor's Theorem for Several Variables:** Approximation with higher-order terms. #### 3. Integration in $\mathbb{R}^n$ (Lebesgue Measure and Integral) - **Jordan Measure and Riemann Integral in $\mathbb{R}^n$:** Generalization of Riemann integration. - **Fubini's Theorem:** Conditions for changing order of integration. - **Change of Variables Formula:** Jacobian determinant, transformations (polar, cylindrical, spherical coordinates). - **Lebesgue Measure and Integration:** More general and powerful theory of integration. - **Measure Spaces:** $\sigma$-algebras, measures. - **Measurable Functions:** Functions preserving measure structure. - **Lebesgue Integral:** Defined for measurable functions, properties (Monotone Convergence Theorem, Dominated Convergence Theorem). #### 4. Vector Calculus - **Vector Fields:** Gradient, divergence, curl. Physical interpretations. - **Line Integrals:** Scalar and vector line integrals, independence of path, conservative vector fields. - **Surface Integrals:** Scalar and vector surface integrals, orientation of surfaces. - **Classical Theorems:** - **Green's Theorem:** Relation between line integral and double integral. - **Stokes' Theorem:** Relation between line integral and surface integral of curl. - **Divergence Theorem (Gauss's Theorem):** Relation between surface integral and triple integral of divergence. ### Linear Algebra: A Rigorous Foundation #### 1. Vector Spaces - **Definition:** Axioms of a vector space over a field (e.g., $\mathbb{R}$ or $\mathbb{C}$). - **Subspaces:** Criteria for a subset to be a subspace. - **Linear Independence, Span, Basis, Dimension:** Formal definitions and properties. - **Direct Sums:** Internal and external direct sums of vector spaces. #### 2. Linear Transformations - **Definition:** Properties (preserves vector addition and scalar multiplication). - **Null Space (Kernel) and Range (Image):** Definitions, properties, Rank-Nullity Theorem. - **Isomorphisms:** Bijections between vector spaces preserving linear structure. - **Matrix Representation of Linear Transformations:** Connection between abstract linear maps and concrete matrices. #### 3. Eigenvalues and Eigenvectors - **Definition:** Characteristic polynomial, algebraic and geometric multiplicity. - **Diagonalization:** Conditions for diagonalizability. - **Cayley-Hamilton Theorem:** Every square matrix satisfies its own characteristic equation. #### 4. Inner Product Spaces - **Definition:** Axioms of an inner product (positive-definiteness, linearity, symmetry). - **Norms and Distances:** Induced by inner product. - **Orthogonality:** Orthonormal bases, Gram-Schmidt orthogonalization. - **Adjoint Operators:** Definition and properties. #### 5. Spectral Theory - **Self-Adjoint Operators (Symmetric Matrices):** Properties, real eigenvalues, orthogonal diagonalization. - **Normal Operators:** Commute with their adjoint. - **Spectral Theorem:** For normal operators on complex inner product spaces, or self-adjoint operators on real inner product spaces. ### Differential Equations: Theoretical Aspects #### 1. Ordinary Differential Equations (ODEs) - **Existence and Uniqueness Theorems:** Picard-Lindelöf Theorem (for $y' = f(t, y)$). - **Linear ODEs:** Homogeneous and non-homogeneous equations, fundamental solutions, Wronskian. - **Systems of ODEs:** Matrix methods, phase plane analysis. - **Stability Theory:** Lyapunov stability, asymptotic stability. #### 2. Partial Differential Equations (PDEs) - **Classification:** Elliptic, parabolic, hyperbolic equations. - **Canonical Examples:** Heat equation, wave equation, Laplace's equation. - **Methods of Solution:** Separation of variables, Fourier series, Green's functions, characteristic curves. - **Well-Posedness:** Existence, uniqueness, and continuous dependence on initial/boundary data. ### Complex Analysis: Functions of a Complex Variable #### 1. Complex Numbers - **Algebra and Geometry of $\mathbb{C}$:** Modulus, argument, polar form, Euler's formula. - **Complex Functions:** Mappings, limits, continuity. #### 2. Holomorphic Functions - **Complex Differentiability:** Definition, Cauchy-Riemann equations. - **Analytic Functions:** Equivalence of holomorphic and analytic functions (power series representation). - **Elementary Functions:** Exponential, trigonometric, logarithmic functions in the complex plane. #### 3. Complex Integration - **Contour Integrals:** Definition, properties. - **Cauchy's Integral Theorem:** For simply connected domains. - **Cauchy's Integral Formula:** For derivatives, applications. - **Morera's Theorem:** Converse of Cauchy's Integral Theorem. #### 4. Series and Singularities - **Taylor Series and Laurent Series:** Expansions around regular points and isolated singularities. - **Classification of Singularities:** Removable, poles, essential singularities. - **Residue Theorem:** Calculation of integrals using residues. - **Argument Principle, Rouche's Theorem:** Counting zeros and poles. #### 5. Conformal Mappings - **Definition:** Angle-preserving transformations. - **Möbius Transformations:** Properties, mapping regions. - **Riemann Mapping Theorem:** Every simply connected domain (not $\mathbb{C}$) is conformally equivalent to the unit disk. ### Abstract Algebra: Structures and Symmetries #### 1. Group Theory - **Definition of a Group:** Axioms, examples (cyclic, dihedral, permutation groups). - **Subgroups:** Cosets, Lagrange's Theorem. - **Normal Subgroups and Quotient Groups:** Homomorphisms, Isomorphism Theorems. - **Group Actions:** Burnside's Lemma, Cayley's Theorem. - **Sylow Theorems:** Powerful tools for analyzing finite groups. #### 2. Ring Theory - **Definition of a Ring:** Axioms, examples (integers, polynomial rings, matrix rings). - **Subrings, Ideals:** Quotient rings. - **Homomorphisms:** Isomorphism Theorems for rings. - **Integral Domains, Fields:** Properties, Field of Fractions. - **Unique Factorization Domains (UFDs), Principal Ideal Domains (PIDs), Euclidean Domains:** Hierarchy and properties. #### 3. Field Theory - **Field Extensions:** Algebraic and transcendental extensions. - **Galois Theory:** Connection between field extensions and group theory, solvability of polynomials by radicals. ### Topology: The Study of Shape Without Measurement #### 1. General Topology (Point-Set Topology) - **Topological Spaces:** Definition, open sets, closed sets, basis for a topology. - **Continuity:** Definition in topological spaces. - **Connectedness:** Connected components, path-connectedness. - **Compactness:** Definition, properties (e.g., continuous image of a compact set is compact). - **Separation Axioms:** $T_0, T_1, T_2$ (Hausdorff), $T_3$ (regular), $T_4$ (normal). - **Product and Quotient Topologies:** Constructions of new topological spaces. #### 2. Algebraic Topology (Introduction) - **Homotopy:** Continuous deformation of paths. - **Fundamental Group:** Group of homotopy classes of loops, a topological invariant. - **Covering Spaces:** Relationship to fundamental group. ### Advanced Mathematical Analysis #### 1. Measure Theory - **Motivation:** Limitations of Riemann integral, need for a more general integral. - **$\sigma$-Algebras:** Collections of measurable sets. - **Measures:** Properties, Lebesgue measure on $\mathbb{R}^n$. - **Measurable Functions:** Properties, approximation by simple functions. - **Lebesgue Integral:** Monotone Convergence Theorem, Dominated Convergence Theorem, Fatou's Lemma. - **Product Measures, Fubini's Theorem (Lebesgue version):** For multiple integrals. #### 2. Functional Analysis - **Normed Vector Spaces and Banach Spaces:** Complete normed vector spaces. - **Inner Product Spaces and Hilbert Spaces:** Complete inner product spaces. - **Bounded Linear Operators:** Properties, norms. - **Fundamental Theorems:** Hahn-Banach Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle. - **Spectral Theory of Operators:** Generalization of eigenvalues to infinite dimensions. #### 3. Distribution Theory (Generalized Functions) - **Test Functions and Distributions:** Formalization of concepts like Dirac delta function. - **Derivatives of Distributions:** Generalization of differentiation. - **Applications:** PDEs, signal processing. ### Differential Geometry: Geometry with Calculus #### 1. Manifolds - **Smooth Manifolds:** Definition, charts, atlases, differentiability. - **Tangent Spaces:** Vector fields, Lie brackets. - **Differential Forms:** Exterior algebra, exterior derivative. - **Stokes' Theorem on Manifolds:** Generalization of classical theorems. #### 2. Riemannian Geometry - **Riemannian Metrics:** Inner products on tangent spaces. - **Connections and Covariant Derivatives:** Parallel transport. - **Curvature:** Riemann curvature tensor, Ricci curvature, scalar curvature. - **Geodesics:** "Straight lines" on manifolds. ### Numerical Analysis: Computational Mathematics #### 1. Error Analysis - **Floating-Point Arithmetic:** Representation, round-off errors. - **Condition Number:** Sensitivity of problems to input changes. - **Stability of Algorithms:** How errors propagate. #### 2. Numerical Linear Algebra - **Solving Linear Systems:** LU decomposition, Gaussian elimination, iterative methods (Jacobi, Gauss-Seidel). - **Eigenvalue Problems:** Power method, QR algorithm. #### 3. Numerical Differentiation and Integration - **Finite Difference Methods:** Approximating derivatives. - **Quadrature Rules:** Trapezoidal rule, Simpson's rule, Gaussian quadrature. #### 4. Numerical Solution of ODEs and PDEs - **Initial Value Problems:** Euler's method, Runge-Kutta methods. - **Boundary Value Problems:** Finite difference method, finite element method. ### Probability Theory: A Measure-Theoretic Approach #### 1. Probability Spaces - **Sample Space, $\sigma$-Algebra of Events:** Formal definition. - **Probability Measure:** Axioms of probability. #### 2. Random Variables - **Definition:** Measurable functions. - **Distribution Functions:** Cumulative distribution function (CDF), probability density function (PDF), probability mass function (PMF). #### 3. Expectation and Moments - **Lebesgue Integral for Expectation:** $\mathbb{E}[X] = \int_\Omega X dP$. - **Variance, Covariance:** Properties. #### 4. Convergence of Random Variables - **Convergence in Probability, Almost Surely, in $L^p$:** Different modes of convergence. - **Law of Large Numbers:** Weak and Strong versions. - **Central Limit Theorem:** Convergence to normal distribution. ### Mathematical Logic and Set Theory #### 1. Foundations of Set Theory - **Axioms of ZFC (Zermelo-Fraenkel with Choice):** The standard axiomatic system for set theory. - **Ordinal and Cardinal Numbers:** Transfinite arithmetic. - **Axiom of Choice:** Equivalents (Zorn's Lemma, Well-Ordering Principle). #### 2. Mathematical Logic - **Propositional and First-Order Logic:** Syntax, semantics, proof systems. - **Completeness and Soundness:** Gödel's Completeness Theorem. - **Gödel's Incompleteness Theorems:** Fundamental limits of formal systems. - **Computability Theory:** Turing machines, undecidability, Halting Problem.