PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD ↓


Analysis


Mathematical analysis is a rigorous branch of mathematics that deals with the study of limits, continuity, derivatives, integrals, and infinite series. It is a foundation for many other areas of mathematics, such as differential equations, functional analysis, and mathematical physics.

Basic concepts

To understand mathematical analysis, we begin by exploring its basic concepts, including functions, limits, and sequences.

Work

A function is a relation between a set of inputs and a set of possible outputs where each input relates to exactly one output. It can be thought of as a machine or rule that takes one number and produces another number.

f(x) = x^2

In this function, when you input a value for x, the output is the square of x.

Limitations

The concept of the limit is fundamental to analysis. It describes the value that a function approaches as it approaches some value of the input.

lim (x → 3) f(x) = ? If f(x) = 2x + 1, then lim (x → 3) f(x) = 2(3) + 1 = 7

Sequences and series

A sequence is an ordered list of numbers, and an infinite series is the sum of the terms of the sequence.

Sequence: {1, 1/2, 1/4, 1/8, ...} Series: 1 + 1/2 + 1/4 + 1/8 + ...

Continuity

A function is continuous at a point if the following three conditions are met:

  1. The function is defined at the point.
  2. The limit of the function is finite as it approaches the point.
  3. The limit is equal to the value of the function at that point.

Derivatives

The derivative of a function describes the rate of change of the function. It is represented geometrically as the slope of the tangent line to the graph of the function.

This graph shows a quadratic function with a tangent line. The slope of the tangent line represents the derivative at that point.

f(x) = x^2 f'(x) = 2x

The derivative function, f'(x), gives the slope of the tangent line at any point x to the original function f(x).

Integrals

Integration is the inverse operation of differentiation. While derivatives give us the slope or rate of change, integrals give the total accumulation or the area under the curve.

This diagram illustrates the concept of the area under a curve, which integration measures.

∫ f(x) dx = (1/3)x^3 + C For f(x) = x^2

The integral accumulates the area under the curve of f(x) = x^2, resulting in the function (1/3)x^3 + C

Real-world applications

Analysis is not just theoretical; it is the backbone of many real-world applications. Here are some examples:

Physics

In physics, mathematical analysis is used to describe momentum, energy, and other physical quantities that vary over time. One branch of analysis, calculus, allows physicists to model the dynamics of particles and predict behavior.

Economics

In economics, the analysis is used to model cost functions, predict economic growth, and optimize profits, where it aids in evaluating changes and trends over time.

Profit = Revenue - Cost

Derivatives can help determine profit maximization by analyzing the slope of the cost and revenue functions.

Advanced topics in analysis

For those studying analysis at a higher academic level, additional concepts such as metric spaces, functional analysis, and complex analysis become important.

Metric space

Metric spaces extend the concept of distance between points in a generalized space, allowing a rigorous treatment of functions and their convergence.

d(x, y) = sqrt((x2 - x1)^2 + (y2 - y1)^2)

This is a basic example of a distance function in Euclidean space used in metric space analysis.

Functional analysis

Functional analysis studies vector spaces with boundaries, often applied to differential equations and quantum mechanics. It generalizes some concepts of linear algebra.

Complex analysis

Complex analysis involves the study of functions of complex numbers. It is widely used in engineering fields and number theory, which has a deep connection with algebraic properties.

Conclusion

Analysis provides powerful tools for understanding and solving a wide range of problems, both theoretical and practical. Its basic ideas - limits, derivatives and integrals - form the basis of much of modern mathematics and its myriad applications in a variety of scientific fields. Studying analysis provides a deep insight into the continuous change that characterizes the real world.


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