Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Real Analysis


Real analysis is a branch of mathematics that deals with the set of real numbers and functions of real numbers. The subject focuses on the development of rigorous methods and precise results regarding quantities that can be represented by the real line.

Real number

To understand real analysis, we first need to understand real numbers. Real numbers are all numbers that can be found on the number line. This includes all rational numbers, such as 2/3 and -5, as well as all irrational numbers, such as √2 and π.

Example: The sequence of numbers 1, 2, 3, 4,... extends infinitely in the positive direction of the number line. Between any two integers, say 3 and 4, there are an infinite number of rational fractions like 3.1, 3.2, 3.3,... etc., each of which corresponds to a point on the real line. Moreover, irrational numbers like √10 or π also lie somewhere between these integers.

-+0

Limitations

In real analysis, the concept of the limit is fundamental. The limit is essentially the value that a function or sequence "approaches" when the input or index approaches a value.

Example: Consider the function f(x) = 1/x. As x approaches 0 from the right, the value of f(x) gets larger and larger, which shows that the limit of f(x) as x approaches 0 is infinite.

Convergence of sequences

A sequence is a set of numbers listed in a certain order. If a sequence has a limit, it is called convergent, otherwise, it is divergent.

Example: The sequence 1, 1/2, 1/3, ..., 1/n converges to 0 as n approaches infinity.

a_n = 1/n
01

Continuity

A function is continuous at a point if the limit of the function at that point is equal to the value of the function. Simply put, a function is continuous if you can draw its graph without lifting your pen from the paper.

Example: The function f(x) = x^2 is continuous at x = 2 because as we approach 2, the value of the function approaches 4, which is f(2).

Discrimination

Differentiation is the process of finding the derivative of a function. The derivative measures how the output value of a function changes when the input changes.

Example: The derivative of f(x) = x^2 is 2x. This tells us how fast the function x^2 is changing at any point x.

f'(x) = 2x

Integration

Integration is essentially the inverse operation of differentiation. While the derivative gives us the rate of change, integration allows us to accumulate values. It can be understood as finding the area under a curve.

Example: The integral of f(x) = x over the interval 0 to 1 is 1/2, which is the area of the triangle under the line.

∫x dx = x^2/2 + C

Series

A series is the sum of the terms of a sequence. An infinite series is the sum of an infinite sequence of numbers.

Example: Consider the infinite series 1 + 1/2 + 1/4 + 1/8 + ... This is a geometric series that converges to 2 as the number of terms increases.

S_n = a / (1 - r)

Uniform continuity

Uniform continuity is a stronger form of standard continuity. A function f is uniformly continuous if the size of the epsilon gap in the definition of continuity can be chosen independently of the point x in the domain.

Example: Consider the function f(x) = x. For any epsilon > 0, we can choose delta = epsilon for all x.

Functions of real numbers

Functions map inputs from a domain to outputs in the codomain. Real analysis involves functions that take real numbers as input and give real numbers as output.

Example: A classic example is f(x) = sin(x), which maps any real number x to a number between -1 and 1.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on the closed interval [a, b], and k is any number between f(a) and f(b), then there is at least one number c in the interval where f(c) = k.

Example: Let's consider the function f(x) = x^3 - x on the interval [-2, 2]. The theorem states that for any value between f(-2) and f(2), the function will hit every value between -10 and 6.

The Baire category theorem

The Baire category theorem is a fundamental result that applies only to complete metric spaces. It states that the intersection of countably many dense open sets is itself dense in a complete metric space.

Understanding such theorems serves as a foundation for more complex concepts in real analysis.

Conclusion

These concepts form the foundation of real analysis. Mastery of sets, sequences, series, limits, continuity, differentiation, integration, and more contributes to a deeper understanding of the behavior of real numbers and real-valued functions. Real analysis is an essential area of mathematics for those who want to pursue higher studies or work in fields that require rigorous mathematical problem-solving techniques.


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