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 → Real Analysis ↓


Riemann Integration


In the field of mathematical analysis, Riemann integration stands as a cornerstone concept. Named after German mathematician Bernhard Riemann, this method revolutionizes the way we understand the area under the curve of a function. Riemann integration is completely called the Riemann method. To understand this, we explore its definition, properties, applications, and examples through a mix of text, mathematical notation, and visual demonstrations.

Defining Riemann integration

At its core, Riemann integration is the process of assigning a number to a certain type of function defined on a closed bounded interval. This number represents the area under the curve of the function on that interval. To understand how it works, read on. Consider a continuous function f defined on a closed interval [a, b].

We start by dividing the interval [a, b] into smaller subintervals. The partition P of [a, b] is a finite sequence of numbers:

P = { x_0, x_1, x_2, ..., x_n } where a = x_0 < x_1 < x_2 < ... < x_n = b

Each sub-interval [x_{i-1}, x_i] is used to approximate the area under the curve by rectangles whose heights are determined by the function values at some points within each sub-interval. These points are usually the left end point, right end point, or any point within the sub-interval.

Riemann sum

The Riemann sum of f with respect to a partition P is defined as:

S(P, f) = Σ f(x_i^*) (x_i - x_{i-1}) for i = 1, 2, ..., n

Here, x_i^* is a point in the subinterval [x_{i-1}, x_i]. The Riemann sum represents the total area of the rectangles and serves as an approximation of the area under the curve.

x_0 x_1 x_2 x_3

As demonstrated, if we increase the number of partitions and refine the width of each rectangle, the Riemann sum approaches more closely the actual area under the curve.

Riemann integral

A function f is called Riemann integrable on [a, b] if the limit of the Riemann sums exists and is the same regardless of the choice of the points x_i^* within each subinterval as the width of the subinterval approaches zero The Riemann integral of f from a to b is then defined as:

∫ a b f(x) dx = lim (||P|| → 0) S(P, f)

where ||P|| denotes the norm of the partition, which is the length of the longest sub-interval in that partition.

Example: Integrating a linear function

Consider the linear function f(x) = x on the interval [0, 1]. Let us find the Riemann integral of f(x) on this interval.

Divide [0, 1] into n equal sub-intervals, where the width of each sub-interval is Δx = 1/n. Choose the right endpoints for simplicity:

x_i^* = i/nf(x_i^*) = i/n

So the Riemann sum is:

S(P, f) = Σ (i/n) * (1/n) = (1/n^2) Σ i

Recognizing that:

Σ i = (n(n + 1))/2

gives us:

S(P, f) = (1/n^2) * (n(n + 1))/2 = (n + 1)/(2n)

Taking the limit when n approaches infinity, we get:

lim (n → ∞) (n + 1)/(2n) = 1/2

So the Riemann integral of f(x) = x from 0 to 1 is 1/2.

Properties of the Riemann integral

Riemann integration is governed by several important properties that simplify its application:

  • Linearity: If f and g are integrable on [a, b] and c is a constant, then: 
    ∫ a b (cf + g) dx = c∫ a b f(x) dx + ∫ a b g(x) dx
  • Monotonicity: If f(x) ≤ g(x) for all x in [a, b], then: 
    ∫ a b f(x) dx ≤ ∫ a b g(x) dx
  • Additivity: If c is a point in [a, b], then: 
    ∫ a b f(x) dx = ∫ a c f(x) dx + ∫ c b f(x) dx
  • Order of integration: Reversing the limits of integration changes the sign of the integral: 
    ∫ a b f(x) dx = -∫ b a f(x) dx

Applications of Riemann integration

Riemann integration is used in many branches of mathematics and its applications extend to physics, engineering, and beyond. Some notable applications include:

  • Calculation of area and volume: Riemann integral is helpful in determining the area under curves and the volume of solids of revolution.
  • Solving differential equations: Many solutions of differential equations are expressed as integrals, and the Riemann integral provides a means of evaluating them.
  • Physics and engineering: in calculations of quantities such as work and energy, where integrals represent cumulative processes over continuous domains.

Conclusion

Riemann integration is a fundamental tool in analysis that enables us to rigorously quantify the notion of area under curves. Through dividing intervals and adding the areas of rectangles, Riemann integration approximates the essence of integral calculus and captures it. Its properties ensure a robust framework for calculations and applications, reinforcing its importance in a variety of mathematical and applied fields.

This brief exploration provides a glimpse into the versatile and indispensable nature of Riemann integration, and invites deeper study and appreciation of its many aspects.


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