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 → Geometry ↓


Euclidean Geometry


Euclidean geometry is a mathematical system attributed to the ancient Greek mathematician Euclid, who introduced it in his work "Elements" around 300 B.C. This form of geometry deals with flat surfaces and is foundational to our understanding of the physical world. It explores points, lines, angles, surfaces, and shapes such as triangles, rectangles, and circles.

Basic concepts and definitions

Before delving deeper, let us understand some basic concepts of Euclidean geometry.

Points and lines

A point refers to a location in space. It has no shape or dimensions. In diagrams, we represent points using dots and label them with capital letters, such as A or B

A line is a straight path that extends to infinity in two directions. It has one dimension, length, but no thickness. In diagrams, we usually represent lines with a straight line and arrows at both ends, labeled by small letters such as line l, or by two points on the line, such as line AB.

Plane

A plane is a flat, two-dimensional surface that extends infinitely in all directions. It is visualized as a flat sheet, like a piece of paper, that never ends. We often label planes with Greek letters such as the plane α.

Angles

An angle is formed by two rays (the sides of the angle) sharing a common endpoint (the vertex). Angles describe the degrees of rotation from one ray to the other.

Practical examples measured in degrees (°) include right angles (90°), acute angles (less than 90°), and obtuse angles (more than 90°).

ABC

The diagram shows the angle ∠ABC with vertex A

Triangle

Triangles are polygons with three sides, and they play an important role in Euclidean geometry. The types of triangles are determined by the lengths of the sides and the internal angles.

Types of triangles

  • Equilateral triangle: All sides and angles are equal, each angle is 60°.
  • Isosceles triangle: Two sides and the angles opposite to them are equal.
  • Scalene triangle: All sides and angles have different measures.

The sum of the interior angles in any triangle is equal to 180°.

ABC

This is a triangle ΔABC, where the sum of the angles ∠A + ∠B + ∠C = 180°.

Circles

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The essential parts include the radius, diameter, and circumference.

Parts of a circle

  • Centre: The fixed point from which distances are measured.
  • Radius: The distance from the center of the circle to any point.
  • Diameter: The longest distance across a circle, twice the radius.
  • Circumference: The distance around a circle, given by the formula C = 2πr.
OA

Line OA is the radius, where O is the center, and A is a point on the circle.

Major theorems and postulates

Euclidean geometry is based on postulates or axioms (assumed truths) and theorems (proven statements).

Euclid's principles

Euclid's five postulates are the foundation of Euclidean geometry:

  1. A straight line can be drawn joining any two points.
  2. A finite line can be extended infinitely in both directions.
  3. A circle can be drawn with any center and radius.
  4. All right angles are equal.
  5. If a line intersects two lines forming an interior angle less than 180°, then those two lines will eventually meet on the side where the angle is less than 180°.

Pythagorean theorem

For right triangles, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides:

a² + b² = c²
ABC

For triangle ΔABC, if ∠C is right angled, then a² + b² = c² where c is the hypotenuse.

Similarity and congruence

Equality

Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are proportional.

Conformity

Two figures are congruent if they have the same shape and size. All corresponding sides and angles are equal.

Applications of euclidean geometry

Despite its ancient origins, Euclidean geometry has many modern applications:

  • Architecture: The design of buildings and structures to ensure that they stand upright and meet spatial requirements.
  • Art: Drawing and painting measure perspective, proportion, and symmetry.
  • Navigation: Used in map making and establishing coordinates and boundaries.

Conclusion

Euclidean geometry, with its axiomatic approach and logical structure, is not only historically important but also forms the backbone of many modern mathematical branches and methods. Its principles continue to inspire and solve real-world problems, proving its eternal relevance.


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Euclidean Geometry

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