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


Geometry is a branch of mathematics that studies the size, shape, properties, and dimensions of objects. It is one of the oldest topics in mathematics and can be found in many aspects of our daily lives. From the smallest molecule to the largest galaxy, geometry gives us the tools to understand the structure of the universe. In undergraduate studies, geometry provides students with a way to develop critical reasoning skills and spatial awareness.

Basic concepts

Geometry is based on basic concepts such as points, lines, and planes. Understanding these concepts forms the basis for exploring more complex geometric shapes and structures.

Score

A point indicates a location in space. It has no size, no width, no length and no depth. It is represented by a dot and labeled with a capital letter.

Lines

A line is a straight one-dimensional figure that extends to infinity in both directions. It has infinite length, but it has no width or thickness. A line is often represented by two points and a line symbol, such as (overleftrightarrow{AB}).

Line view:

  
  
  
  A
  B

Plane

A plane is a flat, two-dimensional surface that extends infinitely in all directions. Planes have length and width but no thickness. A simple example of a plane is the surface of a sheet of paper that extends infinitely.

Plane view:

  
  Plane ABCD

Types of geometry

Geometry can be divided into various subfields, including Euclidean, non-Euclidean, analytic, and differential geometry.

Euclidean geometry

Euclidean geometry is the study of flat space, based on the theories of the ancient Greek mathematician Euclid. It focuses on concepts such as points, lines, and planes in two-dimensional or three-dimensional space.

Non-Euclidean geometry

Non-Euclidean geometry involves the study of curved spaces. The two main types are hyperbolic and elliptical geometry. Unlike Euclidean geometry, non-Euclidean geometry explores spaces where parallel lines may never intersect or may have more than one intersection point.

Analytic geometry

Analytical geometry uses coordinate systems to describe geometric shapes. This approach combines algebra and geometry to solve geometric problems. René Descartes, who developed the Cartesian coordinate system, is a key figure in analytical geometry.

Differential geometry

Differential geometry uses calculus and algebra to study the properties of curves and surfaces. This branch of geometry is essential for understanding the shape of space in physics, especially in the theory of relativity.

Geometric shapes and properties

Angles and their types

An angle is formed when two lines or rays meet at a point. Angles are measured in degrees. The main types of angles are:

  • Acute angle: An angle that is less than 90 degrees.
  • Right angle: An angle that is exactly 90 degrees.
  • Obtuse angle: An angle that is more than 90 degrees but less than 180 degrees.
  • Straight angle: An angle that is exactly 180 degrees.
Angle view:

  
  Intense
  
  Correct
  
  Frustrated

Triangle

Triangles are three-sided polygons and are classified based on their angle types and side lengths.

  • Equilateral triangle: All three sides and angles are equal.
  • Isosceles triangle: Two sides and two angles are equal.
  • Scalene triangle: All sides and angles are different.
Triangle view:

  
  Equilateral
  
  Isosceles
  
  Scalene

Quadrilateral

Quadrilaterals are polygons with four sides. Some common quadrilaterals include:

  • Square: Four equal sides and four right angles.
  • Rectangle: Opposite sides are equal, and all angles are right angles.
  • Parallelogram: Opposite sides are equal and parallel.
  • Rhombus: All sides are equal, and opposite angles are also equal.
  • Trapezoid: Only one pair of parallel sides.
Quadrant view:

  
  Social class
  
  Rectangle
  
  Quadrilateral
  
  Rhombus

Circles

A circle is a figure whose all points on the boundary are the same distance from the center. The main elements of a circle include:

  • Radius: The distance from the center of the circle to any point.
  • Diameter: The distance through the center of the circle (twice the radius).
  • Circumference: The total distance around the circle.
  • Arc: Any part of a circumference.
  • Sector: A slice of a circle, like a slice of pie.

The formula for the circumference of a circle is given as:

    C = 2 * π * r

where r is the radius of the circle and π (pi) is approximately 3.14159.

Transformations

Geometric transformations are operations that change the position, size, or orientation of a shape. Common transformations include:

Translation

Translation involves moving a shape from one place to another without rotating or flipping it. You can think of it as sliding the shape on a plane.

Translation view:

Rotation

Rotation moves a figure around a fixed point called the center of rotation. The figure remains symmetric, maintaining size and shape but the angle varies.

Rotating view:

Reflection

Reflection flips a figure along a line, producing a mirror image. This line is called the line of reflection.

Reflection view:

Scaling

Scaling changes the size of the shape. It can be uniform, which changes the shape's size proportionally, or non-uniform, which changes the dimensions independently.

Scaling visualization:

Conclusion

Geometry provides a rich and varied visual landscape, from simple shapes like points, lines, and circles to more complex shapes like polygons and three-dimensional objects. By understanding the principles of geometry, students can gain a better understanding of the spatial elements they interact with every day. Engaging with geometry not only improves analytical and problem-solving skills but also fosters imagination and creativity, laying the foundation for future studies in mathematics, science, architecture, and beyond. This exploration of geometry is just the beginning of a lifelong journey of understanding the mathematical beauty that shapes our world.


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