Introduction to Geometry and Algebra

• Mathematics is the abstract science of number, quantity, and space, studied either as abstract concepts or as applied to other disciplines such as physics and engineering.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry, based on Euclid's axioms, deals with space and figures, using concepts like points, lines, angles, and planes.
• Non-Euclidean geometries, such as hyperbolic and elliptic geometries, reject the parallel postulate of Euclidean geometry.
• Analytic geometry uses algebraic techniques to study geometric problems, often involving coordinate systems.
• Differential geometry uses calculus to study curves and surfaces.
• Topology studies properties of geometric objects that are preserved under continuous deformations such as stretching, twisting, crumpling and bending.

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in formulas and equations.
• Elementary algebra involves solving equations with variables and using algebraic expressions and formulas.
• Abstract algebra studies algebraic structures such as groups, rings, and fields.
• These structures are defined by sets and operations that satisfy certain axioms.
• Linear algebra deals with vector spaces, linear transformations, and systems of linear equations.
• Commutative algebra focuses on commutative rings and their ideals.

Calculus

• Calculus is the study of continuous change.
• Differential calculus concerns the instantaneous rates of change, and the slopes of curves.
• Key concepts include derivatives, limits, and differentiation rules.
• Integral calculus concerns the accumulation of quantities, and the areas under and between curves.
• Key concepts include integrals, integration techniques, and the fundamental theorem of calculus.
• Multivariable calculus extends the concepts of calculus to functions of multiple variables.
• Differential equations relate functions to their derivatives, and used to model dynamic systems.

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) relate angles to ratios of sides in right triangles.
• The unit circle provides a way to define trigonometric functions for all real numbers.
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.
• Trigonometry is used to solve problems involving angles and distances in fields such as navigation, surveying, and physics.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics involves methods for organizing and summarizing data, including measures of central tendency (mean, median, mode) and measures of dispersion (variance, standard deviation).
• Inferential statistics involves making inferences and generalizations about a population based on a sample of data.
• Hypothesis testing is a process for determining whether there is enough statistical evidence to reject a null hypothesis.
• Regression analysis is a statistical technique for modeling the relationship between a dependent variable and one or more independent variables.
• Probability theory provides the mathematical framework for understanding and analyzing random events.
• Statistical distributions, such as the normal distribution, are used to model the probability of different outcomes.