Vector properties, magnitudes, and algebraic operations including addition and scalar multiplication. Introduces matrix representations, arithmetic, and computational techniques for solving linear systems.
This mathematical physics sequence explores the coordinate systems necessary for solving problems involving complex shapes, moving beyond Cartesian coordinates to General Curvilinear systems. Students derive scale factors, volume elements, and differential operators, culminating in solving Laplace's equation and understanding metric tensors.
A comprehensive exploration of linear recurrence relations, from first-order foundations to complex second-order systems and real-world predator-prey modeling. Undergraduate students transition from recursive thinking to closed-form solutions, applying discrete math to algorithm analysis and biology.
A graduate-level exploration of discrete dynamical systems, moving from linear growth models to the complex, chaotic behavior of the logistic map. Students apply recursive sequences to model biological and economic phenomena, emphasizing stability analysis and bifurcation theory.
An advanced exploration of vector-valued functions and their applications in modeling 2D motion and force, preparing students for multivariable calculus.
A comprehensive unit on parametric equations and their applications in modeling motion. Students move from the basics of parametric curves to advanced calculus concepts like derivatives, concavity, vectors, and arc length.
This advanced sequence explores systems of linear equations through the lens of computational linear algebra. Graduate students will move from theoretical existence and uniqueness in vector spaces to matrix factorizations, algorithmic complexity, numerical stability, and iterative methods for large-scale systems.
This sequence explores the geometric interpretation of matrices, treating them as operators that transform space. Students move from calculation to visual application, using matrices to represent coordinates, perform translations/dilations, and apply rotations/reflections via matrix multiplication.
A comprehensive sequence on matrix operations, focusing on determinants, inverses, and their application in solving linear systems and cryptography. Students move from basic calculations to advanced problem-solving techniques.
A graduate-level sequence exploring the gradient vector as the foundational tool for modern optimization. Students move from the geometric interpretation of multivariate derivatives to the implementation of stochastic algorithms used in machine learning.
This graduate-level sequence explores the distinction between contravariant vectors and covariant co-vectors within the framework of dual spaces, metric tensors, and higher-rank tensor transformations. Students move from rigorous linear algebra definitions to applications in non-Euclidean geometry and physical systems like stress and strain.
A graduate-level exploration of vector quantities as high-dimensional data points. This sequence bridges linear algebra and data science, examining how geometric intuitions like magnitude, direction, and distance evolve and paradoxically degrade in high-dimensional spaces.
A comprehensive advanced calculus unit exploring the use of vector-valued functions to model and analyze motion in 2D and 3D space. Students will master differentiation, integration, and arc length calculations within a kinematic context, culminating in complex projectile modeling.
A 12th-grade inquiry into complex numbers through the lens of geometry and vector operations. Students transition from algebraic rules to visual intuition, exploring rotations, dilations, and translations in the complex plane.
A comprehensive exploration of complex numbers through a geometric lens, bridging algebraic arithmetic with vector transformations and polynomial theory for undergraduate students.
This sequence explores matrices as geometric transformations of vectors. Students learn to visualize and calculate how matrices stretch, rotate, reflect, and shear space, culminating in a project where they design a computer graphics animation sequence.
This sequence explores matrix algebra as a tool for data organization and system solving. Students progress from basic arithmetic to complex operations like multiplication, determinants, and inverses, ultimately applying these skills to solve systems of linear equations.
A project-based sequence for 12th Grade students exploring linear transformations through the lens of computer graphics. Students learn to use 2x2 matrices to scale, reflect, shear, and rotate vectors, culminating in a retro video game animation project.
This sequence guides undergraduate students through the transition from descriptive statistics to predictive modeling. It covers hypothesis testing, linear and multiple regression, model evaluation, and logistic classification, emphasizing both mathematical foundations and practical coding implementation.
A comprehensive undergraduate-level sequence exploring the intrinsic geometry of space curves through the TNB (Tangent, Normal, Binormal) frame, curvature, and torsion. Students move from basic vector functions to advanced structural analysis of curves in 3D space.
An advanced 11th-grade Calculus unit focusing on the integration of parametric and polar coordinate systems. Students analyze motion, calculate complex areas, perform error analysis, and complete a final synthesis project based on particle kinematics.
This sequence introduces students to parametric equations through the lens of particle motion and physics simulations. Students progress from basic plotting and parameter elimination to advanced calculus applications involving derivatives, vectors, and arc length.
A project-based calculus sequence for 12th grade students focusing on the engineering applications of vector-valued functions, including path optimization, differentiability, and arc length.
This sequence applies vector calculus to particle motion in two and three dimensions, interpreting derivatives and integrals as velocity, acceleration, and displacement to model real-world kinematics.
This sequence guides students through the fundamental operations of vector analysis, bridging the gap between geometric visualization and algebraic computation. Students progress from 2D component forms to 3D spatial analysis and complex products, applying their knowledge to physics-based problems like work and torque.
An advanced exploration of vector fields and tensor calculus for graduate students, bridging the gap between vector analysis and general relativity through curvilinear coordinates, transformation rules, and continuum mechanics.
This sequence guides 9th-grade students through the algebraic representation of vectors. Moving from geometric drawings to coordinate components, students use trigonometry and the Pythagorean theorem to decompose, reconstruct, and add vectors with precision.
A comprehensive introduction to vectors through geometric representation, focusing on the distinction between scalars and vectors, visual addition/subtraction, scalar multiplication, and the transition to component form and magnitude calculation.
A comprehensive 11th-grade unit on vector quantities, moving from conceptual geometric representations to complex algebraic modeling in aviation and navigation contexts. Students master component resolution, vector arithmetic, and resultant force calculations.
A foundational sequence for undergraduate students exploring the arithmetic, geometric, and algebraic properties of complex numbers, focusing on the imaginary unit, standard form, operations, and the complex plane.
An advanced trigonometry sequence for 12th-grade students focusing on the Law of Sines and Cosines applied to 3D spatial problems, vector magnitudes, static forces, and structural engineering. Students transition from 2D calculations to analyzing complex 3D systems and physical forces.
This sequence explores the measurement of area and the analysis of forces using general triangles. Students move beyond the basic 1/2 bh formula to discover how sine and perimeter can define area in oblique scenarios, specifically using Heron's Formula and vector analysis.
A comprehensive sequence for undergraduate students covering the calculus of vector-valued functions, from basic visualization to curvature and kinematics. Students analyze space curves, compute arc length, and decompose acceleration into tangential and normal components.
A high-level exploration of stochastic processes, focusing on how random systems reach equilibrium. Students will master Markov chains, steady-state algebra, and real-world applications like Google's PageRank algorithm.
A comprehensive sequence for 12th-grade students on discrete-time Markov chains, covering state diagrams, transition matrices, and n-step probability calculations using matrix algebra.
An advanced exploration of the general second-degree equation, focusing on identifying, rotating, and graphing conics with cross-product terms using both trigonometric and matrix methods.
This sequence explores geometric transformations using the complex plane as a primary framework. Students will learn how complex arithmetic maps to translations, rotations, dilations, and reflections, culminating in an investigation of non-linear mappings like circle inversion and Möbius transformations.
A rigorous exploration of planar geometry through the lens of group theory, covering isometries, the Three Reflections Theorem, and the classification of finite and infinite symmetry groups.
An undergraduate-level exploration of planar transformations using linear algebra. This sequence covers linear mapping, the necessity of homogeneous coordinates for affine transformations, matrix composition, and the geometric interpretations of determinants and eigenvalues.
A rigorous exploration of rigid motions in Euclidean geometry, focusing on isometries, matrix representations, homogeneous coordinates, and formal proofs of congruence for undergraduate students.
This sequence explores geometric congruence and similarity through the lens of linear algebra. Students learn to represent and manipulate shapes using matrices, homogeneous coordinates, and composite transformations, bridging the gap between abstract geometry and computer graphics.
This undergraduate sequence bridges classical geometry and modern algebra by exploring similarity through the lens of complex numbers and linear algebra. Students will master spiral similarities, matrix representations of conformal mappings, and iterative fractal generation.
