Maps scalar inputs to vectors to define curves in two and three-dimensional space. Calculates derivatives and integrals to analyze velocity, acceleration, arc length, and curvature.
An advanced introduction to the metric tensor and non-Euclidean geometry, serving as a primer for General Relativity.
Students solve Laplace's equation for systems with spherical symmetry, introducing Legendre polynomials and Spherical Harmonics.
Students translate the Del operator into general curvilinear coordinates and apply these operators to physical vector fields.
Focusing on integration, students construct volume and area elements (Jacobians) for spherical and cylindrical geometries and practice integrating scalar fields over complex 3D domains.
Students derive basis vectors and scale factors for general orthogonal curvilinear coordinates and learn how to define position vectors in non-Cartesian geometries.
The sequence culminates with a realistic physics modeling lesson. Students set up and analyze parametric equations for projectiles, accounting for gravity and initial velocity vectors.
Students solve complex motion problems, such as finding the time when a particle is moving perpendicular to its position vector or closest to the origin.
A comparative analysis lesson where students rigorously distinguish between the net change in position (displacement vector) and the total scalar distance traveled (integral of speed).
Students extend calculus operations to vector components. They perform component-wise differentiation and integration to find velocity vectors from position and position vectors from velocity.
Students formally define vector-valued functions and explore limits and continuity. They learn to visualize the domain and output as vectors pointing to a path.
Calculates the total distance traveled and arc length of parametric curves by integrating speed.
Applies derivatives to physics, interpreting parametric equations as position vectors and calculating velocity, speed, and acceleration.
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.
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 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.
An advanced graduate sequence exploring vector calculus from 3D fields to differential forms on manifolds, focusing on fluid dynamics and electromagnetic theory. It moves from parameterizing static fields to understanding global topological constraints on curved surfaces.
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 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 explores the intrinsic geometry of curves in 3D space, focusing on arc length parameterization, the unit tangent vector, curvature, the principal normal vector, and torsion. Students will learn to quantify how paths bend and twist using the TNB (Tangent, Normal, Binormal) frame, providing a coordinate-independent description of movement.
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 foundational sequence introduces 12th-grade calculus students to vector-valued functions, bridging parametric equations with 3D vector analysis through the lens of aerospace navigation. Students explore domains, limits, continuity, differentiation, and integration to model and visualize complex space curves.
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.
This sequence explores vector-valued functions, connecting abstract calculus concepts to the physical world through kinematics. Students will master defining space curves, differentiating for velocity, integrating for projectile motion, and decomposing acceleration into tangential and normal components.
Summative assessment covering the entire sequence, including conceptual questions on scale factors, vector operator calculations, and applications to metrics and symmetry.
Discussion and inquiry guide for teachers to explore metric tensors, map distortions, and geodesics through thought experiments and physical demonstrations.
Introductory presentation on the metric tensor, non-Euclidean geometry, geodesics, and the conceptual foundation of General Relativity.
Problem set focused on solving boundary value problems using Legendre polynomials, including hemispherical potential and a sphere in a uniform electric field.
Case study handout connecting the mathematical solutions of Laplace's equation to the physical shapes of atomic orbitals in quantum mechanics.
Presentation on solving Laplace's equation in spherical coordinates using separation of variables and Legendre polynomials, with physical boundary condition examples.
Practice problems for calculating gradient, divergence, curl, and Laplacian in physics-based contexts using curvilinear coordinates.
Comprehensive reference sheet for vector operators (gradient, divergence, curl, Laplacian) in general and specific curvilinear systems.
Visual explanation of vector operators (gradient, divergence, curl, Laplacian) generalized for any orthogonal curvilinear coordinate system.
Detailed answer key for the Jacobian Jigsaw activity, including worked solutions for toroidal volume, mass integration, and surface area.
Problem set for students to calculate volume and area elements for toroidal, spherical, and cylindrical systems, including mass density integration.
Slide deck explaining Jacobian determinants, derivation of volume elements for spherical/cylindrical systems, and integration methods for scalar fields.
