Vector-Valued Functions

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.