Comets are cosmic snowballs of frozen gases, rock and dust that orbit the Sun. Isaac Newton suspected that comets were the origin of the life-supporting component of air and a key source for water replenishment in planetary interiors. A close-up view of comet Hartley 2 was taken by NASA's EPOXI mission during its flyby of the comet, using the spacecraft's medium resolution instrument. Comet Hartley has a novel asymmetric dumbbell-like shape. We employed mathematical models to study comet Hartley. Using calculus-based methods, we estimated various static properties, including the arc lengths (outer boundary length), surface area, and volume of Comet Hartley. Assuming a constant density, we also estimated the mass, center of mass, and moments of inertia for Comet Hartley using triple integration methods using cylindrical coordinates. The center of mass, moments of inertia, and radius of gyration form key inputs for studying the orbital mechanics of the comet in outer space. This research project provides excellent opportunities for hands-on explorations using multivariable calculus studies for engineering and space sciences applications. This research is important as studies of comets unravel secrets about the formation of the solar system.

Gliders are robotic vehicles used in the air and underwater to collect and transmit real-time data. Studies using gliders have important applications in oceanography, engineering, and remote sensing. The goal of this project was to model and identify aspects of a glider’s flight using vector-calculus and matrix-algebra based methods. We employed mathematical models to study the flightpath of a glider using vector valued functions and calculated the osculating plane of the glider. The model parameters were optimized to minimize turbulence. We studied the kinematics of underwater gliders using GPS data reported from gliders deployed by Rutgers University and the University of Washington. We analyzed the reported glider velocity data and applied vector-calculus based methods to calculate the instantaneous and average velocities and acceleration vectors. Additionally, we applied matrix-algebra based methods to translate and rotate the glider to position it at appropriate coordinates underwater for gathering data. This research provided insight into mathematical modeling of real-world data and involved applied optimization and data visualization. These studies provide novel avenues for hands on exploration and application of key mathematical concepts.

STS-121 is a NASA space shuttle mission to the International Space Station (ISS). The ISS is a habitable satellite (Space station) in a low Earth orbit. We employ calculus-based methods to analyze and study the flightpaths, altitude, velocity, and acceleration profiles of the STS121 data reported by NASA as it travelled through outer space. Our studies unravel information about the critical points, local maxima and minima, concavity, and inflection points in the altitude data. The velocity profiles were fitted to polynomial functions using least square data fitting using linear algebra-based methods. The acceleration data involve piecewise functions which is related to the time scales involving burning of the propellent and separation of the external propellant tank as the space shuttle gets ready to move into orbit. We estimated the work done in transferring a load from Earth to the International Space station. We used optimization methods to design an optimal solar panel geometry for a satellite by minimizing the surface area. This research provides novel applications of the fundamental theorems of calculus to study motion in outer space and involves mathematical modeling, optimization, curve fitting, data analysis and data visualization.